SOCP Reformulation for the Generalized Trust Region Subproblem via a Canonical Form of Two Symmetric Matrices

TL;DR

This paper introduces a novel SOCP reformulation for the generalized trust region subproblem by using a canonical form of symmetric matrices, enabling faster solutions and extending classical S-lemma results.

Contribution

It generalizes SOCP reformulation of GTRS beyond simultaneous diagonalizability, using block diagonalization for broader applicability.

Findings

GTRS with bounded optimal value are SOCP representable.

The new SOCP reformulation is computationally faster than SDP approaches.

Extensions to GTRS variants and simplified S-lemma versions are achieved.

Abstract

We investigate in this paper the generalized trust region subproblem (GTRS) of minimizing a general quadratic objective function subject to a general quadratic inequality constraint. By applying a simultaneous block diagonalization approach, we obtain a congruent canonical form for the symmetric matrices in both the objective and constraint functions. By exploiting the block separability of the canonical form, we show that all GTRSs with an optimal value bounded from below are second order cone programming (SOCP) representable. Our result generalizes the recent work of Ben-Tal and Hertog (Math. Program. 143(1-2):1-29, 2014), which establishes the SOCP representability of the GTRS under the assumption of the simultaneous diagonalizability of the two matrices in the objective and constraint functions. Compared with the state-of-the-art approach to reformulate the GTRS as a semi-definite programming problem, our SOCP reformulation delivers a much faster solution algorithm. We further extend our method to two variants of the GTRS in which the inequality constraint is replaced by either an equality constraint or an interval constraint. Our methods also enable us to obtain simplified versions of the classical S-lemma, the S-lemma with equality, and the S-lemma with interval bounds.