Minimax theorems: existence and stability of the saddle point
Department of Optimization and Control,
Institute of Mathematics, VAST Hanoi, Vietnam
We discuss existence and stability conditions for the saddle value and the saddle point of a quasiconvex quasiconcave function depending upon a parameter. It is shown in particular that a straightforward refinement of Tuy's earlier topological minimax theorems can be obtained that yields several new minimax theorems with much weaker compactness and continuity assumptions than usual. An application to Lagrange relaxation is given that provides a rigorous foundation for decomposition methods in nonconvex global optimization.