On the existence of continuous selections of a multivalued mapping related to the problem of minimizing a functional

Abstract

The article considers a parametric problem of the form f(x,y)→"inf",x∈M, where M is a convex closed subset of a Hilbert or uniformly convex space X, y is a parameter belonging to a topological space Y. For this problem, the set of ϵ-optimal points is given by a_ϵ (y)={x∈M|f(x,y)≤〖"inf" 〗┬(x∈M)⁡〖f(x,y)+ϵ〗 }, where ϵ>0. Conditions for the semicontinuity and continuity of the multivalued mapping a_ϵ are discussed. Using gradient projection and linearization methods, we obtain theorems on the existence of continuous selections of the multivalued mapping a_ϵ. One of the main assumptions of these theorems is the convexity of the functional f(x,y) with respect to the variable x on the set M and continuity of the derivative f_x^' (x,y) on the set M×Y. Examples that confirm the significance of the assumptions made are given, as well as examples illustrating the application of the obtained statements to optimization problems.