On standard quadratic programs with exact and inexact doubly nonnegative relaxations (original) (raw)

An alternative perspective on copositive and convex relaxations of nonconvex quadratic programs

Journal of Global Optimization, 2021

We study convex relaxations of nonconvex quadratic programs. We identify a family of so-called feasibility preserving convex relaxations, which includes the well-known copositive and doubly nonnegative relaxations, with the property that the convex relaxation is feasible if and only if the nonconvex quadratic program is feasible. We observe that each convex relaxation in this family implicitly induces a convex underestimator of the objective function on the feasible region of the quadratic program. This alternative perspective on convex relaxations enables us to establish several useful properties of the corresponding convex underestimators. In particular, if the recession cone of the feasible region of the quadratic program does not contain any directions of negative curvature, we show that the convex underestimator arising from the copositive relaxation is precisely the convex envelope of the objective function of the quadratic program, strengthening Burer’s well-known result on t...

A note on set-semidefinite relaxations of nonconvex quadratic programs

Journal of Global Optimization, 2012

We consider semidefinite, copositive, and more general, set-semidefinite programming relaxations of general nonconvex quadratic problems. For the semidefinite case a comparison between the feasible set of the original program and the feasible set of the relaxation has been given by Kojima and Tunçel (SIAM J Optim 10(3):750-778, 2000). In this paper the comparison is presented for set-positive relaxations which contain copositive relaxations as a special case.

A Quadratically Constrained Quadratic Optimization Model for Completely Positive Cone Programming

SIAM Journal on Optimization, 2013

We propose a class of quadratic optimization problems whose exact optimal objective values can be computed by their completely positive cone programming relaxations. The objective function can be any quadratic form. The constraints of each problem are described in terms of quadratic forms with no linear terms, and all constraints are homogeneous equalities, except one inhomogeneous equality where a quadratic form is set to be a positive constant. For the equality constraints, "a hierarchy of copositvity" condition is assumed. This model is a generalization of the standard quadratic optimization problem of minimizing a quadratic form over the standard simplex, and covers many of the existing quadratic optimization problems studied for exact copositive cone and completely positive cone programming relaxations. In particular, it generalizes the recent results on quadratic optimization problems by Burer and the set-semidefinite representation by Eichfelder and Povh.

Semidefinite Programming Relaxation for Nonconvex Quadratic Programs

Journal of Global Optimization, 1997

This paper applies the SDP (semidefinite programming)relaxation originally developed for a 0-1 integer program to ageneral nonconvex QP (quadratic program) having a linear objective functionand quadratic inequality constraints, and presents some fundamental characterizations of the SDP relaxation including its equivalence to arelaxation using convex-quadratic valid inequalities for the feasible regionof the QP.

Convergent LMI relaxations for nonconvex quadratic programs

Proceedings of the 39th IEEE Conference on Decision and Control (Cat. No.00CH37187)

We consider the general nonconvex quadratic programming problem and provide a series of convex positive semidefinite programs (or LMI relaxations) whose sequence of optimal values is monotone and converges to the optimal value of the original problem. It improves and includes as a special case the well-known Shor's LMI formulation. Often, the optimal value is obtained at some particular early relaxation as shown on some nontrivial test problems from Floudas and Pardalos [9].

A Geometrical Analysis of a Class of Nonconvex Conic Programs for Convex Conic Reformulations of Quadratic and Polynomial Optimization Problems

arXiv (Cornell University), 2019

We present a geometrical analysis on the completely positive programming reformulation of quadratic optimization problems and its extension to polynomial optimization problems with a class of geometrically defined nonconvex conic programs and their covexification. The class of nonconvex conic programs is described with a linear objective function in a linear space V, and the constraint set is represented geometrically as the intersection of a nonconvex cone K ⊂ V, a face J of the convex hull of K and a parallel translation L of a supporting hyperplane of the nonconvex cone K. We show that under a moderate assumption, the original nonconvex conic program can equivalently be reformulated as a convex conic program by replacing the constraint set with the intersection of J and the hyperplane L. The replacement procedure is applied to derive the completely positive programming reformulation of quadratic optimization problems and its extension to polynomial optimization problems.

Doubly nonnegative relaxations are equivalent to completely positive reformulations of quadratic optimization problems with block-clique graph structures

Cornell University - arXiv, 2019

We study the equivalence among a nonconvex QOP, its CPP and DNN relaxations under the assumption that the aggregated and correlative sparsity of the data matrices of the CPP relaxation is represented by a block-clique graph G. By exploiting the correlative sparsity, we decompose the CPP relaxation problem into a cliquetree structured family of smaller subproblems. Each subproblem is associated with a node of a clique tree of G. The optimal value can be obtained by applying an algorithm that we propose for solving the subproblems recursively from leaf nodes to the root node of the clique-tree. We establish the equivalence between the QOP and its DNN relaxation from the equivalence between the reduced family of subproblems and their DNN relaxations by applying the known results on: (i) CPP and DNN reformulation of a class of QOPs with linear equality, complementarity and binary constraints in 4 nonnegative variables. (ii) DNN reformulation of a class of quadratically constrained convex QOPs with any size. (iii) DNN reformulation of LPs with any size. As a result, we show that a QOP whose subproblems are the QOPs mentioned in (i), (ii) and (iii) is equivalent to its DNN relaxation, if the subproblems form a clique-tree structured family induced from a block-clique graph.

Polyhedral properties of RLT relaxations of nonconvex quadratic programs and their implications on exact relaxations

Mathematical programming, 2024

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. Using these properties, we present a complete description of the set of instances that admit an exact RLT relaxation. We then give a thorough discussion of how our results can be converted into simple algorithmic procedures to construct instances of quadratic programs with exact, inexact, or unbounded RLT relaxations.

Doubly nonnegative relaxations for quadratic and polynomial optimization problems with binary and box constraints

Mathematical Programming, 2020

We propose a doubly nonnegative (DNN) relaxation for polynomial optimization problems (POPs) with binary and box constraints. This work is an extension of the work by Kim, Kojima and Toh in 2016 from quadratic optimization problems (QOPs) to POPs. The dense and sparse DNN relaxations are reduced to a simple conic optimization problem (COP) to which an accelerated bisection and projection (BP) algorithm is applied. The COP involves a single equality constraint in a matrix variable which is restricted to the intersection of the positive semidefinite cone and a polyhedral cone representing the algebraic properties of binary and box constraints as well as the sparsity structure of the objective polynomial. Our DNN relaxation serves as a variant of the standard Lasserre moment-SOS hierarchy for binary and box constrained POPs as the size of the cones is gradually increased to compute tighter lower bounds for their optimal values. A significant part of the paper is devoted to deriving and analyzing a class of polyhedral cones which strengthen the moment-SOS relaxation and to efficient computation of the metric projections onto these cones in the accelerated BP algorithm. Using the basic DNN relaxation framework, we also show why tight lower bounds of binary and box constrained QOPs were obtained numerically by their Lagrangian-DNN relaxation in the work by the authors in 2016. Numerical results on large scale POPs are provided to demonstrate the efficiency of the proposed method for attaining tight bounds.