A New Sequential Optimality Condition of Cardinality-Constrained Optimization Problems and Application

TL;DR

This paper introduces a stronger sequential optimality condition for cardinality-constrained optimization, improving theoretical understanding and convergence guarantees of augmented Lagrangian algorithms without extra assumptions.

Contribution

It proposes a new sequential optimality condition that is stronger than existing ones and establishes convergence results for augmented Lagrangian methods under tailored constraint qualifications.

Findings

New sequential optimality condition is stronger than existing results.

Feasible accumulation points satisfy the new optimality condition.

Algorithm converges to Mordukhovich-type stationary points under certain qualifications.

Abstract

In this paper, we consider the cardinality-constrained optimization problems and propose a new sequential optimality condition for the continuous relaxation reformulation which is popular recently. It is stronger than the existing results and is still a first-order necessity condition for the cardinality constraint problems without any additional assumptions. Meanwhile, we provide a problem-tailored weaker constraint qualification, which can guarantee that new sequential conditions are Mordukhovich-type stationary points. On the other hand, we improve the theoretical results of the augmented Lagrangian algorithm. Under the same condition as the existing results, we prove that any feasible accumulation point of the iterative sequence generated by the algorithm satisfies the new sequence optimality condition. Furthermore, the algorithm can converge to the Mordukhovich-type (essentially strong) stationary point if the problem-tailored constraint qualification is satisfied.