On the Connectedness of Sublevel Sets in Invex Optimization

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.