New Bounds for Exact Penalized Cardinality-Constrained Optimization with Pseudonormality Conditions

TL;DR

This paper develops new theoretical bounds and conditions for solving cardinality-constrained optimization problems using exact penalty methods, extending pseudonormality concepts.

Contribution

It extends pseudonormality conditions to CCO problems and establishes local exact penalization without requiring Lipschitz continuity.

Findings

Established local exact penalization under pseudonormality conditions.

Provided convergence guarantees for projected subgradient methods.

Derived more precise bounds for iterates and objective values.

Abstract

Cardinality-constrained optimization (CCO) is a popular topic in sparse learning and signal recovery, yet remains challenging due to the inherent nonconvexity and discontinuity of cardinality constraints. This paper investigates the exact penalty theory for CCO problems with general equality and inequality constraints. In particular, we extend the pseudonormality condition to the cardinality-constrained framework and establish the local exact penalization without imposing Lipschitz continuity on the objective function. We further analyze both the projected subgradient method and its stochastic variant with convergence guarantees for the derived exact penalty formulation. Compared with the existing results, we give some more precise bounds of the iterate sequence and the objective function value.