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≤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.