Beyond binarity: Semidefinite programming for ternary quadratic problems

Abstract: We study the ternary quadratic problem (TQP), a quadratic optimization problem with linear constraints where the variables take values in ${0, \pm 1}$. While semidefinite programming (SDP) techniques are well established for ${0,1}$- and ${\pm 1}$-valued quadratic problems, no dedicated integer semidefinite programming framework exists for the ternary case. In this paper, we introduce a ternary SDP formulation for the TQP that forms the basis of an exact solution approach. We derive new theoretical insights in rank-one ternary positive semidefinite matrices, which lead to a basic SDP relaxation that is further strengthened by valid triangle, RLT, split and $k$-gonal inequalities. These are embedded in a tailored branch-and-bound algorithm that iteratively solves strengthened SDPs, separates violated inequalities, applies a ternary branching strategy and computes high-quality feasible solutions. We test our algorithm on TQP variations motivated by practice, including unconstrained, linearly constrained and quadratic ratio problems. Computational results on these instances demonstrate the effectiveness of the proposed algorithm.