Topologically-based characterizations of the existence of solutions of optimization-related problems
Phan Quốc Khánh, Nguyen Hong Quan · Mathematische Nachrichten · 2015
Necessary and sufficient conditions for the existence of solutions of optimization-related problems, defined on general sets, are established by using topologically-based structures, without the usual linear structure. First, we prove that the existence of solutions to a variational relation problem is equivalent to the existence of either a KKM-structure or a connectedness structure, satisfying additionally some verifiable conditions. Then, applying these results, we obtain such full (two-way) characterizations for the existence of invariant points, solutions of quasiequilibrium problems of the Stampacchia-Minty type, saddle points, and Nash equilibria for noncooperative games. The main advantages of our scheme with the mentioned two structures are that full characterizations for existence are obtained in a unified way, with no convexity assumptions, and also that one has flexibility to choose a suitable structure in investigations. (This flexibility is decisive when the natural underlying structure of the given problem is scarcely employed.)