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

Ellen H. Fukuda, Kosuke Okabe

TL;DR

The paper addresses nonlinear SOCP where constraint qualifications may fail, and introduces explicit second-order sequential optimality conditions $AKKT2$ (and $CAKKT2$) to provide robust necessary conditions for optimality. It proves that local minima satisfy $AKKT2$ without CQ, and connects $AKKT2$ to the weak second-order necessary condition ($WSONC$) under CQs such as Robinson CQ and WCR. Two practical algorithms are developed to generate $AKKT2$ sequences: an augmented Lagrangian method with a computable second-order surrogate and a stabilized SQP method with a VOMF multiplier-update scheme, both with convergence guarantees to $AKKT2/CAKKT2$ points. The work offers a concrete framework for second-order optimality in nonlinear conic programming and lays the groundwork for extending AKKT2 to broader conic settings and for refining 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.

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

TL;DR

The paper addresses nonlinear SOCP where constraint qualifications may fail, and introduces explicit second-order sequential optimality conditions (and ) to provide robust necessary conditions for optimality. It proves that local minima satisfy without CQ, and connects to the weak second-order necessary condition () under CQs such as Robinson CQ and WCR. Two practical algorithms are developed to generate sequences: an augmented Lagrangian method with a computable second-order surrogate and a stabilized SQP method with a VOMF multiplier-update scheme, both with convergence guarantees to points. The work offers a concrete framework for second-order optimality in nonlinear conic programming and lays the groundwork for extending AKKT2 to broader conic settings and for refining 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.
Paper Structure (11 sections, 14 theorems, 76 equations, 3 algorithms)

This paper contains 11 sections, 14 theorems, 76 equations, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Existing optimality conditions
  4. Second-order sequential optimality conditions
  5. Validity of the new optimality condition
  6. Strength of the new optimality condition
  7. Numerical example
  8. Algorithms that generate AKKT2 sequences
  9. The augmented Lagrangian method
  10. The sequential quadratic programming method
  11. Conclusion

Key Result

Theorem 2.1

HI84 Let $x^*$ be a local minimizer of a differentiable function $W\colon \mathbb{R}^n \to \mathbb{R}$ and $\nabla W$ be locally Lipschitz. Then, $\nabla W(x^*) = 0$ holds and for each $d \in \mathbb{R}^n$, there exists $M \in \partial^2 W(x^*)$ such that $d^{\top\space}Md \geq 0$.

Theorems & Definitions (31)

  • Theorem 2.1
  • Theorem 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Theorem 3.4
  • Definition 3.5
  • ...and 21 more