On existence of solutions to non-convex minimization problems

TL;DR

This paper introduces a unified framework using asymptotic and retractive cones to analyze the existence of solutions in general non-convex minimization problems, providing new necessary and sufficient conditions.

Contribution

It develops a systematic approach with new conditions for solution existence, introducing cones of retractive directions and exploring their properties and relationships.

Findings

Established basic properties of cones of retractive directions.

Derived necessary and sufficient conditions for solution existence.

Refined conditions for structured non-convex problems.

Abstract

We provide a unified framework for a systematic analysis of the existence of solutions to general nonconvex problems, relying on asymptotic and retractive cones for functions and sets. Using this framework we develop new necessary and sufficient conditions for the existence of solutions to a general problem of minimizing a proper closed function over a closed, possibly unbounded, set. Towards the result, we introduce cones of retractive directions for a set and a function, establishing some basic properties for them. We also investigate the relationships between the cone of retractive directions of a function and the cone of level sets of the function. Using the cones of retractive directions we provide necessary and sufficient conditions for the existence of solutions that require an asymptotically bounded decay of a function, and a relation between the cones of retractive directions of the constraint set and the asymptotic cone of the objective function. Finally we refine these conditions for more structured problems.