On the Stationary Duality of Structural Composite Cardinality Optimization

TL;DR

This paper explores the duality in composite cardinality optimization, simplifying complex counting problems via stationary duality and establishing conditions for solutions.

Contribution

It introduces a dual formulation for CCOP using stationary duality and analyzes solution existence and correspondence between primal and dual problems.

Findings

Dual formulation reduces composite counting to simple counting.

Conditions for global solution existence are validated on examples.

Local solutions are shown to be equivalent to stationary points.

Abstract

Simple cardinality refers to counting nonzero elements of an independent variable satisfying certain properties. Composite cardinality is a simple counting process composited with an affine mapping, and is therefore more complicated than the simple cardinality. We study the composite cardinality optimization problem (CCOP) with structures covering a wide range of applications. Through the use of the stationary duality, we reduce the composite counting to simple counting, and thereby obtain a dual formulation of CCOP. For both primal and dual problems, we investigate the sufficient conditions for the existence of global solutions. Those conditions are validated on representative examples from existing literature. We then show that local solutions of the primal and dual problems are equivalent to their stationary points. This result further helps us establish a one-to-one correspondences between primal and dual local solutions. We also demonstrate that the correspondence holds for a pair of global solutions to the primal and dual problems, provided that the dual weighted parameters are appropriately selected. The reported theoretical results lay foundation for developing numerical algorithms for CCOP in future.