Abstract
Given a factorable function f, we propose a procedure that constructs a concave underestimator of f that is tight at a given point. These underestimators can be used to generate intersection cuts. A peculiarity of these underestimators is that they do not rely on a bounded domain. We propose a strengthening procedure for the intersection cuts that exploits the bounds of the domain. Finally, we propose an extension of monoidal strengthening to take advantage of the integrality of the non-basic variables.
This work has been supported by the Research Campus MODAL Mathematical Optimization and Data Analysis Laboratories funded by the Federal Ministry of Education and Research (BMBF Grant 05M14ZAM). The author thank the Schloss Dagstuhl – Leibniz Center for Informatics for hosting the Seminar 18081 “Designing and Implementing Algorithms for Mixed-Integer Nonlinear Optimization” for providing the environment to develop the ideas in this paper.
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- 1.
An \(\epsilon \)-subgradient of a convex function f at \(y \in \mathrm{dom\,}f\) is v such that \(f(x) \ge f(y) - \epsilon + v^\mathsf {T}(x - y)\) for all \(x \in \mathrm{dom\,}f\).
- 2.
He actually leaves it as an exercise for the reader.
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Acknowledgments
The author would like to thank Stefan Vigerske, Franziska Schlösser, Sven Wiese, Ambros Gleixner, Dan Steffy and Juan Pablo Vielma for helpful discussions, and Leon Eifler, Daniel Rehfeldt for comments that improved the manuscript. He would also like to thank three anonymous reviewers for valuable comments.
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A Proofs
A Proofs
1.1 A.1 Proof of Theorem 1
Proof
Clearly, \(h(\bar{x}) = f(g(\bar{x}))\).
To establish \(h(x) \le f(g(x))\), notice that
Since \(z = g(x)\) is a feasible solution and \(f_{ave}\) is an underestimator of f, we obtain that \(h(x) \le f(g(x))\).
Now, let us prove that h is concave. To this end, we again use the representation (10). To simplify notation, we write \(g_1, g_2\) for \(g_{ave}, g^{vex}\), respectively.
We prove concavity by definition, that is,
Let
By the concavity of \(g_1\) and convexity of \(g_2\) we have \(I \subseteq J\). Therefore,
Since \(f_{ave}\) is concave, the minimum is achieved at the boundary,
Furthermore, \(f_{ave}(\lambda g_i(x_1) + (1 - \lambda ) g_i(x_2)) \ge \lambda f_{ave}(g_i(x_1)) + (1 - \lambda ) f_{ave}(g_i(x_2))\) which implies that
as we wanted to show.
1.2 A.2 Proof of Proposition 1
Proof
The function \(\hat{h}\) is concave since it is the minimum of linear functions. This establishes the convexity of C.
To show that \(C \supseteq \{ x : h_{ave}(x) \ge 0 \}\), notice that \(h_{ave}(x) = \min _{z} h_{ave}(z) + \nabla h_{ave}(z)^\mathsf {T}(x - z)\). The inclusion follows from observing that the objective function in the definition of \(\hat{h}(x)\) is the same as above, but over a smaller domain.
To show that it is S-free, we will show that for every \(x \in [l,u]\) such that \(h(x) \le 0\), \(\hat{h}(x) \le 0\).
Let \(x_0 \in [l,u]\) such that \(h(x_0) \le 0\). Since \(h_{ave}\) is a concave underestimator at \(\bar{x}\), \(h_{ave}(\bar{x}) > 0\) and \(h_{ave}(x_0) \le 0\). If \(h_{ave}(x_0) = 0\), then, by definition, \(\hat{h}(x_0) \le h_{ave}(x_0) = 0\) and we are done. We assume, therefore, that \(h_{ave}(x_0) < 0\).
Consider \(g(\lambda ) = h_{ave}(\bar{x}+ \lambda (x_0 - \bar{x}))\) and let \(\lambda _1 \in (0,1)\) be such that \(g(\lambda _1) = 0\). The existence of \(\lambda _1\) is justified by the continuity of g, \(g(0) > 0\) and \(g(1) < 0\). Equivalently, \(x_1 = \bar{x}+ \lambda _1(x_0 - \bar{x})\) is the intersection point between the segment joining \(x_0\) with \(\bar{x}\) and \(\{x : h_{ave}(x) = 0\}\). The linearization of g at \(\lambda _1\) evaluated at \(\lambda = 1\) is negative, because g is concave, and equals \(h_{ave}(x_1) + \nabla h_{ave}(x_1)^T (x_0 - x_1)\). Finally, given that \(x_1 \in [l,u]\) and \(h_{ave}(x_1) = 0\), \(x_1\) is feasible for (3) and we conclude that \(\hat{h}(x_0) < 0\).
1.3 A.3 Proof of Lemma 1
Proof
If \(z \equiv 0\), then (6) reduces to (4).
Otherwise, let \(y_0 \in Y\) such that \(z(y_0) > 0\), that is, \(z(y_0) \ge 1\). By Remark 3, for every \(y \in Y\), it holds \(1 - \beta (y) > 0\), and so
Therefore, \(\beta (y_0) \ge 1 - z(y_0) (1 - \beta (y_0))\). Since every \(x \ge 0\) satisfying (4) satisfies \(\frac{\nabla h(y_0)^\mathsf {T}x}{\nabla h(y_0)^\mathsf {T}y_0} \ge \beta (y_0)\), we conclude that \(\frac{\nabla h(y_0)^\mathsf {T}x}{\nabla h(y_0)^\mathsf {T}y_0} + z(y_0) (1 - \beta (y_0)) \ge 1\) holds.
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Serrano, F. (2019). Intersection Cuts for Factorable MINLP. In: Lodi, A., Nagarajan, V. (eds) Integer Programming and Combinatorial Optimization. IPCO 2019. Lecture Notes in Computer Science(), vol 11480. Springer, Cham. https://doi.org/10.1007/978-3-030-17953-3_29
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