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-dimensional space X is not stable with respect to a perturbation of the minimized function. Here a generalized solution is defined as an element of suitable completion of X. A necessary and sufficient condition for the completion of X to 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 minimizing filter on X, which generalizes the notion of the minimizing sequence.