Quadratic factorization heuristics for copositive programming

Copositive optimization problems are particular conic programs: optimize linear forms over the copositive cone subject to linear constraints. Every quadratic program with linear constraints can be formulated as a copositive program, even if some of the variables are binary. So this is an NP-hard problem class. While most methods try to approximate the copositive cone from within, we propose a method which approximates this cone from outside. This is achieved by passing to the dual problem, where the feasible set is an affine subspace intersected with the cone of completely positive matrices, and this cone is approximated from within. We consider feasible descent directions in the completely positive cone, and regularized strictly convex subproblems. In essence, we replace the intractable completely positive cone with a nonnegative cone, at the cost of a series of nonconvex quadratic subproblems. Proper adjustment of the regularization parameter results in short steps for the nonconvex quadratic programs. This suggests to approximate their solution by standard linearization techniques. Preliminary numerical results on three different classes of test problems are quite promising.     

A note on set-semidefinite relaxations of nonconvex quadratic programs

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.     

Nonconvex quadratically constrained quadratic programming: best D.C. decompositions and their SDP representations

We propose in this paper a general D.C. decomposition scheme for constructing SDP relaxation formulations for a class of nonconvex quadratic programs with a nonconvex quadratic objective function and convex quadratic constraints. More specifically, we use rank-one matrices and constraint matrices to decompose the indefinite quadratic objective into a D.C. form and underestimate the concave terms in the D.C. decomposition formulation in order to get a convex relaxation of the original problem. We show that the best D.C. decomposition can be identified by solving an SDP problem. By suitably choosing the rank-one matrices and the linear underestimation, we are able to construct convex relaxations that dominate Shor’s SDP relaxation and the strengthened SDP relaxation. We then propose an extension of the D.C. decomposition to generate an SDP bound that is tighter than the SDP+RLT bound when additional box constraints are present. We demonstrate via computational results that the optimal D.C. decomposition schemes can generate both tight SDP bounds and feasible solutions with good approximation ratio for nonconvex quadratically constrained quadratic problems.     

An eigenvalue decomposition based branch-and-bound algorithm for nonconvex quadratic programming problems with convex quadratic constraints

In this paper, we propose a branch-and-bound algorithm for finding a global optimal solution for a nonconvex quadratic program with convex quadratic constraints (NQPCQC). We first reformulate NQPCQC by adding some nonconvex quadratic constraints induced by eigenvectors of negative eigenvalues associated with the nonconvex quadratic objective function to Shor’s semidefinite relaxation. Under the assumption of having a bounded feasible domain, these nonconvex quadratic constraints can be further relaxed into linear ones to form a special semidefinite programming relaxation. Then an efficient branch-and-bound algorithm branching along the eigendirections of negative eigenvalues is designed. The theoretic convergence property and the worst-case complexity of the proposed algorithm are proved. Numerical experiments are conducted on several types of quadratic programs to show the efficiency of the proposed method.     

Burer’s key assumption for semidefinite and doubly nonnegative relaxations

Burer has shown that completely positive relaxations of nonconvex quadratic programs with nonnegative and binary variables are exact when the binary variables satisfy a so-called key assumption. Here we show that introducing binary slack variables to obtain an equivalent problem that satisfies the key assumption will not improve the semidefinite relaxation. In contrast, such slack variables will improve the doubly nonnegative relaxation, but the same improvement can be obtained in a simpler fashion by adding certain linear inequality constraints.     

Optimizing a polyhedral-semidefinite relaxation of completely positive programs

It has recently been shown (Burer, Math Program 120:479–495, 2009) that a large class of NP-hard nonconvex quadratic programs (NQPs) can be modeled as so-called completely positive programs, i.e., the minimization of a linear function over the convex cone of completely positive matrices subject to linear constraints. Such convex programs are NP-hard in general. A basic tractable relaxation is gotten by approximating the completely positive matrices with doubly nonnegative matrices, i.e., matrices which are both nonnegative and positive semidefinite, resulting in a doubly nonnegative program (DNP). Optimizing a DNP, while polynomial, is expensive in practice for interior-point methods. In this paper, we propose a practically efficient decomposition technique, which approximately solves the DNPs while simultaneously producing lower bounds on the original NQP. We illustrate the effectiveness of our approach for solving the basic relaxation of box-constrained NQPs (BoxQPs) and the quadratic assignment problem. For one quadratic assignment instance, a best-known lower bound is obtained. We also incorporate the lower bounds within a branch-and-bound scheme for solving BoxQPs and the quadratic multiple knapsack problem. In particular, to the best of our knowledge, the resulting algorithm for globally solving BoxQPs is the most efficient to date.     

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.     

Nonconvex Piecewise-Quadratic Underestimation for Global Minimization

Motivated by the fact that important real-life problems, such as the protein docking problem, can be accurately modeled by minimizing a nonconvex piecewise-quadratic function, a nonconvex underestimator is constructed as the minimum of a finite number of strictly convex quadratic functions. The nonconvex underestimator is generated by minimizing a linear function on a reverse convex region and utilizes sample points from a given complex function to be minimized. The global solution of the piecewise-quadratic underestimator is known exactly and gives an approximation to the global minimum of the original function. Successive shrinking of the initial search region to which this procedure is applied leads to fairly accurate estimates, within 0.0060%, of the global minima of synthetic nonconvex functions for which the global minima are known. Furthermore, this process can approximate a nonconvex protein docking function global minimum within four-figure relative accuracy in six refinement steps. This is less than half the number of refinement steps required by previous models such as the convex kernel underestimator (Mangasarian et al., Computational Optimization and Applications, to appear) and produces higher accuracy here.     

Decomposition Methods for Solving Nonconvex Quadratic Programs via Branch and Bound*

The aim of this paper is to suggest branch and bound schemes, based on a relaxation of the objective function, to solve nonconvex quadratic programs over a compact polyhedral feasible region. The various schemes are based on different d.c. decomposition methods applied to the quadratic objective function. To improve the tightness of the relaxations, we also suggest solving the relaxed problems with an algorithm based on the so called “optimal level solutions” parametrical approach. *This paper has been partially supported by M.I.U.R. and C.N.R.     

A new global optimization method for univariate constrained twice-differentiable NLP problems

In this paper, a new global optimization method is proposed for an optimization problem with twice-differentiable objective and constraint functions of a single variable. The method employs a difference of convex underestimator and a convex cut function, where the former is a continuous piecewise concave quadratic function, and the latter is a convex quadratic function. The main objectives of this research are to determine a quadratic concave underestimator that does not need an iterative local optimizer to determine the lower bounding value of the objective function and to determine a convex cut function that effectively detects infeasible regions for nonconvex constraints. The proposed method is proven to have a finite ε-convergence to locate the global optimum point. The numerical experiments indicate that the proposed method competes with another covering method, the index branch-and-bound algorithm, which uses the Lipschitz constant.     

Exact Solutions of Some Nonconvex Quadratic Optimization Problems via SDP and SOCP Relaxations

We show that SDP (semidefinite programming) and SOCP (second order cone programming) relaxations provide exact optimal solutions for a class of nonconvex quadratic optimization problems. It is a generalization of the results by S. Zhang for a subclass of quadratic maximization problems that have nonnegative off-diagonal coefficient matrices of quadratic objective functions and diagonal coefficient matrices of quadratic constraint functions. A new SOCP relaxation is proposed for the class of nonconvex quadratic optimization problems by extracting valid quadratic inequalities for positive semidefinite cones. Its effectiveness to obtain optimal values is shown to be the same as the SDP relaxation theoretically. Numerical results are presented to demonstrate that the SOCP relaxation is much more efficient than the SDP relaxation.     

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.     

Convex Underestimation of Twice Continuously Differentiable Functions by Piecewise Quadratic Perturbation: Spline αBB Underestimators

This paper describes the construction of convex underestimators for twice continuously differentiable functions over box domains through piecewise quadratic perturbation functions. A refinement of the classical α BB convex underestimator, the underestimators derived through this approach may be significantly tighter than the classical αBB underestimator. The convex underestimator is the difference of the nonconvex function f and a smooth, piecewise quadratic, perturbation function, q. The convexity of the underestimator is guaranteed through an analysis of the eigenvalues of the Hessian of f over all subdomains of a partition of the original box domain. Smoothness properties of the piecewise quadratic perturbation function are derived in a manner analogous to that of spline construction.     

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.     

Globally solving nonconvex quadratic programming problems with box constraints via integer programming methods

We present effective linear programming based computational techniques for solving nonconvex quadratic programs with box constraints (BoxQP). We first observe that known cutting planes obtained from the Boolean Quadric Polytope (BQP) are computationally effective at reducing the optimality gap of BoxQP. We next show that the Chvátal–Gomory closure of the BQP is given by the odd-cycle inequalities even when the underlying graph is not complete. By using these cutting planes in a spatial branch-and-cut framework, together with a common integrality-based preprocessing technique and a particular convex quadratic relaxation, we develop a solver that can effectively solve a well-known family of test instances. Our linear programming based solver is competitive with SDP-based state of the art solvers on small instances and sparse instances. Most of our computational techniques have been implemented in the recent version of CPLEX and have led to significant performance improvements on nonconvex quadratic programs with linear constraints.     

A new variable reduction technique for convex integer quadratic programs

Convex integer quadratic programming involves minimization of a convex quadratic objective function with affine constraints and is a well-known NP-hard problem with a wide range of applications. We proposed a new variable reduction technique for convex integer quadratic programs (IQP). Based on the optimal values to the continuous relaxation of IQP and a feasible solution to IQP, the proposed technique can be applied to fix some decision variables of an IQP simultaneously at zero without sacrificing optimality. Using this technique, computational effort needed to solve IQP can be greatly reduced. Since a general convex bounded IQP (BIQP) can be transformed to a convex IQP, the proposed technique is also applicable for the convex BIQP. We report a computational study to demonstrate the efficacy of the proposed technique in solving quadratic knapsack problems.     

A New Class of Improved Convex Underestimators for Twice Continuously Differentiable Constrained NLPs

We present a new class of convex underestimators for arbitrarily nonconvex and twice continuously differentiable functions. The underestimators are derived by augmenting the original nonconvex function by a nonlinear relaxation function. The relaxation function is a separable convex function, that involves the sum of univariate parametric exponential functions. An efficient procedure that finds the appropriate values for those parameters is developed. This procedure uses interval arithmetic extensively in order to verify whether the new underestimator is convex. For arbitrarily nonconvex functions it is shown that these convex underestimators are tighter than those generated by the BB method. Computational studies complemented with geometrical interpretations demonstrate the potential benefits of the proposed improved convex underestimators.     

Globally solving nonconvex quadratic programming problems via completely positive programming

Nonconvex quadratic programming (QP) is an NP-hard problem that optimizes a general quadratic function over linear constraints. This paper introduces a new global optimization algorithm for this problem, which combines two ideas from the literature—finite branching based on the first-order KKT conditions and polyhedral-semidefinite relaxations of completely positive (or copositive) programs. Through a series of computational experiments comparing the new algorithm with existing codes on a diverse set of test instances, we demonstrate that the new algorithm is an attractive method for globally solving nonconvex QP.     

稠密k-子图问题的双非负松弛

稠密k-子图问题是组合优化里面一类经典的优化问题,其在通常情况下是非凸且NP-难的。本文给出了求解该问题的一个新凸松弛方法-双非负松弛方法,并建立了问题的相应双非负松弛模型,而且证明了其在一定的条件下等价于一个新的半定松弛模型。最后,我们使用一些随机例子对这些模型进行了数值测试,测试的结果表明双非负松弛的计算效果要优于等价的半定松弛。     

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.