A second-order sequential optimality condition for nonlinear second-order cone programming problems

TL;DR

This paper introduces explicit second-order sequential optimality conditions for nonlinear second-order cone programming, proving their satisfaction at local optima without constraint qualifications and demonstrating convergence of related algorithms.

Contribution

It defines explicit AKKT2 and CAKKT2 conditions for SOCPs and proves their validity at local optima without constraint qualifications.

Findings

AKKT2/CAKKT2 conditions are satisfied at local optima.

Algorithms based on augmented Lagrangian and SQP converge globally to these conditions.

The conditions do not require constraint qualifications.

Abstract

In the last two decades, the sequential optimality conditions, which do not require constraint qualifications and allow improvement on the convergence assumptions of algorithms, had been considered in the literature. It includes the work by Andreani et al. (2017), with a sequential optimality condition for nonlinear programming, that uses the second-order information of the problem. More recently, Fukuda et al. (2023) analyzed the conditions that use second-order information, in particular for nonlinear second-order cone programming problems (SOCP). However, such optimality conditions were not defined explicitly. In this paper, we propose an explicit definition of approximate-Karush-Kuhn-Tucker 2 (AKKT2) and complementary-AKKT2 (CAKKT2) conditions for SOCPs. We prove that the proposed AKKT2/CAKKT2 conditions are satisfied at local optimal points of the SOCP without any constraint qualification. We also present two algorithms that are based on augmented Lagrangian and sequential quadratic programming methods and show their global convergence to points satisfying the proposed conditions.