Error bounds for monomial convexification in polynomial optimization
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 worstcase 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 inequalityMathematics Subject Classification
90C26 65G99 52A271 Introduction
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 nvariate 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 cooccurrence 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].
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 nsimplex 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
Theorem 1.1
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
Corollary 1.2
Proof
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
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
1.1.2 Multilinear monomial
Consider the multilinear monomial Open image in new window .
Theorem 1.3
We conjecture that \({\mathscr {D}_{r,n}\le \mathscr {E}_{r,n}}\) for all r, n 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
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
Observation 2.1
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.
Observation 2.2
For any Open image in new window , we have Open image in new window , with Open image in new window being optimal to Open image in new window if and only if Open image in new window is optimal to Open image in new window .

Open image in new window scale to any box having a vertex at Open image in new window ,

Open image in new window scale to any box for which the ratio between lower and upper bounds is the same positive scalar in each coordinate, and

Open image in new window scale to any box that is symmetric with respect to Open image in new window .
Lemma 2.1
Proof
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.
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 .
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 i, j 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
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 degreedependent 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
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 nonemptiness 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 lowenough and highenough, 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 .
Corollary 3.2
For any Open image in new window with Open image in new window , the upper bound Open image in new window on Open image in new window is tight if and only if Open image in new window . In particular, 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.
Lemma 3.1
Open image in new window for all Open image in new window , and Open image in new window if and only if Open image in new window .
Proof
Proposition 3.2
Proof
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 . \(\square \)
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
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
 1.
 2.
If Open image in new window , then Open image in new window .
 3.
If Open image in new window , then 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.
Lemma 3.2
Proof
Lemma 3.3
We are now ready to state our upper bound on error from the convex underestimator Open image in new window .
Theorem 3.2
Proof
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 degreedependent 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
 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 .
The following technical lemma will be useful. It is proved in “Appendix”.
Lemma 3.4
 1.
 2.
Open image in new window has exactly one root in Open image in new window , denoted Open image in new window , and Open image in new window .
 3.
Open image in new window for all Open image in new window and Open image in new window for all Open image in new window .
 4.
Open image in new window for all Open image in new window , and Open image in new window for all Open image in new window
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 .
Lemma 3.5
Open image in new window for every Open image in new window with Open image in new window . Hence Open image in new window if and only if 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 . \(\square \)
Proposition 3.5
 2.
Open image in new window only if Open image in new window for Open image in new window with Open image in new window , and Open image in new window for Open image in new window with Open image in new window .
 3.
Suppose Open image in new window . Then Open image in new window only if Open image in new window for all Open image in new window .
 4.
If Open image in new window for some Open image in new window , then Open image in new window .
Proof
(1) Observe that showing Open image in new window for all Open image in new window is equivalent to showing Open image in new window for all Open image in new window such that Open image in new window . Indeed, Open image in new window is exact at Open image in new window and for any Open image in new window , Open image in new window implies that Open image in new window which is nonpositive due to Open image in new window and Open image in new window . Therefore to show Open image in new window , we prove Open image in new window for every Open image in new window .
Claim 3.2
For any Open image in new window and Open image in new window , we have Open image in new window for all Open image in new window .
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
(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).
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
Proposition 3.6
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
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
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
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 convexextendable 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 convexextendable 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.
Proposition 4.2
Let Open image in new window be the smallest stationary point of Open image in new window on Open image in new window , and Open image in new window be the global maxima of Open image in new window on Open image in new window . If Open image in new window , then Open image in new window , otherwise if Open image in new window , then Open image in new window , otherwise Open image in new window .
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
4.1.2 Comparing \({\mathscr {D}_{r,n}}\) and \({\mathscr {E}_{r,n}}\)
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.
Proposition 4.3
Open image in new window , and Open image in new window if Open image in new window or Open image in new window .
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].
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
 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 subcasewhere 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$$\begin{aligned} c^\top x&=\left( c_{j_1}\right) \left( \sum _{i\in C}x_i\sum _{i\in B}x_i\right) +\sum _{i\in C}\left( c_i+c_{j_1}\right) x_i+\sum _{i\in B}\left( c_i c_{j_1}\right) x_i\\&\le \left( c_{j_1}\right) (n1)+\sum _{i\in C}\left( c_i+ c_{j_1}\right) +\sum _{i\in B}\left( c_{j_1} c_i\right) \\&=\sum _{i\in C}c_i+\sum _{i\in B}\left(  c_i\right) +2 c_{j_1}\\&=\sum _{i\in B{\setminus } \{j_1\}}\left(  c_i\right) +\sum _{i\in C\cup \{j_1\}} c_i, \end{aligned}$$$$\begin{aligned} c^\top x&= c_{j_1}\left( \sum _{i\in C}x_i\sum _{i\in B}x_i\right) +\sum _{i\in C}\left( c_i c_{j_1}\right) x_i+\sum _{i\in B}\left( c_i+c_{j_1}\right) x_i\\&\le c_{j_1}(n1)+\sum _{i\in C}\left( c_i c_{j_1}\right) +\sum _{i\in B}\left(  c_i c_{j_1}\right) \\&=\sum _{i\in C}c_i+\sum _{i\in B}\left(  c_i\right) 2 c_{j_1}\\&=\sum _{i\in B\cup \{j_1\}}\left(  c_i\right) +\sum _{i\in C{\setminus }\{j_1\}}c_i. \end{aligned}$$
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
 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.
It is an equivalence relation, i.e., a reflexive symmetric transitive relation. This is obvious by construction of Open image in new window .
Proof of Theorem 1.4
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 errorpreserving, 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.
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.
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 vertexextendable 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 N000141612168. The second author was supported in part by ONR Grant N000141612725. We thank two referees whose meticulous reading helped us clarify some of the technical details.
References
 1.Adams, W., Gupte, A., Xu, Y.: An RLT approach for convexifying symmetric multilinear polynomials. Working paper (2017)Google Scholar
 2.AlKhayyal, F., Falk, J.: Jointly constrained biconvex programming. Math. Oper. Res. 8(2), 273–286 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
 3.Bao, X., Khajavirad, A., Sahinidis, N.V., Tawarmalani, M.: Global optimization of nonconvex problems with multilinear intermediates. Math. Program. Comput. 7(1), 1–37 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
 4.Belotti, P., Lee, J., Liberti, L., Margot, F., Wächter, A.: Branching and bounds tightening techniques for nonconvex MINLP. Optim. Methods Softw. 24(4), 597–634 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
 5.Belotti, P., Miller, A.J., Namazifar, M.: Valid inequalities and convex hulls for multilinear functions. Electron. Notes Discrete Math. 36, 805–812 (2010)CrossRefzbMATHGoogle Scholar
 6.Benson, H.P.: Concave envelopes of monomial functions over rectangles. Naval Res. Logist. (NRL) 51(4), 467–476 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
 7.Boland, N., Dey, S.S., Kalinowski, T., Molinaro, M., Rigterink, F.: Bounding the gap between the mccormick relaxation and the convex hull for bilinear functions. Math. Program. 162, 523–535 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
 8.Buchheim, C., D’Ambrosio, C.: Monomialwise optimal separable underestimators for mixedinteger polynomial optimization. J. Glob. Optim. 67(4), 759–786 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
 9.Buchheim, C., Michaels, D., Weismantel, R.: Integer programming subject to monomial constraints. SIAM J. Optim. 20(6), 3297–3311 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
 10.Crama, Y.: Concave extensions for nonlinear 0–1 maximization problems. Math. Program. 61(1–3), 53–60 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
 11.Crama, Y., RodríguezHeck, E.: A class of valid inequalities for multilinear 0–1 optimization problems. Discrete Optim. 25, 28–47 (2017)MathSciNetCrossRefGoogle Scholar
 12.Dalkiran, E., Sherali, H.D.: RLTPOS: reformulationlinearization techniquebased optimization software for solving polynomial programming problems. Math. Program. Comput. 8, 1–39 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
 13.De Klerk, E., Laurent, M.: Error bounds for some semidefinite programming approaches to polynomial minimization on the hypercube. SIAM J. Optim. 20(6), 3104–3120 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
 14.De Klerk, E., Laurent, M., Sun, Z.: An error analysis for polynomial optimization over the simplex based on the multivariate hypergeometric distribution. SIAM J. Optim. 25(3), 1498–1514 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
 15.De Klerk, E., Laurent, M., Sun, Z.: Convergence analysis for Lasserres measurebased hierarchy of upper bounds for polynomial optimization. Math. Program. 162, 1–30 (2016)MathSciNetGoogle Scholar
 16.Del Pia, A., Khajavirad, A.: A polyhedral study of binary polynomial programs. Math. Oper. Res. 42, 389–410 (2016)MathSciNetzbMATHGoogle Scholar
 17.Dey, S.S., Gupte, A.: Analysis of MILP techniques for the pooling problem. Oper. Res. 63(2), 412–427 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
 18.Lasserre, J.B.: Global optimization with polynomials and the problem of moments. SIAM J. Optim. 11(3), 796–817 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
 19.Lasserre, J.B.: An Introduction to Polynomial and Semialgebraic Optimization, vol. 52. Cambridge University Press, Cambridge (2015)CrossRefzbMATHGoogle Scholar
 20.Laurent, M.: Sums of squares, moment matrices and optimization over polynomials. In: Emerging Applications of Algebraic Geometry, pp. 157–270. Springer (2009)Google Scholar
 21.Liberti, L., Pantelides, C.C.: Convex envelopes of monomials of odd degree. J. Glob. Optim. 25(2), 157–168 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
 22.Linderoth, J.: A simplicial branchandbound algorithm for solving quadratically constrained quadratic programs. Math. Program. 103(2), 251–282 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
 23.Locatelli, M.: Polyhedral subdivisions and functional forms for the convex envelopes of bilinear, fractional and other bivariate functions over general polytopes. J. Glob. Optim. Online First (2016). https://doi.org/10.1007/s1089801604184 zbMATHGoogle Scholar
 24.Locatelli, M., Schoen, F.: On convex envelopes for bivariate functions over polytopes. Math. Program. 144(1–2), 65–91 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
 25.Luedtke, J., Namazifar, M., Linderoth, J.: Some results on the strength of relaxations of multilinear functions. Math. Program. 136(2), 325–351 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
 26.McCormick, G.: Computability of global solutions to factorable nonconvex programs: part I. Convex underestimating problems. Math. Program. 10(1), 147–175 (1976)CrossRefzbMATHGoogle Scholar
 27.Meyer, C., Floudas, C.: Trilinear monomials with mixed sign domains: facets of the convex and concave envelopes. J. Glob. Optim. 29(2), 125–155 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
 28.Meyer, C., Floudas, C.: Convex envelopes for edgeconcave functions. Math. Program. 103(2), 207–224 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
 29.Misener, R., Floudas, C.A.: Antigone: algorithms for continuous/integer global optimization of nonlinear equations. J. Glob. Optim. 59(2–3), 503–526 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
 30.Misener, R., Smadbeck, J.B., Floudas, C.A.: Dynamically generated cutting planes for mixedinteger quadratically constrained quadratic programs and their incorporation into GloMIQO 2. Optim. Methods Softw. 30(1), 215–249 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
 31.Pang, J.S.: Error bounds in mathematical programming. Math. Program. 79(1–3), 299–332 (1997)MathSciNetzbMATHGoogle Scholar
 32.Rikun, A.: A convex envelope formula for multilinear functions. J. Glob. Optim. 10(4), 425–437 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
 33.Ryoo, H.S., Sahinidis, N.V.: Analysis of bounds for multilinear functions. J. Glob. Optim. 19(4), 403–424 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
 34.Sherali, H.: Convex envelopes of multilinear functions over a unit hypercube and over special discrete sets. Acta Math. Vietnam. 22(1), 245–270 (1997)MathSciNetzbMATHGoogle Scholar
 35.Sherali, H.D., Dalkiran, E., Liberti, L.: Reduced RLT representations for nonconvex polynomial programming problems. J. Glob. Optim. 52(3), 447–469 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
 36.Speakman, E., Lee, J.: Quantifying double McCormick. Math. Oper. Res. 42(4), 1230–1253 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
 37.Tawarmalani, M., Richard, J.P.P., Xiong, C.: Explicit convex and concave envelopes through polyhedral subdivisions. Math. Program. 138(1–2), 531–577 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
 38.Tawarmalani, M., Sahinidis, N.: Convex extensions and envelopes of lower semicontinuous functions. Math. Program. 93(2), 247–263 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
 39.Tawarmalani, M., Sahinidis, N.: A polyhedral branchandcut approach to global optimization. Math. Program. 103(2), 225–249 (2005)MathSciNetCrossRefzbMATHGoogle Scholar