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.

The first author was partially supported by Centro de Modelamiento Matemático (CMM), ACE210010 and FB210005, BASAL funds for center of excellence and ANID-Chile grant Fondecyt Regular 1240335. The second author has been partially supported by Grant PID2022-136399NB-C21 founded by MICIU/AEI/10.13039/501100011033. The third author was partially supported by Centro de Modelamiento Matemático (CMM), ACE210010 and FB210005, BASAL funds for center of excellence and ANID-Chile grant: MATH-AMSUD 23-MATH-09 and MATH-AMSUD 23-MATH-17, ECOS-ANID ECOS320027, Fondecyt Regular 1220886, Fondecyt Regular 1240335, Fondecyt Regular 1240120 and Exploración 13220097.