A new SOCP relaxation of nonconvex quadratic programming problems with a few negative eigenvalues

Abstract

We present a new second order cone programming (SOCP) relaxation of nonconvex quadratic programs with a few negative eigenvalues (NQP-$r$-NE) by employing the difference of convex (DC) decomposition and simultaneous matrix diagonalization together. The auxiliary variables are bounded from above by one more convex quadratic constraint in the proposed SOCP relaxation than that in the classical SOCP relaxation provided by Kim and Kojima (2001). We prove that the proposed SOCP relaxation is strictly tighter than the classical SOCP relaxation under certain circumstances, especially when there exists one constraint matrix being positive definite in the primal problem. Three types of numerical experiments including the large-scale NQP-$r$-NE problem with the number of negative eigenvalues $r\le 20$, the optimal spectrum sharing problem in MIMO cognitive radio networks and the two-trust-region subproblem are provided to illustrate that the proposed SOCP relaxation could achieve a better lower bound than that of the classical SOCP relaxation. Moreover, when the number of variables is larger than 250, the proposed SOCP relaxation shows its great superiority both in the bound quality and computing efficiency compared to the classical SOCP relaxation.

Introduction

In this paper, we consider a nonconvex generalized quadratic programming problem with convex quadratic constraints in the following form:

$$minf\left(x\right)=x^{\mathrm{T}}Qx+q^{\mathrm{T}}x,$$$$\mathrm{s.t.}{x}^{\mathrm{T}}{Q}_{k}x+q_k^{\mathrm{T}}x\le c_k,k=1,\dots ,K,Ax\le a.$$

$Q$ is an $n\times n$ symmetric indefinite matrix with $r$ negative eigenvalues, $Q_k$ is a positive semidefinite matrix for $k=1,\dots ,K$, $A\in {\mathbb{R}}^{m\times n}$ and $a\in {\mathbb{R}}^m$. The feasible region $\mathcal{F}=\{x\in {\mathbb{R}}^n|x^{\mathrm{T}}{Q}_{k}x+q_k^{\mathrm{T}}x\le c_k,k=1,\dots ,K,Ax\le a\}$ of (1.1) is supposed to be bounded with nonempty relative interior points. Problem (1.1) covers many important engineering problems and combinatorial optimization problems [1], [2], [3]. In this paper, we focus on the (1.1) with a few negative eigenvalues ($r\le 20$). In fact, several well-known problems, such as the concave quadratic programming problems [4], the low rank DC programs [5], [6], the optimal spectrum sharing problems in MIMO cognitive radio networks with multiple-input multiple-output(MIMO) links [7] and the two-trust-region subproblems [8], could be recast into the model of (1.1). (1.1) is NP-hard even with one negative eigenvalue [9]. Cambini et al. [10] stated some theoretical properties and characterized the existence of minimizers of DC programs with one negative eigenvalue and no quadratic constraints. They [5], [6] also proposed global algorithms for DC programs with only a few negative eigenvalues. As a special nonconvex quadratically constrained quadratic programming problem, many researches have paid more attention to the design of convex relaxations of (1.1). In the past few decades, semidefinite programming (SDP) relaxation has been regarded as an attractive technique for solving (1.1) [11], [12]. However, though the SDP relaxation could provide a good lower bound, the computational complexity is high, especially for large scale problems due to the fact that the SDP relaxation enlarges the dimension of the problem by transforming the original $n$-dimensional variable vector to an $(n+1)\times (n+1)$ variable matrix. Therefore, Luo et al. [13] used the classical SOCP relaxation to achieve a lower bound for (1.1) when designing a branch and bound algorithm and they showed that the SOCP relaxation is effective if the number of negative eigenvalues is few.

Note that the convex relaxation effect is critical when designing a global algorithm, thus, this paper aims to present a new SOCP relaxation of (1.1) via a simultaneous diagonalization tool in order to enhance the bound quality of the classical SOCP relaxation. Ben-Tal et al. [14] first introduced the simultaneous diagonalization tool to solve the quadratic programs with one or two parallel quadratic constraints when the objective and constraint matrices are simultaneously diagonalized. They proved that the simultaneous diagonalization based SOCP relaxation is tight since the auxiliary variables could be bounded from above by one convex constraint. Then this tool has been effectively extended to solve the convex quadratic programs with linear complementary constraints and the generalized trust-region problems [15], [16]. All these researches illustrated that the approach of applying the simultaneous diagonalization tool into the SOCP relaxation could improve the relaxation effect more often than not. In this paper, we first add a redundant constraint $x^{\mathrm{T}}\left({\sum }_{k=1}^K{Q}_k\right)x+{\left({\sum }_{k=1}^K{q}_k\right)}^{\mathrm{T}}x\le {\sum }_{k=1}^K{c}_k$ into (1.1). Then the objective matrix $Q$ are decomposed to the subtraction between two semidefinite matrices $Q=Q_{+}-Q_{-}$ according to the signs of its eigenvalues. Afterwards, we find a nonsingular matrix such that $Q_{-}$ and the redundant constraint matrix ${\sum }_{k=1}^K{Q}_k$ are simultaneously diagonalizable. Finally, we design a new SOCP relaxation of (1.1). It is worth mentioning that the auxiliary variables are bounded by one more convex quadratic constraint in the proposed SOCP relaxation than that in the classical SOCP relaxation, thus, the proposed SOCP relaxation offers a better lower bound. Moreover, it can be proved that the proposed SOCP relaxation is strictly tighter than the classical SOCP relaxation when ${\sum }_{k=1}^K{Q}_k$ is positive definite.

The rest of the paper is organized as follows. In Section 2, we design a simultaneous diagonalization based SOCP relaxation of (1.1) and make comparisons with the classical SOCP relaxation. In Section 3, we carry out numerical experiments to show the effectiveness of the proposed SOCP relaxation both in the relaxation effect and computing time. Conclusions are given in Section 4.

Some notations are adopted throughout this paper. $X⪰0$ represents a real symmetric matrix $X$ that is positive semidefinite. Given a vector $h\in {\mathbb{R}}^n$, $\text{diag}\left(h\right)$ indicates an $n\times n$ diagonal matrix with its diagonal elements equal to $h$.