ABSTRACT Quadratically constrained quadratic programming (QCQP) problems appear in a wide range of engineering fields, including computer science, communication engineering, and finance. A key difficulty in solving these problems lies in efficiently finding global solutions, especially for large-scale and nonconvex instances. To address this, Wen & Yin (2013) proposed a method that reformulates semidefinite programming (SDP) relaxations of QCQPs into a nonlinear and nonconvex low-rank problem. This reformulated problem can be efficiently solved using a curvilinear search method combined with Barzilai-Borwein (BB) steps, known as the CSBB algorithm. In this study, we compare two approaches for solving QCQPs: the conventional convex SDP relaxation and the nonconvex low-rank reformulation introduced by Wen and Yin. We propose a set of general QCQP models that are compatible with the low-rank framework and conduct a series of numerical experiments to evaluate their performance. This study evaluates performance using several classes of NP-hard problems, including Binary Integer Quadratic (BIQ) problems, Max-Cut problems, Boolean Least Squares (BLS) problems, and 0-1 Quadratic Knapsack Problems (QKP). The results demonstrate that the low-rank approach offers competitive performance and shows strong potential for solving large-scale QCQPs more efficiently than traditional convex methods.
ABSTRACT Quadratically constrained quadratic programming (QCQP) problems appear in a wide range of engineering fields, including computer science, communication engineering, and finance. A key difficulty in solving these problems lies in efficiently finding global solutions, especially for large-scale and nonconvex instances. To address this, Wen & Yin (2013) proposed a method that reformulates semidefinite programming (SDP) relaxations of QCQPs into a nonlinear and nonconvex low-rank problem. This reformulated problem can be efficiently solved using a curvilinear search method combined with Barzilai-Borwein (BB) steps, known as the CSBB algorithm. In this study, we compare two approaches for solving QCQPs: the conventional convex SDP relaxation and the nonconvex low-rank reformulation introduced by Wen and Yin. We propose a set of general QCQP models that are compatible with the low-rank framework and conduct a series of numerical experiments to evaluate their performance. This study evaluates performance using several classes of NP-hard problems, including Binary Integer Quadratic (BIQ) problems, Max-Cut problems, Boolean Least Squares (BLS) problems, and 0-1 Quadratic Knapsack Problems (QKP). The results demonstrate that the low-rank approach offers competitive performance and shows strong potential for solving large-scale QCQPs more efficiently than traditional convex methods.
We consider low-rank semidefinite programming (LRSDP) relaxations of ±1 quadratic problems that can be formulated as the nonconvex nonlinear programming problem of minimizing a quadratic function subject to separable quadratic equality constraints. We prove the equivalence of the LRSDP problem with the unconstrained minimization of a new merit function and we define an effcient and globally convergent algorithm for finding critical points of the LRSDP problem. Finally, we test our code on an extended set of instances of the Max-Cut problem and we report comparisons with other existing codes.
At the intersection of combinatorial and nonlinear optimization, quadratic programming (QP) plays an important role in practice and optimization theory.With wide-spread real-world applications including, but not limited to, engineering design, game theory, and economics, quadratic programming has many valuable contributions and has attracted considerable interest over the past few decades.As a more general formulation, quadratically constrained quadratic programming (QCQP) problems arise naturally from a lot of applications such as facility location, production planning, economic equilibria and Euclidean distance geometry, etc.Most global optimization approaches for nonconvex QCQPs are based on constructing a convex or linear relaxation of the problem, and embedding it within an exhaustive search mechanism, where a lower bound is calculated as the value of the objective function of this relaxation (for a minimization problem) at each iteration of the search tree.The two principle elements affecting the overall effectiveness of such exhaustive search methods are the tightness of the underlying relaxation and the efficiency of calculating the lower bound.Besides, the performance of a branch-and-bound algorithm also relies on the procedure used to select a branching node and branching variable, which leads to good quality feasible solutions or equivalently an upper bound.vii LIST OF TABLES 5.5 Computational results of 0-1 QPs with n = 40 . . . . . . . . . . . .110 5.6 Computational results of 0-1 QPs with n = 50 . . . . . . . . . . . .
This thesis investigates and develops algorithms based on Lagrangian relaxation and normalized multiparametric disaggregation technique to solve nonconvex mixed-integer quadratically constrained quadratic programming.First, relaxations for quadratic programming and related problem classes are reviewed.Then, the normalized multiparametric disaggregation technique is improved to a reformulated version, in which the size of the generated subproblems are reduced in the number of binary variables.Furthermore, issues related to the use of the Lagrangian relaxation to solve nonconvex problems are addressed by replacing the dual subproblems with convex relaxations.This method is compared to commercial and open source off-the-shelf global solvers using randomly generated instances.The proposed method converged in 35 of 36 instances, while Baron, the benchmark solver that obtained the best results only converged in 4 of 36.Additionally, even for the one instance the methods did not converge, it achieved relative gaps below 1% in all instances, while Baron achieved relative gaps between 10% and 30% in most of them.
Quadratic programming (QP) is a well-studied fundamental NP-hard optimization problem which optimizes a quadratic objective over a set of linear constraints. In this paper, we reformulate QPs as a mixed-integer linear problem (MILP). This is done via the reformulation of QP as a linear complementary problem, and the use of binary variables and big-M constraints, to model the complementary constraints. To obtain such reformulation, we show how to impose bounds on the dual variables without eliminating all the (globally) optimal primal solutions; using some fundamental results on the solution of perturbed linear systems. Reformulating non-convex QPs as MILPs provides an advantageous way to obtain global solutions as it allows the use of current state-of-the-art MILP solvers. To illustrate this, we compare the performance of our solution approach, labeled quadprogIP, with the current benchmark global QP solver quadprogBB, as well as with BARON, one of the leading non-linear programming (NLP) solvers, and CPLEX's non-convex QP solver, on a large variety of QP test instances. In practice, quadprogIP is shown to typically outperform by orders of magnitude quadprogBB, BARON, and CPLEX on standard QPs. For general QPs, quadprogIP outperforms quadprogBB, outperforms BARON in most instances, while CPLEX performs the best on these instances. For box-constrained QPs, quadprogIP has a comparable performance to quadprogBB and BARON in small- to medium-scale instances, but is outperformed by these solvers on large-scale instances; while CPLEX performs the best on box-constrained QP instances. Also, unlike quadprogBB, the solution approach proposed here is able to solve QP instances whose dual feasible set is unbounded. The MATLAB code, called quadprogIP, and the instances used to perform these numerical experiments are publicly available at https://github.com/xiawei918/quadprogIP.
The class of mixed-integer quadratically constrained quadratic programs (QCQP) consists of minimizing a quadratic function under quadratic constraints where the variables could be integer or continuous. On a previous paper we introduced a method called MIQCR for solving QCQPs with the following restriction: all quadratic sub-functions of purely continuous variables are already convex. In this paper, we propose an extension of MIQCR which applies to any QCQP. Let (P) be a QCQP. Our approach to solve (P) is first to build an equivalent mixed-integer quadratic problem (P∗). This equivalent problem (P∗) has a quadratic convex objective function, linear constraints, and additional variables y that are meant to satisfy the additional quadratic constraints y=xxT, where x are the initial variables of problem (P). We then propose to solve (P∗) by a branch-and-bound algorithm based on the relaxation of the additional quadratic constraints and of the integrality constraints. This type of branching is known as spatial branch-and-bound. Computational experiences are carried out on a total of 325 instances. The results show that the solution time of most of the considered instances is improved by our method in comparison with the recent implementation of QuadProgBB, and with the solvers Cplex, Couenne, Scip, BARON and GloMIQO.
The technique of semidefinite programming (SDP) relaxation can be used to obtain a nontrivial bound on the optimal value of a nonconvex quadratically constrained quadratic program (QCQP). We explore concave quadratic inequalities that hold for any vector in the integer lattice Zn, and show that adding these inequalities to a mixed-integer nonconvex QCQP can improve the SDP-based bound on the optimal value. This scheme is tested using several numerical problem instances of the max-cut problem and the integer least squares problem.