A necessary and sufficient condition of convexity for SOC reformulation of trust-region subproblem with two intersecting cuts

Abstract

We consider the extended trust-region subproblem with two linear inequalities. In the “nonintersecting” case of this problem, Burer et al. have proved that its semi-definite programming relaxation with second-order-cone reformulation (SDPR-SOCR) is a tight relaxation. In the more complicated “intersecting” case, which is discussed in this paper, so far there is no result except for a counterexample for the SDPR-SOCR. We present a necessary and sufficient condition for the SDPR-SOCR to be a tight relaxation in both the “nonintersecting” and “intersecting” cases. As an application of this condition, it is verified easily that the “nonintersecting” SDPR-SOCR is a tight relaxation indeed. Furthermore, as another application of the condition, we prove that there exist at least three regions among the four regions in the trust-region ball divided by the two intersecting linear cuts, on which the SDPR-SOCR must be a tight relaxation. Finally, the results of numerical experiments show that the SDPR-SOCR can work efficiently in decreasing or even eliminating the duality gap of the nonconvex extended trust-region subproblem with two intersecting linear inequalities indeed.