Strengthened SDP Relaxation for an Extended Trust Region Subproblem with an Application to Optimal Power Flow

TL;DR

This paper introduces enhanced SDP relaxations for an extended trust region subproblem with applications to optimal power flow, providing new valid cuts that improve solution quality and computational efficiency.

Contribution

It develops a class of polynomial-time separable valid cuts for the extended trust region subproblem, connecting and generalizing existing OPF cuts, and introduces new SOC cuts for nonconvex quadratic programs.

Findings

Cuts significantly close the SDP relaxation gap in low-dimensional instances.

New SOC cuts are derived for nonconvex quadratic programs over polyhedral-conic sets.

Computational results show improved relaxation bounds on random instances.

Abstract

We study an extended trust region subproblem minimizing a nonconvex function over the hollow ball $r \le \|x\| \le R$ intersected with a full-dimensional second order cone (SOC) constraint of the form $\|x - c\| \le b^T x - a$. In particular, we present a class of valid cuts that improve existing semidefinite programming (SDP) relaxations and are separable in polynomial time. We connect our cuts to the literature on the optimal power flow (OPF) problem by demonstrating that previously derived cuts capturing a convex hull important for OPF are actually just special cases of our cuts. In addition, we apply our methodology to derive a new class of closed-form, locally valid, SOC cuts for nonconvex quadratic programs over the mixed polyhedral-conic set $\{x \ge 0 : \| x \| \le 1 \}$. Finally, we show computationally on randomly generated instances that our cuts are effective in further closing the gap of the strongest SDP relaxations in the literature, especially in low dimensions.