Convergence Properties of Local Solutions of Sequences of Mathematical Programming Problems in General Spaces

摘要

The paper gives several sets of sufficient conditions that a local solution x sup k exists of the problem min sub(R sup K) (f sup k)(x) k = 1,2,..., such that x sup k has cluster points that are local solutions of a problem of the form min(sub R)f(x). The underlying space will generally be assumed to be any space on which there has been defined a notion of convergence. The results are based on a well-known concept of topological, or pointwise, convergence of the sets R sup k to R. Similar conditions have been obtained by others for characterizing the relationship of global solutions of the problems (P sup k) to Problem P, utilizing more elaborate constructs, e.g., point-to-set mappings, to define the constraint sets and the minimizing sets. Such results have been used to construct and validate large classes of mathematical programming methods based on successive approximations of the problem functions. They are also directly applicable to parametric and sensitivity analysis, and provide additional characterizations of optimality for large classes of nonlinear programming problems. (Author)