Topologically‐based characterizations of the existence of solutions of optimization‐related problems |
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Authors: | Phan Quoc Khanh Nguyen Hong Quan |
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Institution: | 1. Department of Mathematics, International University, Vietnam National University Hochiminh City, Linh Trung, Thu Duc, Hochiminh City, Vietnam;2. Department of Scientific Fundamentals, Posts and Telecommunications Institute of Technology, Hochiminh City, Vietnam |
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Abstract: | 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.) |
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Keywords: | KKM‐structures connectedness structures variational relation problems invariant points the Stampacchia‐Minty type of quasiequilibria minimax inequalities saddle points Nash equilibria 91A06 91A10 54H25 49J53 |
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