A strong second-order sequential optimality condition for nonlinear programming problems
Huimin Li, Yuya Yamakawa, Ellen H. Fukuda, Nobuo Yamashita
TL;DR
Addresses nonlinear programming without constraint qualifications by proposing a strong second-order sequential necessary condition, $\\tilde{S}$-SAKKT2. The paper proves $\\tilde{S}$-SAKKT2 is stronger than both AKKT2 and $\\tilde{C}$-SAKKT2 and is a genuine necessary condition for local minima. It develops a quartic-penalty based penalty method and an augmented-Lagrangian method that generate sequences whose accumulation points satisfy $\\tilde{S}$-SAKKT2 under mild assumptions. These results broaden CQ-free optimality guarantees and suggest future work on SQP and CAKKT-based variants.
Abstract
Most numerical methods developed for solving nonlinear programming problems are designed to find points that satisfy certain optimality conditions. While the Karush-Kuhn-Tucker conditions are well-known, they become invalid when constraint qualifications (CQ) are not met. Recent advances in sequential optimality conditions address this limitation in both first- and second-order cases, providing genuine optimality guarantees at local optima, even when CQs do not hold. However, some second-order sequential optimality conditions still require some restrictive conditions on constraints in the recent literature. In this paper, we propose a new strong second-order sequential optimality condition without CQs. We also show that a penalty-type method and an augmented Lagrangian method generate points satisfying these new optimality conditions.