Exact SDP relaxations of quadratically constrained quadratic programs with forest structures

Journal of Global Optimization - We study the exactness of the semidefinite programming (SDP) relaxation of quadratically constrained quadratic programs (QCQPs). With the aggregate sparsity matrix...     

On convex relaxations for quadratically constrained quadratic programming

We consider convex relaxations for the problem of minimizing a (possibly nonconvex) quadratic objective subject to linear and (possibly nonconvex) quadratic constraints. Let $\mathcal{F }$ denote the feasible region for the linear constraints. We first show that replacing the quadratic objective and constraint functions with their convex lower envelopes on $\mathcal{F }$ is dominated by an alternative methodology based on convexifying the range of the quadratic form $\genfrac(){0.0pt}{}{1}{x}\genfrac(){0.0pt}{}{1}{x}^T$ for $x\in \mathcal{F }$ . We next show that the use of ?? $\alpha $ BB?? underestimators as computable estimates of convex lower envelopes is dominated by a relaxation of the convex hull of the quadratic form that imposes semidefiniteness and linear constraints on diagonal terms. Finally, we show that the use of a large class of D.C. (??difference of convex??) underestimators is dominated by a relaxation that combines semidefiniteness with RLT constraints.     

Controlled perturbations for quadratically constrained quadratic programs

Consider a minimization problem of a convex quadratic function of several variables over a set of inequality constraints of the same type of function. The duel program is a maximization problem with a concave objective function and a set of constrains that are essentially linear. However, the objective function is not differentiable over the constraint region. In this paper, we study a general theory of dual perturbations and derive a fundamental relationship between a perturbed dual program and the original problem. Based on this relationship, we establish a perturbation theory to display that a well-controlled perturbation on the dual program can overcome the nondifferentiability issue and generate an ε-optimal dual solution for an arbitrarily small number ε. A simple linear program is then constructed to make an easy conversion from the dual solution to a corresponding ε-optimal primal solution. Moreover, a numerical example is included to illustrate the potential of this controlled perturbation scheme.     

A relaxation method for nonconvex quadratically constrained quadratic programs

We present an algorithm for finding approximate global solutions to quadratically constrained quadratic programming problems. The method is based on outer approximation (linearization) and branch and bound with linear programming subproblems. When the feasible set is non-convex, the infinite process can be terminated with an approximate (possibly infeasible) optimal solution. We provide error bounds that can be used to ensure stopping within a prespecified feasibility tolerance. A numerical example illustrates the procedure. Computational experiments with an implementation of the procedure are reported on bilinearly constrained test problems with up to sixteen decision variables and eight constraints.This research was supported in part by National Science Foundation Grant DDM-91-14489.     

A simplicial branch-and-bound algorithm for solving quadratically constrained quadratic programs

We propose a branch-and-bound algorithm for solving nonconvex quadratically-constrained quadratic programs. The algorithm is novel in that branching is done by partitioning the feasible region into the Cartesian product of two-dimensional triangles and rectangles. Explicit formulae for the convex and concave envelopes of bilinear functions over triangles and rectangles are derived and shown to be second-order cone representable. The usefulness of these new relaxations is demonstrated both theoretically and computationally.     

Semidefinite relaxations for quadratically constrained quadratic programming: A review and comparisons

At the intersection of nonlinear and combinatorial optimization, quadratic programming has attracted significant interest over the past several decades. A variety of relaxations for quadratically constrained quadratic programming (QCQP) can be formulated as semidefinite programs (SDPs). The primary purpose of this paper is to present a systematic comparison of SDP relaxations for QCQP. Using theoretical analysis, it is shown that the recently developed doubly nonnegative relaxation is equivalent to the Shor relaxation, when the latter is enhanced with a partial first-order relaxation-linearization technique. These two relaxations are shown to theoretically dominate six other SDP relaxations. A computational comparison reveals that the two dominant relaxations require three orders of magnitude more computational time than the weaker relaxations, while providing relaxation gaps averaging 3% as opposed to gaps of up to 19% for weaker relaxations, on 700 randomly generated problems with up to 60 variables. An SDP relaxation derived from Lagrangian relaxation, after the addition of redundant nonlinear constraints to the primal, achieves gaps averaging 13% in a few CPU seconds.     

A distributed algorithm for high-dimension convex quadratically constrained quadratic programs

Computational Optimization and Applications - We propose a Jacobi-style distributed algorithm to solve convex, quadratically constrained quadratic programs (QCQPs), which arise from a broad range...     

Solution existence and stability of quadratically constrained convex quadratic programs

We propose verifiable necessary and sufficient conditions for the solution existence of a convex quadratic program whose constraint set is defined by finitely many convex linear-quadratic inequalities. A related stability analysis of the problem is also considered.     

A modified alternating direction method for convex quadratically constrained quadratic semidefinite programs

We propose a modified alternating direction method for solving convex quadratically constrained quadratic semidefinite optimization problems. The method is a first-order method, therefore requires much less computational effort per iteration than the second-order approaches such as the interior point methods or the smoothing Newton methods. In fact, only a single inexact metric projection onto the positive semidefinite cone is required at each iteration. We prove global convergence and provide numerical evidence to show the effectiveness of this method.     

Convex relaxations of non-convex mixed integer quadratically constrained programs: extended formulations

This paper addresses the problem of generating strong convex relaxations of Mixed Integer Quadratically Constrained Programming (MIQCP) problems. MIQCP problems are very difficult because they combine two kinds of non- convexities: integer variables and non-convex quadratic constraints. To produce strong relaxations of MIQCP problems, we use techniques from disjunctive programming and the lift-and-project methodology. In particular, we propose new methods for generating valid inequalities from the equation Y =  x x ^T . We use the non-convex constraint ${ Y - x x^T \preccurlyeq 0}$ to derive disjunctions of two types. The first ones are directly derived from the eigenvectors of the matrix Y ? x x ^T with positive eigenvalues, the second type of disjunctions are obtained by combining several eigenvectors in order to minimize the width of the disjunction. We also use the convex SDP constraint ${ Y - x x^T \succcurlyeq 0}$ to derive convex quadratic cuts, and we combine both approaches in a cutting plane algorithm. We present computational results to illustrate our findings.     

Convex relaxations of non-convex mixed integer quadratically constrained programs: projected formulations

A common way to produce a convex relaxation of a Mixed Integer Quadratically Constrained Program (MIQCP) is to lift the problem into a higher-dimensional space by introducing variables Y _ij to represent each of the products x _i x _j of variables appearing in a quadratic form. One advantage of such extended relaxations is that they can be efficiently strengthened by using the (convex) SDP constraint Y - x x^T \succeq 0{Y - x x^T \succeq 0} and disjunctive programming. On the other hand, the main drawback of such an extended formulation is its huge size, even for problems for which the number of x _i variables is moderate. In this paper, we study methods to build low-dimensional relaxations of MIQCP that capture the strength of the extended formulations. To do so, we use projection techniques pioneered in the context of the lift-and-project methodology. We show how the extended formulation can be algorithmically projected to the original space by solving linear programs. Furthermore, we extend the technique to project the SDP relaxation by solving SDPs. In the case of an MIQCP with a single quadratic constraint, we propose a subgradient-based heuristic to efficiently solve these SDPs. We also propose a new eigen-reformulation for MIQCP, and a cut generation technique to strengthen this reformulation using polarity. We present extensive computational results to illustrate the efficiency of the proposed techniques. Our computational results have two highlights. First, on the GLOBALLib instances, we are able to generate relaxations that are almost as strong as those proposed in our companion paper even though our computing times are about 100 times smaller, on average. Second, on box-QP instances, the strengthened relaxations generated by our code are almost as strong as the well-studied SDP+RLT relaxations and can be solved in less than 2 s, even for large instances with 100 variables; the SDP+RLT relaxations for the same set of instances can take up to a couple of hours to solve using a state-of-the-art SDP solver.     

Convex relaxations for nonconvex quadratically constrained quadratic programming: matrix cone decomposition and polyhedral approximation

We present a decomposition-approximation method for generating convex relaxations for nonconvex quadratically constrained quadratic programming (QCQP). We first develop a general conic program relaxation for QCQP based on a matrix decomposition scheme and polyhedral (piecewise linear) underestimation. By employing suitable matrix cones, we then show that the convex conic relaxation can be reduced to a semidefinite programming (SDP) problem. In particular, we investigate polyhedral underestimations for several classes of matrix cones, including the cones of rank-1 and rank-2 matrices, the cone generated by the coefficient matrices, the cone of positive semidefinite matrices and the cones induced by rank-2 semidefinite inequalities. We demonstrate that in general the new SDP relaxations can generate lower bounds at least as tight as the best known SDP relaxations for QCQP. Moreover, we give examples for which tighter lower bounds can be generated by the new SDP relaxations. We also report comparison results of different convex relaxation schemes for nonconvex QCQP with convex quadratic/linear constraints, nonconvex quadratic constraints and 0–1 constraints.     

Robust convex quadratically constrained programs

In this paper we study robust convex quadratically constrained programs, a subset of the class of robust convex programs introduced by Ben-Tal and Nemirovski [4]. In contrast to [4], where it is shown that such robust problems can be formulated as semidefinite programs, our focus in this paper is to identify uncertainty sets that allow this class of problems to be formulated as second-order cone programs (SOCP). We propose three classes of uncertainty sets for which the robust problem can be reformulated as an explicit SOCP and present examples where these classes of uncertainty sets are natural. Research partially supported by DOE grant GE-FG01-92ER-25126, NSF grants DMS-94-14438, CDA-97-26385, DMS-01-04282 and ONR grant N000140310514.Research partially supported by NSF grants CCR-00-09972, DMS-01-04282 and ONR grant N000140310514.     

New semidefinite programming relaxations for box constrained quadratic program

We establish in this paper optimal parametric Lagrangian dual models for box constrained quadratic program based on the generalized D.C. (difference between convex) optimization approach, which can be reformulated as semidefinite programming problems. As an application, we propose new valid linear constraints for rank-one relaxation.     

Hidden convexity in some nonconvex quadratically constrained quadratic programming

We consider the problem of minimizing an indefinite quadratic objective function subject to twosided indefinite quadratic constraints. Under a suitable simultaneous diagonalization assumption (which trivially holds for trust region type problems), we prove that the original problem is equivalent to a convex minimization problem with simple linear constraints. We then consider a special problem of minimizing a concave quadratic function subject to finitely many convex quadratic constraints, which is also shown to be equivalent to a minimax convex problem. In both cases we derive the explicit nonlinear transformations which allow for recovering the optimal solution of the nonconvex problems via their equivalent convex counterparts. Special cases and applications are also discussed. We outline interior-point polynomial-time algorithms for the solution of the equivalent convex programs. This author's work was partially supported by GIF, the German-Israeli Foundation for Scientific Research and Development and by the Binational Science Foundation. This author's work was partially supported by National Science Foundation Grants DMS-9201297 and DMS-9401871.     

Robust solutions of uncertain complex-valued quadratically constrained programs

In this paper, we discuss complex convex quadratically constrained optimization with uncertain data. Using S-Lemma, we show that the robust counterpart of complex convex quadratically constrained optimization with ellipsoidal or intersection-of-two-ellipsoids uncertainty set leads to a complex semidefinite program. By exploring the approximate S-Lemma, we give a complex semidefinite program which approximates the NP-hard robust counterpart of complex convex quadratic optimization with intersection-of-ellipsoids uncertainty set.     

A sequential quadratically constrained quadratic programming method for unconstrained minimax problems

In this paper, a sequential quadratically constrained quadratic programming (SQCQP) method for unconstrained minimax problems is presented. At each iteration the SQCQP method solves a subproblem that involves convex quadratic inequality constraints and a convex quadratic objective function. The global convergence of the method is obtained under much weaker conditions without any constraint qualification. Under reasonable assumptions, we prove the strong convergence, superlinearly and quadratic convergence rate.     

A branch and cut algorithm for nonconvex quadratically constrained quadratic programming

We present a branch and cut algorithm that yields in finite time, a globally ε-optimal solution (with respect to feasibility and optimality) of the nonconvex quadratically constrained quadratic programming problem. The idea is to estimate all quadratic terms by successive linearizations within a branching tree using Reformulation-Linearization Techniques (RLT). To do so, four classes of linearizations (cuts), depending on one to three parameters, are detailed. For each class, we show how to select the best member with respect to a precise criterion. The cuts introduced at any node of the tree are valid in the whole tree, and not only within the subtree rooted at that node. In order to enhance the computational speed, the structure created at any node of the tree is flexible enough to be used at other nodes. Computational results are reported that include standard test problems taken from the literature. Some of these problems are solved for the first time with a proof of global optimality. Received December 19, 1997 / Revised version received July 26, 1999?Published online November 9, 1999     

New reformulations for probabilistically constrained quadratic programs

The mixed integer quadratic programming (MIQP) reformulation by Zheng, Sun, Li, and Cui (2012) for probabilistically constrained quadratic programs (PCQP) recently published in EJOR significantly dominates the standard MIQP formulation ( and ) which has been widely adopted in the literature. Stimulated by the dimensionality problem which Zheng et al. (2012) acknowledge themselves for their reformulations, we study further the characteristics of PCQP and develop new MIQP reformulations for PCQP with fewer variables and constraints. The results from numerical tests demonstrate that our reformulations clearly outperform the state-of-the-art MIQP in Zheng et al. (2012).