Evaluation of the epi-convergence

Abstract

The paper presents an equivalent characterization of the epi convergence of lower semicontinuous functions The proposed measures naturally estimate the distance between optimal values of two optimization programs A comparison of optimal solutions is more complex We propose an estimation for a given function which can be taken as the excess of a set over another being in metric space Hence we receive an estimate of the distance between optimal solutions The concept of the epi convergence Looking for the global minimum of a given function is the central problem discussed in the literature and solved in optimization theory The lack of complete information on the objective function is the main trouble met in the practice We have to work with approximations and therefore we naturally ask about the stability of our optimization problem provided varying objective function The epi convergence looks to be the most e cient tool for that see or We present an equivalent description of the epi convergence which can be used to derive estimate of the distance between optimal values Treating of the optimal solutions is more di cult We o er a procedure which is natural in the case of metric spaces as the example illustrates Let us specify the subject of our treatment We work on the Hausdor topological space X Therefore we will employ open closed and compact sets of X the notion of nets cluster points limit points Kuratowski Painlev e convergence of sets etc For convenience we are giving the list of the used notation in Appendix We consider the function f X R Our interest is focused in con tinuity property of its optimal value f infx X f x and its set of optimal solu tions f fx X f x f g Let us note that the problem of optimal solutions do not need special treating It is su cient to consider truncated function f maxff f g as we do in the example Known observation is that f is closed set provided f is l s c lower semicon tinuous To avoid any misunderstanding let us recall that f X R is l s c if lim infy x f y f x for each x X Let us denote the set of all l s c functions on X by LSC X We consider the space LSC X with the epi convergence De nition Let be a directed set f LSC X for each and f LSC X We say that the net f epi converges to f provided epi f K lim epi f Recall epi f f x X R f x g and the de nition of K lim is remembered in Appendix The research is supported by Deutsche Forschungsgemeinschaft Projekt Nr Ro the Czech Republic Grant and the Charles University Grant The paper has partially been written during a stay at the Humboldt University of Berlin The epi convergence admits a helpful characterization by means of the nets Proposition Let f LSC X for each belonging to the directed set and f LSC X The net f epi converges to f if and only if at each point x X both of the following conditions hold Let be a directed set and be monotone i e and con nal i e for each there exists such that Then we have lim inf f x f x whenever lim x x There exists directed set and x X for each such that lim x x and lim f x f x Proof Fix the point x X Let f epi converges to f a Let be a directed set and be monotone and con nal Assume lim x x and let us denote lim inf f x Take the set G G X R with x G According to the de nition of the lim inf we have for each some such that and x f x G Then x Ls epi f epi f because of the epi convergence Therefore f x b According to the de nition of the epi convergence Li epi f epi f and hence x f x Li epi f Therefore for each G G X R with x f x G there is such that G epi f for each Let us take x G G G epi f for and de ne x G x G f x whenever The set n G G X R x f x G o is directed by inclusion i e