Versions of the Weierstrass Theorem for Bifunctions and Solution Existence in Optimization
Phan Quốc Khánh, Nguyen Hong Quan · SIAM Journal on Optimization · 2019
We propose an approach to a unified study of existence based on using new purely set-theoretic notions of anti-cyclicity and cyclic anti-quasimonotonicity together with a given topological structure, without linear or (generalized) convexity structures. We first establish an extension of the classical Weierstrass extreme-value theorem to the bifunction case and its equivalent versions (with quite different formulations). Next, we apply these results to obtain sufficient conditions for the solution existence of various optimization-related problems. The proof technique of the results is simple and elementary, but seems to also be applicable for many other problems.