Approximate optimality conditions and duality results for non-smooth semi-infinite programming problems

This paper concentrates on investigating a semi-infinite optimization problem with cone constraints. Under the attainment of Abadie constraint qualification, an approximate necessary optimality condition by employing Clarke subdifferential for the semi-infinite optimization problem having cone constraints is developed. A new class of functions namely Q-quasiconvex functions is introduced and in the light of the approximate pseudoconvexity and Q-quasiconvexity assumptions, an approximate sufficient optimality condition is investigated using locally Lipschitz functions. Additionally, we formulate the approximate Wolfe's and approximate Mond-Weir dual problems for the non-smooth semi-infinite optimization problem. Subsequently, duality relations in terms of weak, strong and converse results between the semi-infinite optimization model and the aforementioned dual problems are established under the approximate pseudoconvexity and Q-quasiconvexity assumptions. Moreover, to justify the main results of the paper, numerical examples have been shown at suitable places.