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Error bounds for monomial convexification in polynomial optimization

  • Warren Adams
  • Akshay Gupte
  • Yibo Xu
Full Length Paper Series A

Abstract

Convex hulls of monomials have been widely studied in the literature, and monomial convexifications are implemented in global optimization software for relaxing polynomials. However, there has been no study of the error in the global optimum from such approaches. We give bounds on the worst-case error for convexifying a monomial over subsets of Open image in new window . This implies additive error bounds for relaxing a polynomial optimization problem by convexifying each monomial separately. Our main error bounds depend primarily on the degree of the monomial, making them easy to compute. Since monomial convexification studies depend on the bounds on the associated variables, in the second part, we conduct an error analysis for a multilinear monomial over two different types of box constraints. As part of this analysis, we also derive the convex hull of a multilinear monomial over Open image in new window .

Keywords

Polynomial optimization Monomial Multilinear Convex hull Error analysis Means inequality 

Mathematics Subject Classification

90C26 65G99 52A27 

1 Introduction

A polynomial \({p\in \mathbb {R}[x]}\), where \({\mathbb {R}[x]=\mathbb {R}[x_{1},\dots ,x_{n}]}\) is the ring of n-variate polynomials, is a linear combination of monomials and is expressed as Open image in new window where the sum is finite, Open image in new window is a monomial, and every Open image in new window is a nonnegative integer. A polynomial optimization problem is
$$\begin{aligned} z^{*}_{ S}= \min \,\{p(x)\mid x\in S\} \end{aligned}$$
for a compact convex set S and \({p \in \mathbb {R}[x]}\). It is common to assume that the degree of the polynomial is bounded by some constant m and this is denoted by \({p\in \mathbb {R}[x]_{m}}\). Polynomials, in general, are nonconvex functions, thereby necessitating the use of global optimization algorithms for optimizing them. Strong and efficiently computable convex relaxations are a major component of these algorithms, making them a subject of ongoing research. One approach for devising good relaxations is based on taking the convex envelope of each polynomial p(x) over S. However, since this computation is NP-hard even in the most basic cases having \({m=2}\) and Open image in new window or S being a standard simplex, a main emphasis of the envelope studies has been on finding the envelope either under structural assumptions on S or by considering only a subset of all the monomials appearing in p(x). Also, one is interested in obtaining polyhedral relaxations of the envelope so that lower bounds can be computed cheaply by solving linear programs (LPs) iteratively [24, 28, 35, 37]. If p(x) is a multilinear polynomial (i.e. Open image in new window for all j) and S is a box, then the envelopes are polyhedral and we know exponential sized extended formulations [32, 34], as well as valid inequalities [11, 16] and efficient cutting planes [3, 30] in projected spaces. A second method for obtaining lower bounds on the polynomial optimization problem has been to use the moments approach and Lasserre [18] hierarchy of semidefinite relaxations (SDPs) that converges to the global optimum [19, 20]. All of these techniques can of course also be used for relaxing a optimization problem that has polynomials in both the objective and constraints.

For a general polynomial Open image in new window , given that it is hard to find the envelope explicitly and that computability of the SDP bounds does not scale well, a common relaxation technique, motivated by the classical work of McCormick [26], has been to replace each monomial Open image in new window with a continuous variable, say w, and then add inequalities to convexify the graph of Open image in new window over S, which is the set Open image in new window . This is referred to as monomial convexification, and it typically yields a weaker relaxation than the envelope of the polynomial due to the fact that the envelope operator does not distribute over sums in general. However, because they may be cheaper and easier to generate than convexification of the entire polynomial, convex hulls of monomials have received significant attention [3, 4, 8, 21] and are also routinely implemented in leading global optimization software [12, 29, 39]. We still do not know an explicit form for the convex hull of a general monomial, but a number of results are available for bivariate monomials [23] and n-variate multilinear monomials [2, 5, 6, 10, 25, 27, 33]. Moreover, there also exist challenging applications [9] where the constraints can be formulated as having only monomial terms, thereby making monomial convexifications necessary for obtaining strong relaxations.

To quantify the strength of a relaxation of p(x), one is interested in bounding the error produced with respect to the global optimum Open image in new window by optimizing over this relaxation. Error bounds for converging solutions of iterative optimization algorithms have been the subject of study before [31], but since these are not suited for studying relaxation strengths, different error measures have been proposed. Luedtke et al. [25] studied a relative error measure for the relaxation of a bilinear polynomial \({p\in \mathbb {R}[x]_{2}}\) over Open image in new window obtained by convexifying each monomial with its McCormick envelopes. They showed that for every Open image in new window , the ratio of the difference between the McCormick overestimator and underestimator values at x and the difference between the concave and convex envelope values at x can be bounded by a constant that is solely in terms of the chromatic number of the co-occurrence graph of the bilinear polynomial. Recently, Boland et al. [7] showed that this same ratio cannot be bounded by a constant independent of n. Another, and somewhat natural, way of measuring the error from a relaxation is to bound the absolute gap Open image in new window , where \({\tilde{z}_{ S}}\) is a lower bound on Open image in new window due to some convex relaxation of \({\{(x,w)\in S\times \mathbb {R}\mid w = p(x)\}}\). Such a bound helps determine how close one is to optimality in a global optimization algorithm. Also, there are examples (cf. Open image in new window over Open image in new window in [25, pp. 332]) where the relative error gap of McCormick relaxation goes to \({\infty }\), while this can never happen with the absolute gap. The only result that we know of on bounding absolute gaps for general polynomials is due to De Klerk and Laurent [13] who used Bernstein approximation of polynomials for a hierarchy of LP and SDP relaxations. (On the contrary, [14, 15] bound the absolute error from upper bounds on Open image in new window ). We mention that the absolute errors arising from piecewise linear relaxations of bilinear monomials appearing in a specific application were studied by Dey and Gupte [17]. Finally, a third error measure is based on comparing the volume of a convex relaxation to the volume of the convex hull. This has been done for McCormick relaxations of a trilinear monomial over a box by Speakman and Lee [36].

Our contribution In this paper, we bound the absolute gap to Open image in new window from monomial convexification and thereby add to the small number of explicit error bounds for polynomial optimization. To bound this gap, we analyze the error in relaxing a monomial with its convex hull. This error analysis not only implies a bound on the absolute gap to Open image in new window but it also can be used for bounding the error in relaxing any optimization problem with polynomials in both the objective and constraints. Our error measure is the maximum absolute deviation between the actual value and the approximate value of the monomial. Thus for any set X in the (xw)-space, we denote the error of X with respect to Open image in new window by Open image in new window , which is defined asWe will mostly be interested in the error Open image in new window for the convex hull of the graph of Open image in new window and for the convex and concave envelopes of Open image in new window . As mentioned earlier, monomial convexification errors have gone largely unnoticed in the literature, the only results being for the bilinear monomial \({x_{1}x_{2}}\). The folklore result [2, cf.] for \({x_{1}x_{2}}\) over a rectangle Open image in new window states that the convex hull and envelope errors are attained at Open image in new window , which is the midpoint of the two diagonals of the box. Linderoth [22] derived error formulae for \({x_{1}x_{2}}\) over triangles created by the two diagonals of Open image in new window . Since convex hull and envelope results for a bilinear polynomial are invariant to affine transformations, it is equivalent to consider \({x_{1}x_{2}}\) over Open image in new window . Substituting \({n=2}\) and Open image in new window in our forthcoming error bounds recover these known errors.

Notation The vector of ones is Open image in new window , the ith unit coordinate vector is Open image in new window , and the vector of zeros is Open image in new window ; the dimensions will be apparent from the context in which these vectors are used. The convex hull of a set X is Open image in new window and the relative interior of Open image in new window is Open image in new window . A nonempty box in Open image in new window is Open image in new window . The standard boxes that we focus on in this paper are Open image in new window , and \({[1,r]^{n}}\), for arbitrary scalar \({r>1}\). Another compact convex set of interest to us is the standard n-simplex Open image in new window . For convenience, we write Open image in new window , Open image in new window , Open image in new window . The convex envelope of Open image in new window over S, which is defined as the pointwise supremum of all convex underestimators of Open image in new window over S, is denoted by Open image in new window . The concave envelope, which is analogously defined, is Open image in new window . The graph of a function g(x) with domain S is denoted by Open image in new window . The graphs of the monomial and its envelopes are Open image in new window , Open image in new window and Open image in new window . Two special types of monomials are the symmetric monomial and the multilinear monomial. The former has Open image in new window for some Open image in new window , and the latter, denoted by Open image in new window , is a special case of the former with Open image in new window . For Open image in new window , we denote Open image in new window .

1.1 Main results

We obtain strong and explicit upper bounds on Open image in new window for different types of monomials. In the polynomial optimization literature, it is common to assume, upto scaling and translation, that the domain S of the problem is a subset of Open image in new window . When analyzing a single monomial, this assumption is not without loss of generality since the monomial basis of Open image in new window is not closed upto translating and scaling the variables. Hence we divide our analysis into two parts. First, we consider a general monomial Open image in new window over a compact convex set Open image in new window , and bound the errors without using explicit analytic forms of the envelopes, which are hard to compute and unknown in closed form for arbitrary S. The concave error is bounded by computing the error from a specific concave overestimator that is precisely the concave envelope of Open image in new window over Open image in new window . On the convex side, we bound the error for any convex underestimator given as the pointwise supremum of (possibly uncountably many) linear functions, each of which underestimates Open image in new window over S. Thus our error analysis has a distinctly polyhedral flavor.

In the second part, we limit our attention to a multilinear monomial Open image in new window , but the domain S is either a box with constant ratio or a symmetric box. By a box with constant ratio, we mean any box Open image in new window for which there exists a scalar \({r > 1}\) such that Open image in new window for all i with Open image in new window , and Open image in new window for all i with Open image in new window . By a symmetric box, we mean any box Open image in new window that has Open image in new window for all i. Since these boxes are simple scalings of Open image in new window and Open image in new window , respectively, and our error measure Open image in new window scales, we restrict our attention to only Open image in new window and Open image in new window . Contrary to the first part, here we first derive explicit polyhedral characterizations of the envelopes and convex hulls over Open image in new window and Open image in new window and use them to perform a tight error analysis. The polyhedral representations for the Open image in new window case follow from the literature, whereas those over Open image in new window are established in this paper.

1.1.1 General monomial

Consider a monomial Open image in new window with Open image in new window for all j. The degree of this monomial is Open image in new window . The following constants will be useful throughout the paper:

Theorem 1.1

For the monomial Open image in new window over Open image in new window , we havewhere for \({\sigma _{j} := 1 - \max \{x_{j}\mid x \in S\}}\), we defineIf Open image in new window , then Open image in new window .

The monotonicity of Open image in new window and Open image in new window with respect to Open image in new window suggests the intuitive result that convexifying higher degree monomials will likely produce greater errors. As Open image in new window , we have Open image in new window and Open image in new window .

The bounds Open image in new window and Open image in new window depend only on the degree of the monomial. They are a consequence of some general error bounds, established in Theorem 3.1 for the concave error and in Theorem 3.2 for the convex error, that depend on how the monomial behaves over the domain S. The arguments used in proving Theorem 1.1 also imply that a family of convex relaxations of Open image in new window has error equal to Open image in new window . We show this in Proposition 3.6. We also guarantee in Corollary 3.4 that the convex envelope error bound Open image in new window is tight for Open image in new window over Open image in new window .

Theorem 1.1 has two immediate implications. First, we obtain the error in convexifying a monomial over Open image in new window .

Corollary 1.1

Open image in new window .

Second, we obtain an additive error bound on polynomial optimization over subsets of Open image in new window . For a polynomial Open image in new window , denoteLet Open image in new window be the lower bound1 from monomial convexification on the global optimum Open image in new window .

Corollary 1.2

For any \({p\in \mathbb {R}[x]_{m}}\) and compact convex Open image in new window ,

Proof

We have Open image in new window . Therefore,Applying Theorem 1.1 and the construction of Open image in new window gives us Open image in new window . Since Open image in new window , there are at most \({\left( {\begin{array}{c}n+m\\ n\end{array}}\right) }\) monomials in p(x), leading to the claimed error bound. \(\square \)

Computing \({L^{\prime }(p)}\) may get tedious if p(x) has a large number of monomials. A cheaper bound is possible by considering only the largest coefficient in p(x).

Corollary 1.3

For any \({p\in \mathbb {R}[x]_{m}}\) and compact convex Open image in new window ,

Proof

Follows from Corollary 1.2 after using Open image in new window and Open image in new window being monotone in Open image in new window . \(\square \)

The bounds from Theorem 1.1, although applicable to arbitrary Open image in new window , can be weak if Open image in new window and Open image in new window . To emphasize this, we consider a monomial over the standard simplex \({\Delta _{n}}\) and obtain error bounds that depend on not just the degree of the monomial but also the exponent of each variable. These bounds are stronger than the bounds Open image in new window and Open image in new window .

Theorem 1.2

All of the above bounds are tight for a symmetric monomial.

1.1.2 Multilinear monomial

Consider the multilinear monomial Open image in new window .

Theorem 1.3

Denote
$$\begin{aligned} \begin{aligned} \mathscr {D}_{r,n}&:= \max _{i=1,\dots , n-1}\, \left\{ \left( 1+\frac{i}{n}(r-1)\right) ^n - r^{i}\right\} ,\\ \mathscr {E}_{r,n}&:= 1+\frac{r^n-1}{(r-1)}\left[ \frac{n-1}{n}{\left( \frac{r^n-1}{n(r-1)}\right) }^{\frac{1}{n-1}}-1\right] . \end{aligned} \end{aligned}$$
For Open image in new window over Open image in new window ,All bounds are attained only on Open image in new window .

We conjecture that \({\mathscr {D}_{r,n}\le \mathscr {E}_{r,n}}\) for all rn and provide a strong empirical evidence in support of this claim. We prove this conjecture to be asymptotically true by showing that \({\lim _{n\rightarrow \infty }\mathscr {D}_{r,n}/\mathscr {E}_{r,n}\le 1/e}\).

For Open image in new window , we characterize the convex hull in Theorem 4.1 and show that it has the following errors.

Theorem 1.4

For Open image in new window over Open image in new window ,This maximum error is attained at all the \({2^{n}}\) reflections of the point Open image in new window .

The exact description of the reflected points will be provided when we prove this theorem. Taking \({n\rightarrow \infty }\), this error approaches \({1 + 1/e^{2}}\) from below.

1.1.3 Outline

Our analysis begins with some preliminaries on the error measure. We observe that the error scales with the box and present a lower bound on the error, which we remark is also the proposed upper bound for the two cases Open image in new window and Open image in new window . We also formally note the intuition that the convex hull error can be computed as the maximum of the two envelope errors, due to which our error analysis in the remainder of the paper involves analyzing the concave envelope and the convex envelope separately. Sects. 3.1 and 3.2 analyze these errors for a general monomial Open image in new window over Open image in new window . The main error bounds presented in Sect. 1.1.1 are proved in Sect. 3.3 and we compare them to those from literature in Sect. 3.4. The multilinear monomial over Open image in new window and Open image in new window is analyzed in Sects. 4.1 and 4.2.

2 Preliminaries on Open image in new window

The error defined in (1) is obviously monotone with respect to set inclusion: Open image in new window for any \({X_{1}\subseteq X_{2}}\). This enables us to upper bound the convex hull error by using Open image in new window for any convex relaxation X of Open image in new window , and also implies that the convex hull error over a smaller variable domain is upper bounded by the convex hull error over a larger domain. Another property we observe is that computing the convex hull error is equivalent to computing the error due to the convex envelope Open image in new window and that due to the concave envelope Open image in new window . This intuitively seems correct given the well-known fact that Open image in new window , and the fact that the monomial convexification and envelope errors are

Observation 2.1

Let \({X := \{(x,w)\in S\times \mathbb {R}\mid f_{1}(x) \le w \le f_{2}(x) \}}\), where \({f_{1}}\) and \({f_{2}}\) are, respectively, convex and concave continuous functions with Open image in new window for all \({x\in S}\). Thenand equality holds if Open image in new window and Open image in new window .

The proof is straightforward and is left to the reader. Based on this observation, our error analysis in the rest of the paper involves analyzing the concave envelope and the convex envelope separately.

A third and final property we note is that the error scales with the box. For \({c\in \mathbb {R}^{n}_{\ne 0}}\),is the coordinate-wise scaled version of Open image in new window . The bijective linear map Open image in new window gives us the relation Open image in new window . Denote Open image in new window .
Observation 2.2 allows us to focus on boxes with specific bounds Open image in new window and Open image in new window , and to then extend to slightly more general boxes via scalings. In particular, error results for
Finally, we observe a lower bound on Open image in new window , and hence on Open image in new window , when S contains two points on the ray Open image in new window , which happens for example when Open image in new window for some \({t_{1} < t_{2}}\) with \({t_{2} > 0}\).

Lemma 2.1

Suppose Open image in new window and let \({t_{1},t_{2}\ge 0}\) be the minimum and maximum values such that Open image in new window . Let \({\tilde{f}}\) be a concave overestimator of Open image in new window on S and let X be a convex relaxation of Open image in new window . Then, Open image in new window and Open image in new window , where Open image in new window and

Proof

The assumption Open image in new window implies Open image in new window . Convexity of X and Open image in new window lead to Open image in new window for all \({\xi \in [0,1]}\). Thereforewhere the equality is due to \({t_{1},t_{2}\ge 0}\) and convexity of the function Open image in new window on Open image in new window . Since \({\phi (0)=\phi (1)=0}\), by Rolle’s theorem, there exists a stationary point in [0, 1] and this point is exactly \({\xi ^{\prime }}\) stated above. Since \({\phi }\) is concave, \({\xi ^{\prime }}\) must be a maxima. For a concave overestimator \({\tilde{f}}\), we have\(\square \)

Remark 1

For lower bounding Open image in new window , the above proof really only requires Open image in new window . The stronger assumption Open image in new window is made for convenience.

Remark 2

The above method of lower bounding the error can also be utilized by considering arbitrary \({l,u\in S}\) with Open image in new window . This generalization is made possible by the observation that the function Open image in new window is convex over Open image in new window . Since the derivation gets extremely tedious and does not yield new insight, we omit the general case here.

Substituting \({t_{1}=0,t_{2}=1}\) in Lemma 2.1 yields the critical point to be Open image in new window so that Open image in new window , where the constant Open image in new window was introduced in Eq. (2). Thus the significance of the lower bound from this lemma is that we prove in Theorem 1.1 that it is indeed equal to the maximum error of the convex hull when Open image in new window . For a multilinear monomial over Open image in new window for some \({r>1}\), or equivalently Open image in new window using the scaling from Observation 2.2, the constant \({\mathscr {E}_{r,n}}\) defined in the statement of Theorem 1.3 is exactly the lower bound obtained from Lemma 2.1 by substituting \({t_{1}=1,t_{2}=r}\) and we prove that this is the maximum concave envelope error and conjecture, with strong empirical evidence in support, that it is also the maximum convex hull error.

3 Monomial over Open image in new window

This section considers a general multivariate monomial Open image in new window , for some Open image in new window , over a nonempty compact convex set Open image in new window . It follows that Open image in new window . Our main error bounds on Open image in new window depend only on the degree Open image in new window of the monomial and therefore are independent of how the monomial behaves on its domain S. However, en route to deriving these formulas, we establish tighter bounds that depend on the minimum and maximum value of Open image in new window over S and thus are expensive to compute in general. The error formulas for the multilinear case will follow after substituting Open image in new window . Motivated by Observation 2.1, we bound the convex hull error by bounding the envelope errors separately.

Before we begin, we recall that the envelopes of Open image in new window were shown by Crama [10] to be

Remark 3

The envelopes of Open image in new window over a box Open image in new window having one of its vertices at the origin, i.e., Open image in new window for all j, can be obtained by scaling the variables in (4a) as Open image in new window for \({j\in J_{1}:= \{j\mid \L _{j}=0\}}\), Open image in new window for Open image in new window , and Open image in new window .

The concave envelope in (4a) is also the concave envelope of Open image in new window over Open image in new window for every Open image in new window , i.e. This is because Open image in new window for Open image in new window and a monomial Open image in new window with Open image in new window is known to be concave-extendable from the vertices of Open image in new window (meaning that Open image in new window can be obtained by looking at the values of Open image in new window solely at \({\{0,1\}^{n}}\)); see [38]. One can also establish this fact independently without using concave-extendability of Open image in new window .
For notational convenience throughout this section, we denoteThat is, \({E_{0}}\) is the union of all the coordinate plane facets of Open image in new window and \({E_{j}}\) is the jth edge of Open image in new window that is incident to the vertex Open image in new window .

3.1 Concave overestimator error

Throughout, we consider the piecewise linear concave function Open image in new window , which we noted in (4b) to be the concave envelope of Open image in new window over Open image in new window . First, we treat the general case where S is any subset of Open image in new window , and later we consider the case of S being a standard simplex.

3.1.1 General case

For arbitrary Open image in new window , we have Open image in new window due to Open image in new window and Open image in new window implying Open image in new window for all j. We observe that this overestimator is exact only on \({E_{0}}\) or on edges \({E_{i}}\)’s along which the monomial is linear.

Proposition 3.1

Open image in new window if and only if Open image in new window or \({x\in E_{0}}\) or \({x\in E_{i}}\) for some i with Open image in new window .

Proof

For Open image in new window and Open image in new window , Open image in new window follows from the facts and Open image in new window . The equalities Open image in new window and Open image in new window , for all \({x\in E_{0}}\), are obvious. For any Open image in new window , \({x_{i} \in (0,1)}\) and \({x_{j}=1 \ \forall j\ne i}\) give us Open image in new window and Open image in new window . Thus it is obvious that for Open image in new window , Open image in new window if and only if Open image in new window . Now let x be any point in S that does not belong to a coordinate plane nor to any edge \({E_{i}}\). Then there exist distinct indices ij with \({x_{i},x_{j}\in (0,1)}\) and \({ x_{i} \le x_{j} \le x_{k} \ \forall k\ne i,j}\). Therefore Open image in new window . \(\square \)

Since Open image in new window , the error due to Open image in new window , which is the maximum value of the difference Open image in new window over S, provides an upper bound on the error from Open image in new window . Proposition 3.1 tells us that this maximum difference occurs either in the interior of Open image in new window or in the relative interior of some face of Open image in new window passing through Open image in new window . In the following result, we give a tight upper bound on Open image in new window that is attained at a specific point on the diagonal between Open image in new window and Open image in new window . This is our main error bound for Open image in new window .

Theorem 3.1

Open image in new window , where Open image in new window . This bound can be attained only at the point Open image in new window and hence is tight if and only if Open image in new window .

Proof

Since Open image in new window and Open image in new window , we have Open image in new window . This implies Open image in new window for Open image in new window , which leads toSince f(x) is a continuous function with minimum and maximum values Open image in new window and Open image in new window on the closed convex set S, the intermediate value theorem implies thatWe have Open image in new window due to Open image in new window . Elementary calculus tells us that the function Open image in new window is concave on [0, 1] with a unique stationary point at Open image in new window and is increasing on \({[0,\xi _{0})}\) and decreasing on \({(\xi _{0},1]}\). Hence the maximum value of this function on Open image in new window is Open image in new window , where Open image in new window . Combining this with (5) gives us the desired upper bound.
Now we claim that this bound can be tight only on Open image in new window . Suppose this is not true and there exists a Open image in new window such that Open image in new window . The fact that Open image in new window and Open image in new window makes it obvious that Open image in new window . Thus Open image in new window . Since Open image in new window and Open image in new window is the maximum value of the right hand side in (5), we haveimplying that equality holds throughout. Hence Open image in new window . However this is a contradiction to Open image in new window because observe that for any Open image in new window , Open image in new window if and only if \({x_{1}=x_{2}=\dots =x_{n}}\), which is equivalent to Open image in new window . Therefore Open image in new window is necessary for the proposed upper bound to be tight.

Suppose that Open image in new window for some \({0\le \xi _{1}\le \xi _{2}\le 1}\). On Open image in new window , the function Open image in new window transforms to the univariate concave function Open image in new window for \({\xi \in (0,1)}\), which has a unique stationary point at Open image in new window , giving us Open image in new window , if Open image in new window . The function Open image in new window is increasing on \({(0,\tilde{\xi })}\) and decreasing on \({(\tilde{\xi },1)}\). By construction of Open image in new window and Open image in new window , it follows that Open image in new window . Therefore Open image in new window for some \({x\in S}\) if and only if Open image in new window and Open image in new window . \(\square \)

The upper bound presented in Theorem 3.1 depends on the minimum and maximum values of the monomial over S, which can be hard to compute for arbitrary S, and not just on the degree of the monomial. However, an immediate consequence is that the constant Open image in new window , defined as Open image in new window in Eq. (2), is a degree-dependent bound on the error from Open image in new window .

Corollary 3.1

Open image in new window , and this bound is tight if Open image in new window and only if Open image in new window .

Proof

The function Open image in new window attains its maxima over [0, 1] uniquely at Open image in new window . The definition of \({\xi ^{\prime }}\) then gives usand subsequently, Theorem 3.1 leads to Open image in new window being an upper bound on Open image in new window . The uniqueness of the maxima of Open image in new window also implies that for Open image in new window to be a tight bound, we must have Open image in new window , which is equivalent to Open image in new window . \(\square \)

Notice that the necessity of Open image in new window in the above corollary is not immediate from the statement of Theorem 3.1. This can be explained as follows. Denote Open image in new window for some \({0\le \xi _{1}\le \xi _{2}\le 1}\). Since we showed that Open image in new window is an upper bound on Open image in new window , Theorem 3.1 implies that if Open image in new window is a tight bound then Open image in new window . By construction, Open image in new window and Open image in new window . So, by Theorem 3.1, it is possible to have Open image in new window or Open image in new window , if Open image in new window is tight. However, Corollary 3.1 rules out this possibility. Furthermore, the condition Open image in new window is not sufficient to guarantee tightness of Open image in new window . The reason being that this condition does not enforce non-emptiness of Open image in new window , which we know to be necessary from Theorem 3.1.

If the minimum and maximum values of Open image in new window over S are low-enough and high-enough, respectively, as per Corollary 3.1, then we have a precise characterization of when Open image in new window is a tight bound on Open image in new window .

Proof

The assumptions of Open image in new window and Open image in new window imply Open image in new window in Theorem 3.1, thereby leading to the first claim. Since Open image in new window , Open image in new window , and Open image in new window , the second claim follows from the first part and Open image in new window from (4b). \(\square \)

For the simplex Open image in new window , clearly, Open image in new window for any Open image in new window . This simplex can be described as Open image in new window . When Open image in new window , i.e., multilinear monomial, it is easy to verify graphically that Open image in new window so that the point Open image in new window does not belong to Open image in new window . However, the function Open image in new window being monotone in Open image in new window , for large enough values of Open image in new window , we have Open image in new window , as can be verified numerically, and consequently, Open image in new window . Hence, the bound Open image in new window from Corollary 3.2 is tight for arbitrary Open image in new window when the monomial degree is large.

3.1.2 Standard simplex

For monomials considered over the standard Open image in new window -simplex Open image in new window , we obtain a bound in Proposition 3.2 that is tight only for symmetric monomials. The proof of this result uses the following lemma which will be useful also in proving Theorem 1.1 later in Sect. 3.3.

Proof

Obviously Open image in new window . Since Open image in new window , binomial expansion gives usThis is equivalent to Open image in new window . Clearly, equality holds for Open image in new window . For Open image in new window , binomial expansion gives usthereby leading to Open image in new window . \(\square \)

Proposition 3.2

and this bound is tight if and only if Open image in new window .

Proof

Open image in new window because Open image in new window . The maximum value of Open image in new window over Open image in new window is obviously attained in the relative interior of the face defined by the plane Open image in new window . Solving the KKT system for Open image in new window gives us Open image in new window . For fixed integers Open image in new window , it is easy to argue thatusing the convexity of Open image in new window and the integrality of the polytope Open image in new window . Therefore for fixed Open image in new window , the maximum value of Open image in new window is achieved with Open image in new window and is equal to Open image in new window . Thus, Open image in new window . By Lemma 3.1, we have Open image in new window and soThis implies that Open image in new window in Theorem 3.1, thereby giving us the proposed upper bound on Open image in new window . Theorem 3.1 also tells us that this bound is tight if and only if Open image in new window , which is equivalent to showing Open image in new window . Observe the following.

Claim 3.1

Open image in new window for Open image in new window , with equality holding if and only if Open image in new window .

Proof of Claim

This inequality is obtained by applying Jensen’s inequality to the convex function Open image in new window with the n points being Open image in new window and the convex combination weights being all equal to 1 / n. The equality condition is due to \({t\log {t}}\) being strictly convex. \(\square \)

Therefore our bound is tight if and only if the monomial is symmetric. Open image in new window

3.2 Convex underestimator error

We address the case of a simplex first because it is easy.

Proposition 3.3

Suppose that S is a \({0\backslash 1}\) polytope with Open image in new window . Then Open image in new window . In particular, Open image in new window , and the error due to this envelope is equal to Open image in new window .

Proof

Observe the following fact which is an immediate consequence of applying Jensen’s inequality to the definition of convex envelope: for a continuous function Open image in new window for some finite Open image in new window and bounded polyhedral domain Open image in new window , if Open image in new window for every vertex Open image in new window of Open image in new window , then Open image in new window . Since Open image in new window for Open image in new window and Open image in new window for Open image in new window , it follows from the assumption on S that Open image in new window . The standard Open image in new window -simplex Open image in new window satisfies the assumption on S and so the convex envelope over it is the zero function, thereby making the error equal to Open image in new window . This value was argued in the proof of Proposition 3.2 to be equal to Open image in new window . \(\square \)

Hereafter, we let S be an arbitrary subset of Open image in new window , with a special interest in Open image in new window , or more generally Open image in new window , where is a Open image in new window -simplex cornered at Open image in new window . For convenience, we write Open image in new window simply as Open image in new window . The motivation for studying the case Open image in new window is clear from Proposition 3.3 which highlights the significance of the vertex Open image in new window belonging to S. Also note that the polytope Open image in new window , the complement of Open image in new window defined as is a Open image in new window polytope not containing Open image in new window . Note that Open image in new window is not the simplex cornered at Open image in new window , which was defined in Sect. 1 to be Open image in new window . If Open image in new window , Open image in new window for all Open image in new window , and therefore one would be interested in finding strong convex underestimators of Open image in new window over Open image in new window . We will derive a piecewise linear convex underestimator later in Proposition 3.5.

We begin by establishing an error bound in Theorem 3.2. This bound does not have an explicit expression or formula, rather it is stated as the infimum of a certain function. However, it serves as a stepping stone towards deriving explicit error bounds in Sect. 3.2.2 that depend only on the degree of the polynomial, and hence towards proving our main result in Sect. 3.3.

3.2.1 Implicit bound

Unlike Sect. 3.1 where we calculate the error from a specific concave overestimator, here we consider a general convex underestimator defined as the pointwise supremum of a family of affine functions, for some nonempty (possibly countably infinite) set Open image in new window , wherefor each Open image in new window to ensure that the linear function Open image in new window underestimates and touches the graph of Open image in new window . For finite Open image in new window , Open image in new window is a piecewise linear convex underestimator, otherwise Open image in new window could represent the convex envelope of Open image in new window over S. The assumption of nonnegativity on \({\beta }\) is due to the fact that the gradient of Open image in new window at any point in Open image in new window is a nonnegative vector. For convenience, we allow only positive \({\beta }\) and scale it greater than equal to 1 by assuming Open image in new window . The multilinear monomial with Open image in new window would have Open image in new window (cf. (4a)) with Open image in new window and Open image in new window .

Towards proving our main error bound in terms of only the degree of the monomial, we first obtain in Theorem 3.2 a error bound that depends on \({\sigma (\beta )}\)’s. We make some remarks on \({\sigma (\beta )}\) here. An explicit formula for \({\sigma (\beta )}\) for arbitrary S seems hard and the function is expected to be nonconvex (\({\sigma (\beta )}\) is a translate of the negative of the Fenchel conjugate of Open image in new window ). However, it is possible to find bounds on it, which we state next.

Proposition 3.4

We have the following for \({\sigma (\beta )}\) when Open image in new window :Let Open image in new window be the permutation that sorts \({\beta }\) as Open image in new window .

The proof is moved to “Appendix”. The case Open image in new window is not covered in the above proposition since the error over Open image in new window was already dealt with in Proposition 3.3 and hence we would have no use of the bounds on \({\sigma (\beta )}\) in this case.

To establish an upper bound on Open image in new window , we define the following constants for every linear underestimator Open image in new window :It is clear that Open image in new window and so Open image in new window . Since Open image in new window by Proposition 3.4, we have Open image in new window . For any Open image in new window , Open image in new window is a nonincreasing function and so Open image in new window is also a nonincreasing function:We do not know how Open image in new window behaves. The significance of the scalar Open image in new window is as follows.

Proof

Since Open image in new window , Open image in new window is convex over Open image in new window . It is decreasing only over Open image in new window , where Open image in new window is the unique stationary point of Open image in new window . Note that Open image in new window and observe that Open image in new window , which lies in Open image in new window , is the unique fixed point of Open image in new window on Open image in new window . Hence Open image in new window if and only if Open image in new window . The assumption Open image in new window is equivalent to Open image in new window . Therefore Open image in new window . We claim thatThe first inequality is obvious whereas the second is due to the monotonicity of the function Open image in new window on Open image in new window . Thus we have argued that Open image in new window . Now the monotone behavior of Open image in new window on Open image in new window means that Open image in new window because otherwise we would have the contradiction Open image in new window . This implies that the maximum value of Open image in new window on the Open image in new window interval occurs at Open image in new window and, since this is a fixed point, it is equal to Open image in new window . \(\square \)
Since we need Open image in new window in the above lemma and forthcoming results, defineThe assumption Open image in new window makes it obvious that Open image in new window . The structure of Open image in new window discussed in the proof of Lemma 3.2 implies the following claim.

We are now ready to state our upper bound on error from the convex underestimator Open image in new window .

Theorem 3.2

where Open image in new window is a maximal element of Open image in new window under the partial order Open image in new window . In particular, if there exists some Open image in new window such that Open image in new window for all Open image in new window , thenand this bound is tight only if Open image in new window and is attained only at the point Open image in new window .

Proof

Choose some Open image in new window . For every Open image in new window and Open image in new window , Open image in new window gives us Open image in new window and Open image in new window gives us Open image in new window . Thus The generalized arithmetic-geometric means inequality tells us that Open image in new window , which combined with (11a) leads to Open image in new window . Thereforewhich leads toSince Open image in new window is a continuous function with minimum and maximum values Open image in new window on S, the intermediate value theorem implies that (11b) transforms towhere Open image in new window as in Lemma 3.2. Lemma 3.3 leads to Open image in new window . Since Open image in new window was arbitrarily chosen in Open image in new window and we know from (9) that Open image in new window is a nonincreasing function for every \({\beta }\), we may set Open image in new window equal to a maximal Open image in new window to obtain Open image in new window . If Open image in new window for some Open image in new window , then Open image in new window is the unique maximal element in Open image in new window and setting Open image in new window yields the upper bound Open image in new window .

The bound Open image in new window is tight if and only if there is equality throughout in (11a) with Open image in new window , and in the means inequality Open image in new window . Equation (11a) is an equality if and only if Open image in new window , implying that Open image in new window is a necessary condition for tightness. The means inequality is an equality if and only if Open image in new window and hence the bound can be attained only at Open image in new window . \(\square \)

Remark 4

We will show in Proposition 3.5 that Open image in new window , implying that Open image in new window , which is exactly the constant Open image in new window defined in (2), and therefore the above bound is attained only at Open image in new window .

Any polyhedral relaxation of the epigraph of Open image in new window can be encoded by the set Open image in new window in Eq. (7a). Hence Theorem 3.2 yields an upper bound on the error from any polyhedral relaxation that is chosen apriori. Since we do not know the behavior of Open image in new window , a analytic expression for the infimum in Theorem 3.2 does not seem possible in general. Even if Open image in new window is finite, Open image in new window requires the computation of \({\sigma (\beta )}\), which we know to be hard in general. However, one may derive upper bounds on the error using the lower bounds on \({\sigma (\beta )}\) from Proposition 3.4. Note though that this does not help for Open image in new window because the lower bound of Open image in new window on \({\sigma (\beta )}\) gives a trivial upper bound of 1 on the error.

We use the bound in Theorem 3.2 to derive a degree-dependent bound on the convex envelope error. To do so, let us view this upper bound from a different perspective. By construction of Open image in new window , in order to obtain a smaller error bound, we would intuitively want to pick Open image in new window such that it contains only those \({\beta }\) that make \({\sigma (\beta )}\) to be as high as possible. For Open image in new window , or more generally S containing Open image in new window , we know the highest that \({\sigma (\beta )}\) can be is 1. Hence we could do the following reverse construction—instead of choosing a set Open image in new window and then computing \({\sigma (\beta )}\) for each Open image in new window as done before, we could fix Open image in new window and find the values of Open image in new window that enable Open image in new window to be a valid linear underestimator (cf. Eq. (7a)) to Open image in new window over S. This would alleviate the issue of having to compute \({\sigma (\beta )}\) for Open image in new window and could possibly lead to simpler and explicit error bounds that depend only on exponent Open image in new window and degree Open image in new window . We follow this path for the rest of this section. Note also that the convex envelope of the multilinear monomial Open image in new window over Open image in new window is Open image in new window , meaning that there is only one \({\beta }\), the vector Open image in new window , with Open image in new window . Thus our forthcoming derivation implies the error from the convex envelope of a multilinear monomial over Open image in new window .

3.2.2 Explicit bounds

DenoteThis linear function is exact at Open image in new window : Open image in new window . The convex underestimator on Open image in new window is Open image in new window is a closed convex set,2 due to linearity of Open image in new window in \({\beta }\) for fixed Open image in new window , and it represents all the linear functions that are exact at Open image in new window and underestimate Open image in new window everywhere on Open image in new window . Clearly, Open image in new window implies Open image in new window for all Open image in new window , and so Open image in new window implies Open image in new window . But then we could simply delete such a Open image in new window from Open image in new window without affecting the supremum in Open image in new window . Hence we define the nondominated subset of Open image in new window to be the following:so thatA strong error bound from Open image in new window would obviously depend on the elements in Open image in new window (cf. Theorem 3.2), making it important to obtain a (partial) characterization of Open image in new window and Open image in new window based on the structure of S. We mention two cases where Open image in new window is easily seen to be equal to Open image in new window , the most trivial value.
Multilinear over Open image in new window

Here Open image in new window and Eq. (4a) tells us Open image in new window , and therefore Open image in new window .

We will generalize this in Proposition 3.5 by showing that Open image in new window when Open image in new window .

Subsets of Open image in new window

Here Open image in new window is arbitrary and Open image in new window . We know that Open image in new window is valid to S if and only if Open image in new window , where Open image in new window . Clearly Open image in new window is valid to S if it is valid to Open image in new window . We argued in Proposition 3.4 that Open image in new window for Open image in new window and since Open image in new window by assumption, it follows that Open image in new window is valid to S for all Open image in new window . Therefore Open image in new window .

For an arbitrary integer exponent Open image in new window and Open image in new window , it is not at all obvious what the set Open image in new window should be. Note that this includes the case of a monomial over Open image in new window . As a generalization of the multilinear case, is it true that Open image in new window ? The function Open image in new window is Taylor’s first-order approximation of Open image in new window at the point Open image in new window . Having Open image in new window would mean that the gradient inequality at Open image in new window holds true, which is not at all obvious since Open image in new window is a nonconvex function. We show in Proposition 3.5 that Open image in new window is always true, regardless of S, and in fact construct a Open image in new window with Open image in new window , so that Open image in new window in general. This \({\beta }\) depends on S and is constructed by taking projections of S onto each coordinate. We also present some conditions under which Open image in new window can be (partially) characterized.

The following technical lemma will be useful. It is proved in “Appendix”.

Remark 5

Finding an analytic expression for the root Open image in new window seems difficult, and an algebraic root may not even exist, as can be verified using computational algebra software for the polynomial Open image in new window , whose roots are in bijection to that of Open image in new window under the mapping Open image in new window . However, our forthcoming analysis circumvents this issue since it does not depend on the exact value of Open image in new window .

We also need to introduce some notation. For every Open image in new window , denote the projection of S onto the Open image in new window -subspace by
$$\begin{aligned}{{\mathrm{Proj}}}_{x_{i}} S := [1-\sigma ^{1}_{i},1-\sigma ^{2}_{i}],\quad \text { for some } 0 \le \sigma ^{2}_{i}\le \sigma ^{1}_{i}\le 1, \end{aligned}$$
and defineThis \({\gamma }\) is exactly the \({\gamma }\) from the statement of Theorem 1.1 in Sect. 1.1.1. Note that if \({ S\cap E_{i}\ne \emptyset }\), \({ S\cap E_{j}\ne \emptyset }\) for distinct ij, then Open image in new window .

Proof

Open image in new window is obvious due to Open image in new window and Open image in new window . Since Open image in new window , we have Open image in new window , making Open image in new window an increasing function on Open image in new window . Hence, by complementing to Open image in new window , Open image in new window is a decreasing function on Open image in new window . L’Hôpital’s rule gives Open image in new window . \(\square \)

Proof of Claim

If Open image in new window , then Open image in new window and applying the first item in Lemma 3.4 with Open image in new window and Open image in new window tells us Open image in new window for all Open image in new window . Otherwise Open image in new window and Lemma 3.5 allows us to apply Lemma 3.4 with Open image in new window and Open image in new window . It is readily seen from the construction of Open image in new window in (15) that Open image in new window is a root of Open image in new window and by the second item of Lemma 3.4, it is the unique root in Open image in new window . Now Open image in new window and the fourth item of Lemma 3.4 yield Open image in new window for all Open image in new window . Open image in new window

Assume that the inequality is true for Open image in new window and let us argue it for Open image in new window . The induction hypothesis gives usLet Open image in new window for some Open image in new window ; such a Open image in new window exists because Open image in new window for Open image in new window . Hence, the induction hypothesis becomes
$$\begin{aligned} 1+\chi - \gamma _{k+1}\epsilon _{k+1} \ge 1 + \sum _{i=1}^{n}\gamma _{i}(x_{i}-1). \end{aligned}$$
Now,where the inequality is by applying Claim 3.2 to \({i=k+1}\), and using \({1+\chi > 0}\). Since \({(1+\chi )(1-\gamma _{k+1}\epsilon _{k+1}) = 1 + \chi -\gamma _{k+1}\epsilon _{k+1} -\gamma _{k+1}\epsilon _{k+1}\chi }\) and \(\chi < 0, \gamma _{k+1},\epsilon _{k+1} > 0\), we havewhere Open image in new window is from the induction hypothesis. This finishes our inductive proof for showing Open image in new window . Thus every Open image in new window with Open image in new window has Open image in new window . The closedness of Open image in new window under monotonicity and Open image in new window give us Open image in new window .

(2) Choose some Open image in new window . If Open image in new window , then there is nothing to prove because Open image in new window and Open image in new window . So assume Open image in new window . Consider a point Open image in new window , which can be written as Open image in new window and Open image in new window , where Open image in new window if Open image in new window , otherwise Open image in new window is a small positive real. Note that Open image in new window and Open image in new window . The second and third items of Lemma 3.4 with Open image in new window tell us that Open image in new window if Open image in new window . This means that Open image in new window , which rearranges to Open image in new window , is necessary for Open image in new window to be a valid linear underestimator.

(3) It is easy to see that the convexity of S makes Open image in new window equivalent to Open image in new window for all Open image in new window . We also have Open image in new window implying Open image in new window . Therefore Open image in new window . Now (2) gives us Open image in new window for Open image in new window .

(4) The assumption Open image in new window implies Open image in new window for all Open image in new window , Open image in new window , Open image in new window and hence Open image in new window . The claim then follows from (3). \(\square \)

Remark 6

Due to the functions Open image in new window and Open image in new window being nonincreasing and nonpositive, respectively, over Open image in new window , it follows that Open image in new window in Proposition 3.5, meaning that the lower bound on Open image in new window with Open image in new window is weaker than the lower bound on Open image in new window with Open image in new window . This happens because while arguing this part, we used a lower bound on the root of Open image in new window in Open image in new window from Lemma 3.4, since finding a analytic expression for the root seems difficult (cf. Remark 5). Therefore if Open image in new window , then there is no guarantee that Open image in new window is a nondominated point in Open image in new window .

Remark 7

The second item in Proposition 3.5 indicates that a tight lower bound on a valid \({\beta }\) can get arbitrarily close to Open image in new window .

The vector Open image in new window in (15) can be constructed only when projections of S are readily available or can be computed quickly. When these projections are difficult to compute, we could use the first claim of Proposition 3.5 telling us that Open image in new window is a underestimator of Open image in new window . The last item in this proposition provides a clean and simple expression for Open image in new window in (14).

The preceding results on Open image in new window and Open image in new window , combined with Theorem 3.2, imply explicit bounds on the error from the convex underestimator Open image in new window . Recall the constants from (8). Denoting Open image in new window simply as Open image in new window , we have for Open image in new window and Open image in new window , respectively,:where we recall that Open image in new window was defined in (2) and Open image in new window .

Corollary 3.3

Open image in new window , and equality holds throughout if Open image in new window and Open image in new window for some Open image in new window .

Proof

We first observe that Open image in new window . This is obtained by applying Theorem 3.2 with Open image in new window replaced by Open image in new window and noting that Open image in new window is a maximal element of Open image in new window . Since Open image in new window by Proposition 3.5, Open image in new window and hence Open image in new window . Since Open image in new window is concave increasing over Open image in new window and Open image in new window by construction, we get Open image in new window . If Open image in new window , then Open image in new window and the last claim in Proposition 3.5 tells us Open image in new window and Open image in new window . Now recall Theorem 3.2. We have Open image in new window due to Open image in new window . This theorem tells us that the bound on Open image in new window can be attained only at Open image in new window . The assumptions Open image in new window and Open image in new window lead to Open image in new window and therefore Open image in new window . \(\square \)

A direct implication is a tight bound on the error of the convex envelope of a multilinear monomial considered over Open image in new window .

Corollary 3.4

We have Open image in new window . In particular, for a multilinear monomial, Open image in new window .

Proof

Since Open image in new window , the last item in Proposition 3.5 tells us Open image in new window and then the first equality follows immediately from Corollary 3.3. For a multilinear monomial, Eq. (4a) gives us Open image in new window . The claimed error follows by using Open image in new window in the expression for Open image in new window . \(\square \)

3.3 Convex hull error

Proof of Theorem 1.1

Since Open image in new window for Open image in new window , the upper bound of Open image in new window on Open image in new window is due to Open image in new window from Corollary 3.1. Similarly the upper bounds on Open image in new window are due to Open image in new window and Corollary 3.3. By Observation 2.1, we then have that Open image in new window . To show this error is upper bounded by Open image in new window , we argue the following.

Claim 3.3

Open image in new window for Open image in new window and equality holds if and only if Open image in new window .

Proof of Claim

The two constants are Open image in new window and Open image in new window . Therefore the following equivalence holds:Lemma 3.1 proves the last inequality and that it holds at equality only when Open image in new window . Thus we have Open image in new window for any Open image in new window .
If Open image in new window , then setting Open image in new window in Lemma 2.1 yields the critical point to be Open image in new window so thatTherefore the convex hull error and the concave envelope error are lower bounded by Open image in new window , making each of them equal to Open image in new window . \(\square \)
The arguments used in proving Theorem 1.1 also imply that a family of convex relaxations of Open image in new window has error equal to Open image in new window . Recall the convex underestimator Open image in new window from (7a) for any Open image in new window and consider the convex relaxationNote that Open image in new window is not restricted to be in S here. Assume Open image in new window . Also assume Open image in new window so that Open image in new window for every Open image in new window , as per Proposition 3.4. We claim that

Proposition 3.6

Open image in new window .

Proof

The proof of Open image in new window is the same as that in Theorem 1.1, along with using the assumption Open image in new window to get Open image in new window . Tightness of this bound is obtained by applying Lemma 2.1 and Remark 1 after noting that Open image in new window . The point Open image in new window belongs to Open image in new window because Open image in new window , and Open image in new window , which is equal to Open image in new window since Proposition 3.4 states that Open image in new window . The point Open image in new window belongs to Open image in new window because Open image in new window , and Open image in new window , which is less than equal to 1 due to Open image in new window . \(\square \)

The next proof is that of the error bounds over a simplex.

Proof of Theorem 1.2

The concave envelope error bound is from Proposition 3.2 and the fact that Open image in new window . The convex envelope error bound was observed in Proposition 3.3. To upper bound Open image in new window , we note thatDenoting Open image in new window , we have Open image in new window . Thus it suffices to show that Open image in new window , equivalently, Open image in new window . Since Open image in new window due to Open image in new window ,where the last inequality is from binomial expansion. Open image in new window

We end by mentioning that for Open image in new window , or equivalently for Open image in new window upto scaling, our upper bounds on the convex hull error are the same as those in Theorem 1.1 whereas a lower bound can be obtained by setting Open image in new window in Lemma 2.1. However these bounds are not tight, which is not all that surprising since we do not know the exact form of the envelopes of a general monomial over Open image in new window . In Sect. 4, we consider a multilinear monomial over Open image in new window and use the explicit characterization of its envelopes to derive tight error bounds. It so happens that in the multilinear case, the lower bound from Lemma 2.1 with Open image in new window seems to be the convex hull error, a claim that is verified empirically for random Open image in new window and shown to be true for every Open image in new window as Open image in new window .

3.4 Comparison with another error bound

For the problem of optimizing Open image in new window over Open image in new window : Open image in new window , De Klerk and Laurent [13] present a LP and a SDP relaxation of Open image in new window based on two different positivstellensatz and also give a common error bound for these relaxations. Their bound is [13, Theorem 1.4]:where Open image in new window is either of their two relaxations, Open image in new window is an integer with Open image in new window being a degree bound on polynomials in the positivstellensatz, andAs Open image in new window , the two relaxations converge to Open image in new window (the SDP relaxation has finite convergence). Corollary 1.2 states that the monomial convexification approach would yield a error bound, as per our analysis, of Open image in new window for Open image in new window defined in (3). This bound was weakened subsequently in Corollary 1.3 for ease of computation.

We note that for the LP and SDP relaxations to provide a better worst case guarantee, the degrees of the polynomials considered in the respective positivstellensatz must grow cubic in the degree of Open image in new window .

Proposition 3.7

For Open image in new window with Open image in new window , and fixed Open image in new window , the worst case error bound from Open image in new window is better than the worst case error bound from Open image in new window only if Open image in new window .

Proof

The assumption Open image in new window implies Open image in new window , which is lower bounded by Open image in new window . We have Open image in new window from Corollary 1.3. Therefore, the LP and SDP relaxations of [13] give better error bounds than monomial convexification only if
$$\begin{aligned} \delta \ge \hat{\delta } := \frac{\left( {\begin{array}{c}m+1\\ 3\end{array}}\right) n^{m}}{m!\mathscr {C}^{1}_{m}\left( {\begin{array}{c}n+m\\ n\end{array}}\right) } = \frac{m^{2}(m+1)}{6 m^{\frac{1}{1-m}} (1+\frac{m}{n})\cdots (1+\frac{1}{n})} = {\Omega }(m^{3})\quad \text {for fixed } n. \end{aligned}$$
\(\square \)

4 Multilinear monomial

Here we consider a multilinear monomial Open image in new window over either a box with constant ratio or a symmetric box. Since these boxes are simple scalings of Open image in new window and Open image in new window , respectively, and our error measure Open image in new window scales as noted in Observation 2.2, we henceforth restrict our attention to only Open image in new window and Open image in new window . As in Sect. 3, the convex hull error is computed by bounding the convex and concave envelope errors separately.

4.1 Box with constant ratio

Proposition 4.1

Proof

To obtain Open image in new window , we simply substitute Open image in new window in [6, Theorem 1] which states Open image in new window for arbitrary Open image in new window with Open image in new window . The convex envelope can be derived from [37, Theorem 4.6]. This theorem gives a piecewise linear function with Open image in new window pieces as the convex envelope of a function Open image in new window when Open image in new window is convex-extendable from Open image in new window and there exists a convex function Open image in new window such that Open image in new window for every Open image in new window . Consider Open image in new window . Writing Open image in new window , the multilinear term becomes Open image in new window for Open image in new window . Since this Open image in new window is a multilinear function of Open image in new window , it is convex-extendable from Open image in new window . Furthermore, for Open image in new window , Open image in new window , and obviously Open image in new window is convex over Open image in new window . Therefore, the convex envelope formula follows from [37, Theorem 4.6]. \(\square \)

Applying a straightforward scaling argument, similar to the one used for the Open image in new window box at the beginning of Sect. 3, gives us the convex hull of Open image in new window when Open image in new window for all Open image in new window and for some Open image in new window , Open image in new window for all Open image in new window with Open image in new window and Open image in new window for all Open image in new window with Open image in new window . We omit the details.

Before proving Theorem 1.3 which claims that \({\mathscr {D}_{r,n}}\) and \({\mathscr {E}_{r,n}}\) are the maximum envelope errors for Open image in new window over Open image in new window , we provide some background on these two constants. The value \({\mathscr {E}_{r,n}}\) is obtained by applying Lemma 2.1: set Open image in new window to get Open image in new window and Open image in new window , upon simplification, becomes equal to \({\mathscr {E}_{r,n}}\). There is no simple explicit closed form formula for \({\mathscr {D}_{r,n}}\). However, \({\mathscr {D}_{r,n}}\) can be bounded as follows. After replacing Open image in new window , the formula for \({\mathscr {D}_{r,n}}\) requires solving an integer program:
$$\begin{aligned} \mathscr {D}_{r,n}= \max \{\psi (t) \mid t \in \{1/n,2/n,\dots ,1 \} \}, \quad \text { where } \psi (t) = (1 + (r-1)t)^{n} - r^{nt}. \end{aligned}$$
Note that Open image in new window is a difference of two convex increasing functions Open image in new window and Open image in new window . After separating the maximizations over Open image in new window and Open image in new window , we obtain the trivial upper bound Open image in new window . But this bound can be very weak. A tighter bound can be derived by considering the continuous relaxation of the problem:
$$\begin{aligned} \mathscr {D}_{r,n}\le \max \{\psi (t)\mid t \in [0,1]\}. \end{aligned}$$
Since Open image in new window is differentiable with Open image in new window , by Rolle’s theorem, there exists at least one stationary point of Open image in new window in Open image in new window . Based on these stationary points, we can say the following.

The proof is in “Appendix”. Obviously, Open image in new window . We conjecture that Open image in new window .

We now prove our main result in this section.

4.1.1 Proof of Theorem 1.3

Proof

We only prove the maximum errors for the envelopes, the formula for Open image in new window follows subsequently from Observation 2.1. Consider the concave envelope first. We noted earlier that the value \({\mathscr {E}_{r,n}}\) comes from applying Lemma 2.1 with Open image in new window . Hence to prove that the maximum concave envelope error is equal to \({\mathscr {E}_{r,n}}\), it suffices to argue that there exists a point in Open image in new window which maximizes this error. Suppose, for sake of contradiction, that this is not the case. Since Open image in new window and Open image in new window , we know that these two points do not maximize the error. Then our assumption means that for every maximizer Open image in new window there exists some index Open image in new window such that Open image in new window , where Open image in new window is the permutation that permutes variables as Open image in new window . Since Open image in new window , for every Open image in new window , the minimum over Open image in new window in the expression for Open image in new window , which is given in Proposition 4.1, occurs at a permutation Open image in new window such that Open image in new window . Therefore, Open image in new window . In particular, Open image in new window , and the maximum error is Open image in new window . Now consider two points Open image in new window and Open image in new window obtained from Open image in new window by setting, respectively, Open image in new window and Open image in new window . Since the error at these points cannot be larger than Open image in new window , we have Open image in new window and Open image in new window , and consequently, Open image in new window and Open image in new window . Hence
$$\begin{aligned} r^{n-i}\le r^{n-i}x^{*}_{(i)} \le \prod _{j=1}^{n}x^{*}_{j} \le r^{n-i-1}x^{*}_{(i+1)}\le r^{n-i}. \end{aligned}$$
Equality holds in above if and only if Open image in new window and Open image in new window . Therefore Open image in new window , Open image in new window , but at such a point, the error is zero due toThus we have reached a contradiction to Open image in new window being a maximizer. Hence it must be that the error is maximized on Open image in new window .
Now consider the convex envelope. We follow similar steps as in the proof of Theorem 3.2.where we employ the arithmetic-geometric means inequality. By regarding Open image in new window as a scalar variable Open image in new window , we getwhere Open image in new window is a convex function on Open image in new window . Therefore we have to find the maximum value of the pointwise minimum function Open image in new window on the interval Open image in new window and it is apparent that this maximum value is attained at a breakpoint of the function, i.e., at a Open image in new window such that Open image in new window for some Open image in new window . For any Open image in new window , solving for Open image in new window in Open image in new window means that we must find Open image in new window satisfying Open image in new window , which upon canceling and rearranging terms leads to Open image in new window . Therefore Open image in new window and hence, Open image in new window . Substituting this breakpoint Open image in new window into Open image in new window yieldsThe maximum, with respect to Open image in new window , over all such values is the maximum of Open image in new window and notice that this maximum over Open image in new window is exactly the constant \({\mathscr {D}_{r,n}}\). Hence \({\mathscr {D}_{r,n}}\) is a upper bound on the convex envelope error. This bound is tight because the means inequality is an equality when all the Open image in new window ’s are equal to each other, and hence this error is attained on Open image in new window . \(\square \)

4.1.2 Comparing \({\mathscr {D}_{r,n}}\) and \({\mathscr {E}_{r,n}}\)

We conjecture that Open image in new window for every Open image in new window , which would imply that the convex hull error is equal to \({\mathscr {E}_{r,n}}\). Although we were unable to prove this in general due to the extremely complicated forms for \({\mathscr {E}_{r,n}}\), and more specifically, for \({\mathscr {D}_{r,n}}\), we ran some simulations, graphed in Fig. 1, to support our claim. For every Open image in new window and Open image in new window , we computed the ratio Open image in new window and plotted it in Fig. 1a. We also plotted the ratio between the error of the relaxed convex envelope (the relaxation is obtained by taking the maximum over Open image in new window in the expression for Open image in new window ) and \({\mathscr {E}_{r,n}}\), see Fig. 1b. As can be seen in these figures, the ratios are never larger than 1, thereby establishing a strong empirical basis in support of our conjecture that the error from the concave envelope dominates that from the convex envelope, and possibly even from the relaxed convex envelope.
Fig. 1

Error comparisons for Open image in new window over Open image in new window . a Concave and convex envelopes. b Concave and relaxed convex envelopes

Asymptotically, \({\mathscr {E}_{r,n}}\) dominates \({\mathscr {D}_{r,n}}\) in the following sense. Recall Open image in new window and Open image in new window defined in Proposition 4.2.

Proof We have
$$\begin{aligned} \lim _{n\rightarrow \infty }\frac{\mathscr {E}_{r,n}}{r^{n}-1}= & {} \frac{1}{r-1}\left[ \lim _{n\rightarrow \infty } \frac{n-1}{n}\,\lim _{n\rightarrow \infty } {\left( \frac{r^n-1}{n(r-1)}\right) }^{\frac{1}{n-1}}-1\right] \\= & {} \frac{1}{r-1}\left[ 1\cdot r - 1 \right] = 1. \end{aligned}$$
Proposition 4.2 gives two bounds on \({\mathscr {D}_{r,n}}\). If Open image in new window , then
$$\begin{aligned} \lim _{n\rightarrow \infty }\frac{\mathscr {D}_{r,n}}{r^{n}-1}= & {} \lim _{n\rightarrow \infty }\frac{\mathscr {D}_{r,n}}{r^{n}}\,\lim _{n\rightarrow \infty }\frac{r^{n}}{r^{n}-1} \\= & {} \lim _{n\rightarrow \infty }\left( \frac{1}{r} + \left( 1-\frac{1}{n}\right) \left( 1-\frac{1}{r}\right) \right) ^{n} - \frac{1}{r} \\= & {} \lim _{n\rightarrow \infty }\left( 1 -\frac{1}{n}+\frac{1}{nr} \right) ^{n} - \frac{1}{r}\\= & {} e^{\frac{1}{r}-1}-\frac{1}{r} . \end{aligned}$$
The above function of Open image in new window is increasing over Open image in new window and converges to Open image in new window as Open image in new window . The limit on the other value of \({\mathscr {D}_{r,n}}\) is \({\frac{\ln r}{r-1}-\frac{1}{r}}\) as \({n\rightarrow \infty }\), and the value of this function of Open image in new window never exceeds 0.22. \(\square \)

Thus \({\mathscr {E}_{r,n}}\) seems to grow much more rapidly than \({\mathscr {D}_{r,n}}\) in some cases.

4.2 Symmetric box

4.2.1 Convex hull

Luedtke et al. [25] showed that the recursive McCormick relaxation, which Ryoo and Sahinidis [33] had used to obtain an extended formulation of Open image in new window , yields a compact extended formulation of Open image in new window . However, to the best of our knowledge, there is no known characterization of this convex hull in the Open image in new window -space. We provide this next. A different proof based on constructive arguments is presented in a companion paper [1].

Before presenting our proof, we provide an intuition behind the proposed convex hull description. Denote \({x_{n+1}=w}\) to get Open image in new window . It is well-known [32, 34] that for any box Open image in new window , the extreme points of Open image in new window are in bijection with the extreme points of Open image in new window (this is also true for a multilinear polynomial). Hence the set of extreme points of Open image in new window is equal to Open image in new window . A point in Open image in new window violates Open image in new window if and only if the set Open image in new window has odd cardinality. Every such inadmissible point in Open image in new window can be cut off using the “no-good” inequality
$$\begin{aligned} \sum _{i\in I}(x_{i}-(-1)) \,+\, \sum _{i\in \{1,\dots ,n+1\}{\setminus } I}(1-x_{i}) \ge 2 \end{aligned}$$
for some odd subset \({I\subseteq \{1,\dots ,n+1\}}\). The no-good cut for subset I is valid to every point in \({\{-1,1\}^{n+1}}\), except that point which takes the value \({-1}\) at exactly those elements indexed by I. This cut rearranges to
$$\begin{aligned} \sum _{i\in I}x_{i} \,-\, \sum _{\{1,\dots ,n+1\}{\setminus } I}x_{i}\ge -(n-1). \end{aligned}$$
(17)
Hence Open image in new window . Consider the polytopewhich is the LP relaxation of Open image in new window . By construction, this polytope has the property that Open image in new window . We will show in the proof of Theorem 4.1 that the extreme points of Open image in new window are in Open image in new window , thereby implying that Open image in new window . This equality, along with the following claim that is straightforward to verify, gives us the statement of Theorem 4.1.

Observation 4.1

After denoting Open image in new window , each of the convex hull descriptions in Theorem 4.1 becomes equal to the polytope Open image in new window .

Proof of Theorem 4.1

We show that for any Open image in new window , the linear program Open image in new window has an optimal solution in Open image in new window . We proceed by considering cases that are defined using Open image in new window . Note two things: (1) Open image in new window due to Open image in new window for every feasible Open image in new window , (2) any Open image in new window belongs to Open image in new window if and only if Open image in new window has even cardinality.
Case 1    Open image in new window is even.

Since Open image in new window has even cardinality, the point Open image in new window with Open image in new window for Open image in new window and Open image in new window for Open image in new window belongs to Open image in new window . This Open image in new window is optimal to Open image in new window because Open image in new window .

Case 2    Open image in new window is odd and Open image in new window .

Choose an arbitrary Open image in new window and set Open image in new window for Open image in new window and Open image in new window for Open image in new window . This Open image in new window belongs to Open image in new window because Open image in new window is even and is optimal to Open image in new window because Open image in new window .

Case 3    Open image in new window is odd and Open image in new window .
Let Open image in new window . There are two subcases. When Open image in new window , i.e. Open image in new window , the point Open image in new window for Open image in new window and Open image in new window for Open image in new window is optimal with value Open image in new window , because in this subcase
$$\begin{aligned} c^\top x&=\left( -c_{j_1}\right) \left( \sum _{i\in C}x_i-\sum _{i\in B}x_i\right) +\sum _{i\in C}\left( c_i+c_{j_1}\right) x_i+\sum _{i\in B}\left( c_i- c_{j_1}\right) x_i\\&\le \left( -c_{j_1}\right) (n-1)+\sum _{i\in C}\left( c_i+ c_{j_1}\right) +\sum _{i\in B}\left( c_{j_1}- c_i\right) \\&=\sum _{i\in C}c_i+\sum _{i\in B}\left( - c_i\right) +2 c_{j_1}\\&=\sum _{i\in B{\setminus } \{j_1\}}\left( - c_i\right) +\sum _{i\in C\cup \{j_1\}} c_i, \end{aligned}$$
where the Open image in new window inequality is obtained by applying (17) with Open image in new window . When Open image in new window , i.e. Open image in new window , then the point Open image in new window for Open image in new window and Open image in new window for Open image in new window is optimal with value Open image in new window , because in this subcase
$$\begin{aligned} c^\top x&= c_{j_1}\left( \sum _{i\in C}x_i-\sum _{i\in B}x_i\right) +\sum _{i\in C}\left( c_i- c_{j_1}\right) x_i+\sum _{i\in B}\left( c_i+c_{j_1}\right) x_i\\&\le c_{j_1}(n-1)+\sum _{i\in C}\left( c_i- c_{j_1}\right) +\sum _{i\in B}\left( - c_i- c_{j_1}\right) \\&=\sum _{i\in C}c_i+\sum _{i\in B}\left( - c_i\right) -2 c_{j_1}\\&=\sum _{i\in B\cup \{j_1\}}\left( - c_i\right) +\sum _{i\in C{\setminus }\{j_1\}}c_i. \end{aligned}$$
This completes our proof for showing that Open image in new window has extreme points in Open image in new window . \(\square \)

A scaling argument yields Open image in new window when Open image in new window for all Open image in new window .

4.2.2 Errors

In order to prove Theorem 1.4, we make use of the reflection symmetry in the sets Open image in new window and Open image in new window , as described next. Let Open image in new window denote the sign of a scalar, with Open image in new window considered positive. A point Open image in new window is said to have compatible signs if Open image in new window , i.e., Open image in new window is negative if and only if Open image in new window has no zero entries and has an odd number of negative entries. Define the following binary relation on Open image in new window : Open image in new window if (i) Open image in new window and Open image in new window for all Open image in new window , and (ii) both Open image in new window and Open image in new window have compatible signs or both Open image in new window and Open image in new window do not have compatible signs. Thus Open image in new window if and only if Open image in new window is obtained from Open image in new window by reversing signs on odd (even) many entries of Open image in new window and setting Open image in new window ( Open image in new window ). This binary relation has two important properties.
  1. 1.

    It preserves the error measure Open image in new window . Indeed, one can easily argue that Open image in new window if Open image in new window .

     
  2. 2.

    It is an equivalence relation, i.e., a reflexive symmetric transitive relation. This is obvious by construction of Open image in new window .

     
Now consider Open image in new window , the equivalence class of Open image in new window induced by \({\sim }\). Since \({\sim }\) is an equivalence relation on Open image in new window and Open image in new window and Open image in new window are subsets of Open image in new window , each of these sets is partitioned by Open image in new window . Observe that the definition of Open image in new window means that for every Open image in new window having compatible (incompatible) signs, there exists Open image in new window such that Open image in new window and Open image in new window ( Open image in new window ). Now, because every point in Open image in new window has compatible signs and Open image in new window trivially implies Open image in new window , we have To make a similar statement for Open image in new window , we need a small modification because the convex hull contains points with both compatible and incompatible signs. In particular, we must drop the nonnegativity requirement on Open image in new window . Also, if Open image in new window , then using the fact that Open image in new window is a convex combination of points in Open image in new window , all of which have compatible signs, we get that Open image in new window . Thus we have the following:Now, the fact that Open image in new window is error-preserving leads tomeaning that we only need to consider nonnegative values of Open image in new window when computing the convex hull error.

Proof of Theorem 1.4

To upper bound the convex hull error. We only present arguments for when Open image in new window is odd, since the even case is almost exactly the same due to similar characterizations of the convex hulls in Theorem 4.1. By Eq. (19c), we consider only Open image in new window with Open image in new window . Thus, Open image in new window is equal to the maximum of the maximum errors of Open image in new window and Open image in new window calculated over Open image in new window .where Open image in new window has given us the second equality, and the inequality in the last step from applying the arithmetic-geometric means inequality. Therefore, after regarding Open image in new window as a scalar variable Open image in new window , we get Open image in new window to be an upper bound on the convex envelope error, where Open image in new window . The function Open image in new window is convex decreasing on Open image in new window whereas Open image in new window is convex increasing on Open image in new window , and hence the maximum value of Open image in new window on Open image in new window occurs at a breakpoint where the two functions have equal value. Solving for Open image in new window yields Open image in new window , and so the upper bound is Open image in new window . This bound is tight since it is attained at Open image in new window where Open image in new window . On the concave side, we have Open image in new window and since Open image in new window for Open image in new window , the concave envelope error is upper bounded by Open image in new window . Thus, Open image in new window .

To find the points where this bound is attained, we already observed the point Open image in new window . Since our relation Open image in new window is error-preserving, all points in the equivalence class of Open image in new window have the same error, and there are Open image in new window many such points. Finally, note that for any point Open image in new window , the above bounds on the envelopes would be reversed so that both the envelopes have the same maximum error over the entire Open image in new window box. \(\square \)

Footnotes

  1. 1.

    To avoid tediousness and with a slight abuse of notation, for each monomial we write Open image in new window with the understanding that those \({x_{j}}\) that appear in the monomial are included.

  2. 2.

    It does not seem that Open image in new window will be a polyhedron even for Open image in new window . Since general monomials are not vertex-extendable over Open image in new window , it is not clear whether the validity of Open image in new window over the entire box can be certified by checking at only a finite number of points.

Notes

Acknowledgements

The first author was supported in part by ONR Grant N00014-16-1-2168. The second author was supported in part by ONR Grant N00014-16-1-2725. We thank two referees whose meticulous reading helped us clarify some of the technical details.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society 2018

Authors and Affiliations

  1. 1.Department of Mathematical SciencesClemson UniversityClemsonUSA

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