A Quadratically Convergent Sequential Programming Method for Second-Order Cone Programs Capable of Warm Starts

TL;DR

This paper introduces a new sequential quadratic programming method for second-order cone programs that leverages warm starts and polyhedral approximations to achieve quadratic convergence and improved accuracy over interior point methods.

Contribution

The paper presents a novel SQP-based approach for second-order cone programs capable of warm starts and quadratic convergence, with theoretical guarantees and practical advantages.

Findings

Achieves local quadratic convergence.

Can utilize warm starts effectively.

Outperforms interior point solvers in accuracy.

Abstract

We propose a new method for linear second-order cone programs. It is based on the sequential quadratic programming framework for nonlinear programming. In contrast to interior point methods, it can capitalize on the warm-start capabilities of active-set quadratic programming subproblem solvers and achieve a local quadratic rate of convergence. In order to overcome the non-differentiability or singularity observed in nonlinear formulations of the conic constraints, the subproblems approximate the cones with polyhedral outer approximations that are refined throughout the iterations. For nondegenerate instances, the algorithm implicitly identifies the set of cones for which the optimal solution lies at the extreme points. As a consequence, the final steps are identical to regular sequential quadratic programming steps for a differentiable nonlinear optimization problem, yielding local quadratic convergence. We prove the global and local convergence guarantees of the method and present numerical experiments that confirm that the method can take advantage of good starting points and can achieve higher accuracy compared to a state-of-the-art interior point solver.