A generalized solution of a nonconvex minimization problem and its stability

It is well known that the set of the solutions of a minimization problem on an infinite-dimen­ sional space X is not stable with respect to a perturbation of the minimized function. Here a generalized solution is defined as an element of a suitable completion of X. A necessary and sufficient condition for the completion of X\o guarantee the stability of the set of the generalized solutions is given. It is shown that the generalized solution can be considered as a certain mini­ mizing filter on X, which generalizes the notion of the minimizing sequence. Hereinafter, we denote its infimum by inf / = inf f(X) and the set of the (classical) solutions by Arginf/ = {xeX; f(x) = inf/}. In case when X is a reflexive Banach space it is well known that either some type of convexity or the finite dimension of X results, roughly speaking, in a "good behaviour" of the mapping /i-> Arginf/ with respect to perturbations of/ However, in a nonconvex infinite-dimensional case the solution of the minimization problem may fail to exist, although for a "near" minimized function it may exist. This pathology implies that the mapping /H* f-> Arginf/ cannot be stable (i.e. upper semi-continuous) in any separated topology, and it also indicates that the classical solution is not a "natural" notion, at least for nonconvex functions. Besides, optimization algorithms do not yield any solution of the minimization problem (except some special problems), but only a minimizing sequence. This was probably the reason that Golshtein (6) introduced the minimizing sequences as generalized solutions; see also (8). Motivated by it, we try to introduce a "natural" This paper was prepared while the author was with the General Computing Centre of the