Second Order Optimality Conditions and Improved Convergence Results for a Scholtes-type Regularization for a Continuous Reformulation of Cardinality Constrained Optimization Problems

TL;DR

This paper develops second order optimality conditions for a continuous reformulation of cardinality constrained optimization problems and uses them to establish convergence guarantees for a Scholtes-type relaxation method.

Contribution

It introduces second order necessary and sufficient conditions for optimality and extends local convergence theory for a relaxation method in cardinality constrained optimization.

Findings

Guarantees local uniqueness of M-stationary points under second order conditions.

Provides convergence guarantees for a Scholtes-type relaxation method.

Establishes existence and convergence of iterates under suitable assumptions.

Abstract

We consider nonlinear optimization problems with cardinality constraints. Based on a continuous reformulation we introduce second order necessary and sufficient optimality conditions. Under such a second order condition, we can guarantee local uniqueness of M-stationary points. Finally, we use this observation to provide extended local convergence theory for a Scholtes-type relaxation method for cardinality constrained optimization problems, which guarantees the existence and convergence of the iterates under suitable assumptions.