Approximations and solution estimates in optimization
The aim of this paper is to study approximations of lower semicontinuous (lsc) functions and of the solutions of optimization problems in metric spaces. The main tool is the Attouch-Wets distance of two lsc functions, which is defined in terms of their epigraphs. Among the obtained results are an estimate of the difference of the optimal values of two optimization problems, an estimate of the distance of their respective sets of solutions, an approximation of near-optimal, near-feasible solutions, an approximation of the level sets, etc. The last section is devoted to epi-splines, which are a particular kind of approximating functions defined by a finite number of parameters. These are extended real-valued, and constant on each element of a given partition of a metric space. It is shown that every lsc function can be approximated by a sequence of epi-splines, epi-converging and pointwise converging to the function. Actually, the set of rational-valued epi-splines is dense in the space of lsc functions. An estimation of the rate of convergence of epi-splines to lsc functions is obtained.
scientific article; zbMATH DE number 6923437
scientific article