On the Connectedness of Sublevel Sets in Invex Optimization

TL;DR

This paper investigates the connectedness of sublevel sets in invex functions, revealing conditions under which these sets are connected, which aids in understanding non-convex optimization landscapes.

Contribution

It introduces a topological toolkit based on the mountain pass theorem to prove connectedness of sublevel sets in invex functions and related problems.

Findings

Sublevel sets of invex functions are connected under mild assumptions.

Connectedness results extend to solution sets in invex-incave minimax problems.

The toolkit provides new insights into the topology of non-convex optimization landscapes.

Abstract

Understanding the topology of sublevel sets yields crucial insights into the optimization landscape of non-convex functions. If sublevel sets are connected, local search algorithms are less likely to be trapped in isolated valleys, facilitating convergence to global minimizers. However, few results exist to establish connectedness in the nonconvex setting. In this work, we present a mathematical toolkit based on the topological mountain pass theorem and use it to study invex functions, a class of functions that includes those satisfying the Polyak-{\L}ojasiewicz inequality and generalizations thereof. We show that their sublevel sets are connected under mild assumptions. We further leverage our result to establish the connectedness of different solution sets for invex-incave minimax problems and incave games.