On Nonsmooth Semi-Infinite Minimax Programming Problem with\((\mathrm{\Phi },\rho )\)-Invexity

Abstract

We are interested in a nonsmooth minimax programming Problem (SIP). Firstly, we establish the necessary optimality conditions theorems for Problem (SIP) when using the well-known Caratheodory's theorem. Under the Lipschitz $(\mathrm{\Phi },\rho )$-invexity assumptions, we derive the sufficiency of the necessary optimality conditions for the same problem. We also formulate dual and establish weak, strong, and strict converse duality theorems for Problem (SIP) and its dual. These results extend several known results to a wider class of problems.