Quadratic optimization (QO) has been studied extensively in the literature due to its applicability in many practical problems. While practical, it is known that QO problems are generally NP-hard. So, researchers developed many approximation methods to find good solutions. In this paper, we analyze QO problems using robust optimization techniques. To this end, we first … Read more
Standard quadratic optimization problems (StQPs) provide a versatile modelling tool in various applications. In this paper, we consider StQPs with a hard sparsity constraint, referred to as sparse StQPs. We focus on various tractable convex relaxations of sparse StQPs arising from a mixed-binary quadratic formulation, namely, the linear optimization relaxation given by the reformulation-linearization technique, … Read more
In this paper, we propose a new global optimization approach for solving nonconvex optimization problems in which the nonconvex components are sums of products of convex functions. A broad class of nonconvex problems can be written in this way, such as concave minimization problems, difference of convex problems, and fractional optimization problems. Our approach exploits … Read more
We study linear programming relaxations of nonconvex quadratic programs given by the reformulation-linearization technique (RLT), referred to as RLT relaxations. We investigate the relations between the polyhedral properties of the feasible regions of a quadratic program and its RLT relaxation. We establish various connections between recession directions, boundedness, and vertices of the two feasible regions. … Read more
Quadratic programs with box constraints involve minimizing a possibly nonconvex quadratic function subject to lower and upper bounds on each variable. This is a well-known NP-hard problem that frequently arises in various applications. We focus on two convex relaxations, namely the RLT (Reformulation-Linearization Technique) relaxation and the SDP-RLT relaxation obtained by adding semidefinite constraints to … Read more
We introduce a novel Reformulation-Perspectification Technique (RPT) to obtain convex approximations of nonconvex continuous optimization problems. RPT consists of two steps, those are, a reformulation step and a perspectification step. The reformulation step generates redundant nonconvex constraints from pairwise multiplication of the existing constraints. The perspectification step then convexifies the nonconvex components by using perspective … Read more
It is well-known that the convex and concave envelope of a multilinear polynomial over a box are polyhedral functions. Exponential-sized extended and projected formulations for these envelopes are also known. We consider the convexification question for multilinear polynomials that are symmetric with respect to permutations of variables. Such a permutation-invariant structure naturally implies a quadratic-sized … Read more
We study non-convex optimization problems over simplices. We show that for a large class of objective functions, the convex approximation obtained from the Reformulation-Linearization Technique (RLT) admits optimal solutions that exhibit a sparsity pattern. This characteristic of the optimal solutions allows us to conclude that (i) a linear matrix inequality constraint, which is often added … Read more
Adaptive robust optimization is a modelling paradigm for multistage optimization under uncertainty where one seeks decisions that minimize the worst-case cost with respect to all possible scenarios in a prescribed uncertainty set. However, optimal policies for adaptive robust optimization problems are difficult to compute. Therefore, one often restricts to the class of affine policies which … Read more
In this paper, we focus on a subclass of quadratic optimization problems, that is, disjoint bilinear optimization problems. We first show that disjoint bilinear optimization problems can be cast as two-stage robust linear optimization problems with fixed-recourse and right-hand-side uncertainty, which enables us to apply robust optimization techniques to solve the resulting problems. To this … Read more