Solving Two-Trust-Region Subproblems Using Semidefinite Optimization with Eigenvector Branching

Abstract

Semidefinite programming (SDP) problems typically utilize a constraint of the form X⪰xxT to obtain a convex relaxation of the condition X=xxT, where x∈Rn. In this paper, we consider a new hyperplane branching method for SDP based on using an eigenvector of X-xxT. This branching technique is related to previous work of Saxeena et al. (Math Prog Ser B 124:383–411, 2010, https://doi.org/10.1007/s10107-010-0371-9) who used such an eigenvector to derive a disjunctive cut. We obtain excellent computational results applying the new branching technique to difficult instances of the two-trust-region subproblem.