A SOCP relaxation based branch-and-bound method for generalized trust-region subproblem

Abstract

This paper proposes a second-order cone programming (SOCP) relaxation for the generalized trust-region problem by exploiting the property that any symmetric matrix and identity matrix can be simultaneously diagonalizable. We show that our proposed SOCP relaxation can provide a lower bound as tight as that of the standard semidefinite programming (SDP) relaxation. Moreover, we provide a sufficient condition under which the proposed SOCP relaxation is exact. Since the standard SDP relaxation suffers from a much heavier computing burden, the proposed SOCP relaxation has a much higher efficiency in solving process. Then we design a branch-and-bound algorithm based on this SOCP relaxation to obtain the global optimal solution for a general problem. Three types of numerical experiments are carried out to demonstrate the effectiveness and efficiency of our proposed SOCP relaxation.1. Introduction. In this paper, we consider a generalized trust-region problem in the following form:

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