Log-exponential approximation in semi-infinite programming: A variational approach

In this paper, we propose an approximation technique for a function which is the supremum of a general family of functions by means of a function, of LogExp-type, involving a finite number of the data functions.A study of variational properties of such an approximating function is carried out in the paper.In particular, the epigraphic convergence of these approximating functions to the supremum function is proven.Moreover, refined calculus specifying the relations among the subdifferentials (regular and general) of the approximating and the data functions are provided in the first part of the paper.In the second part, we propose to approximate a general semi-infinite programming problem by a simple problem with a single constraint.Applying the results of the first part we establish, under standard hypotheses, the convergence of optimal values and solutions of these problems to the corresponding ones of the original problem.The Lagrangian duality of this problem is also studied but restricted to the semi-infinite convex programming problem, and optimal dual solutions are built by means of a sequential procedure.