Relaxations of Extremal Problems

Abstract

In this chapter, some corollaries of general constructions of sets of asymptotic attainability are considered. These corollaries mean peculiar applications to optimization theory. In essence, we consider regularizations of extremal problems on the basis of well-posed extensions. The sense of these regularizations was partly explained in Section 1.5. Questions on extensions of extremal problems were considered in [1, 15, 20, 33, 34] and in many other papers. Separately, we note the extension constructions of dynamic game problems [24, 30, 33, 34]. The constructions of extensions and relaxations in [13] are closer in manner to the present study and, at the same time, are very general from the point of view of optimization theory. The monograph [13] is a continuation of a whole series of journal articles; in this connection, see for example [4]–[12], [29]. In the present chapter, we consider the setting of a problem which is more general in relation to [13, ch. VIII]. We consistently analyze a general problem of the asymptotic optimization in a preordered topological space. Elements of this problem were considered in [13, ch. II]. Here we need, to a more general extent, the constructions of compactifications. In this connection, see Sections 3.7, 3.8. The procedures for compactifications considered below differ from those used in general topology. In the next section, we shall briefly recall and supplement the definitions and statements of [13, ch. II], orienting towards their employment with the goal of constructing an extension of extremal problems with integral constraints.