arXiv subjects

Samuel Burer

Publications and source records attributed to Samuel Burer.

12 recordsLinked to original sources

arXiv2025

Optimization over Trained Neural Networks: Going Large with Gradient-Based Algorithms

When optimizing a nonlinear objective, one can employ a neural network as a surrogate for the nonlinear function. However, the resulting optimization model can be time-consuming to solve globally with exact methods. As a result, local search that exploits the neural-network structure has been employed to find good solutions within a reasonable time limit. For such methods, a lower per-iteration cost is advantageous when solving larger models. The contribution of this paper is two-fold. First, we propose a gradient-based algorithm with lower per-iteration cost than existing methods. Second, we further adapt this algorithm to exploit the piecewise-linear structure of neural networks that use Rectified Linear Units (ReLUs). In line with prior research, our methods become competitive with -- and then dominant over -- other local search methods as the optimization models become larger.

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arXiv2025

On the Semidefinite Representability of Continuous Quadratic Submodular Minimization With Applications to Pricing and Moment Problems

We study continuous quadratic submodular minimization with bounds and provide a polynomially sized semidefinite relaxation, which is provably tight for dimension n \leq 3. Via an explicit counterexample for n = 4, we show that the relaxation is not tight in general, although it remains empirically tight on randomly generated instances. We apply the relaxation to multi-product pricing and two moment problems arising in distributionally robust optimization and the computation of covariance bounds. Accordingly, this research advances the ongoing study of continuous submodular minimization and opens new application areas therein.

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arXiv2024

An Extended Validity Domain for Constraint Learning

We consider embedding a predictive machine-learning model within a prescriptive optimization problem. In this setting, called constraint learning, we study the concept of a validity domain, i.e., a constraint added to the feasible set, which keeps the optimization close to the training data, thus helping to ensure that the computed optimal solution exhibits less prediction error. In particular, we propose a new validity domain which uses a standard convex-hull idea but in an extended space. We investigate its properties and compare it empirically with existing validity domains on a set of test problems for which the ground truth is known. Results show that our extended convex hull routinely outperforms existing validity domains, especially in terms of the function value error, that is, it exhibits closer agreement between the true function value and the predicted function value at the computed optimal solution. We also consider our approach within two stylized optimization models, which show that our method reduces feasibility error, as well as a real-world pricing case study.

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arXiv2023

A Slightly Lifted Convex Relaxation for Nonconvex Quadratic Programming with Ball Constraints

Globally optimizing a nonconvex quadratic over the intersection of $m$ balls in $\mathbb{R}^n$ is known to be polynomial-time solvable for fixed $m$. Moreover, when $m=1$, the standard semidefinite relaxation is exact. When $m=2$, it has been shown recently that an exact relaxation can be constructed using a disjunctive semidefinite formulation based essentially on two copies of the $m=1$ case. However, there is no known explicit, tractable, exact convex representation for $m \ge 3$. In this paper, we construct a new, polynomially sized semidefinite relaxation for all $m$, which does not employ a disjunctive approach. We show that our relaxation is exact for $m=2$. Then, for $m \ge 3$, we demonstrate empirically that it is fast and strong compared to existing relaxations. The key idea of the relaxation is a simple lifting of the original problem into dimension $n+1$. Extending this construction: (i) we show that nonconvex quadratic programming over $\|x\| \le \min \{ 1, g + h^T x \}$ has an exact semidefinite representation; and (ii) we construct a new relaxation for quadratic programming over the intersection of two ellipsoids, which globally solves all instances of a benchmark collection from the literature.

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arXiv2022

A Strengthened SDP Relaxation for Quadratic Optimization Over the Stiefel Manifold

We study semidefinite programming (SDP) relaxations for the NP-hard problem of globally optimizing a quadratic function over the Stiefel manifold. We introduce a strengthened relaxation based on two recent ideas in the literature: (i) a tailored SDP for objectives with a block-diagonal Hessian; (ii) and the use of the Kronecker matrix product to construct SDP relaxations. Using synthetic instances on four problem classes, we show that, in general, our relaxation significantly strengthens existing relaxations, although at the expense of longer solution times.

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arXiv2021

Strengthened SDP Relaxation for an Extended Trust Region Subproblem with an Application to Optimal Power Flow

We study an extended trust region subproblem minimizing a nonconvex function over the hollow ball $r \le \|x\| \le R$ intersected with a full-dimensional second order cone (SOC) constraint of the form $\|x - c\| \le b^T x - a$. In particular, we present a class of valid cuts that improve existing semidefinite programming (SDP) relaxations and are separable in polynomial time. We connect our cuts to the literature on the optimal power flow (OPF) problem by demonstrating that previously derived cuts capturing a convex hull important for OPF are actually just special cases of our cuts. In addition, we apply our methodology to derive a new class of closed-form, locally valid, SOC cuts for nonconvex quadratic programs over the mixed polyhedral-conic set $\{x \ge 0 : \| x \| \le 1 \}$. Finally, we show computationally on randomly generated instances that our cuts are effective in further closing the gap of the strongest SDP relaxations in the literature, especially in low dimensions.

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arXiv2020

Convex Hull Representations for Bounded Products of Variables

It is well known that the convex hull of $\{(x,y,xy)\}$, where $(x,y)$ is constrained to lie in a box, is given by the Reformulation-Linearization Technique (RLT) constraints. Belotti {\em et al.\,}(2010) and Miller {\em et al.\,}(2011) showed that if there are additional upper and/or lower bounds on the product $z=xy$, then the convex hull can be represented by adding an infinite family of inequalities, requiring a separation algorithm to implement. Nguyen {\em et al.\,}(2018) derived convex hulls with bounds on $z$ for the more general case of $z=x^{b_1} y^{b_2}$, where $b_1\ge 1$, $b_2\ge 1$. We focus on the most important case where $b_1=b_2=1$ and show that the convex hull with either an upper bound or lower bound on the product is given by RLT constraints, the bound on $z$ and a single Second-Order Cone (SOC) constraint. With both upper and lower bounds on the product, the convex hull can be represented using no more than three SOC constraints, each applicable on a subset of $(x,y)$ values. In addition to the convex hull characterizations, volumes of the convex hulls with either an upper or lower bound on $z$ are calculated and compared to the relaxation that imposes only the RLT constraints. As an application of these volume results, we show how spatial branching can be applied to the product variable so as to minimize the sum of the volumes for the two resulting subproblems.

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arXiv2020

Quadratic Optimization with Switching Variables: The Convex Hull for $n = 2$

We consider quadratic optimization in variables $(x,y)$ where $0\le x\le y$, and $y\in\{0,1\}^n$. Such binary $y$ are commonly refered to as "indicator" or "switching" variables and occur commonly in applications. One approach to such problems is based on representing or approximating the convex hull of the set $\{ (x,xx^T, yy^T) : 0\le x\le y\in\{0,1\}^n\}$. A representation for the case $n=1$ is known and has been widely used. We give an exact representation for the case $n=2$ by starting with a disjunctive representation for the convex hull and then eliminating auxilliary variables and constraints that do not change the projection onto the original variables. An alternative derivation for this representation leads to an appealing conjecture for a simplified representation of the convex hull for $n=2$ when the product term $y_1y_2$ is ignored.

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arXiv2018

Exact Semidefinite Formulations for a Class of (Random and Non-Random) Nonconvex Quadratic Programs

We study a class of quadratically constrained quadratic programs (QCQPs), called {\em diagonal QCQPs\/}, which contain no off-diagonal terms $x_j x_k$ for $j \ne k$, and we provide a sufficient condition on the problem data guaranteeing that the basic Shor semidefinite relaxation is exact. Our condition complements and refines those already present in the literature and can be checked in polynomial time. We then extend our analysis from diagonal QCQPs to general QCQPs, i.e., ones with no particular structure. By reformulating a general QCQP into diagonal form, we establish new, polynomial-time-checkable sufficient conditions for the semidefinite relaxations of general QCQPs to be exact. Finally, these ideas are extended to show that a class of random general QCQPs has exact semidefinite relaxations with high probability as long as the number of constraints grows no faster than a fixed polynomial in the number of variables. To the best of our knowledge, this is the first result establishing the exactness of the semidefinite relaxation for random general QCQPs.

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arXiv2017

A Data-Driven Distributionally Robust Bound on the Expected Optimal Value of Uncertain Mixed 0-1 Linear Programming

This paper studies the expected optimal value of a mixed 0-1 programming problem with uncertain objective coefficients following a joint distribution. We assume that the true distribution is not known exactly, but a set of independent samples can be observed. Using the Wasserstein metric, we construct an ambiguity set centered at the empirical distribution from the observed samples and containing the true distribution with a high statistical guarantee. The problem of interest is to investigate the bound on the expected optimal value over the Wasserstein ambiguity set. Under standard assumptions, we reformulate the problem into a copositive program, which naturally leads to a tractable semidefinite-based approximation. We compare our approach with a moment-based approach from the literature on three applications. Numerical results illustrate the effectiveness of our approach.

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arXiv2017

A Copositive Approach for Two-Stage Adjustable Robust Optimization with Uncertain Right-Hand Sides

We study two-stage adjustable robust linear programming in which the right-hand sides are uncertain and belong to a convex, compact uncertainty set. This problem is NP-hard, and the affine policy is a popular, tractable approximation. We prove that under standard and simple conditions, the two-stage problem can be reformulated as a copositive optimization problem, which in turn leads to a class of tractable, semidefinite-based approximations that are at least as strong as the affine policy. We investigate several examples from the literature demonstrating that our tractable approximations significantly improve the affine policy. In particular, our approach solves exactly in polynomial time a class of instances of increasing size for which the affine policy admits an arbitrarily large gap.

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arXiv2015

Robust Sensitivity Analysis of the Optimal Value of Linear Programming

We propose a framework for sensitivity analysis of linear programs (LPs) in minimization form, allowing for simultaneous perturbations in the objective coefficients and right-hand sides, where the perturbations are modeled in a compact, convex uncertainty set. This framework unifies and extends multiple approaches for LP sensitivity analysis in the literature and has close ties to worst-case linear optimization and two-stage adaptive optimization. We define the minimum (best-case) and maximum (worst-case) LP optimal values, p- and p+, over the uncertainty set, and we discuss issues of finiteness, attainability, and computational complexity. While p- and p+ are difficult to compute in general, we prove that they equal the optimal values of two separate, but related, copositive programs. We then develop tight, tractable conic relaxations to provide lower and upper bounds on p- and p+, respectively. We also develop techniques to assess the quality of the bounds, and we validate our approach computationally on several examples from--and inspired by--the literature. We find that the bounds on p- and p+ are very strong in practice and, in particular, are at least as strong as known results for specific cases from the literature.

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