A note on maximum-type functions (Q949232)

In various fields of applied mathematics, for example, optimal control and extremum problem theories, in game theory, and operation analysis, one often deals with maximum-type functions, namely, the functions of the form \[ g(x)=\max_{y\in Y}f(x,y),\tag{1} \] where \(x\in X\), \(X\) being a nonempty open convex set from the Euclidean space \(\mathbb{R}^m\); \(y\in Y\), \(Y\) being a nonempty compact subset of the Euclidean space \(\mathbb{R}^n\); and the function \(f(x,y)\) is, at least, continuous on \(X \times Y\). The symbol \(\mathbb{R}^k(k\geq 1)\) denotes a real arithmetic Euclidean space whose elements are ordered sets of \(k\) numbers written in the form of columns. The scalar product of vectors \(a\) and \(b\) from \(\mathbb{R}^k\) is defined by \(\langle a,b\rangle= \sum^k_{i=1}a_kb_i,\) where \(a_i\) and \(b_i\) are the coordinates of the vectors \(a\) and \(b\), respectively. In the analysis of maximum-type functions, the study of their differential properties in terms of the properties of the original function \(f(x,y)\) is of particular interest. The paper deals with precisely this question. Problem: Let the function \(f(x,y)\) be continuous on \(X\times Y\) and continuously differentiable with respect to \(x\) on the same set and the function \(g(x)\) be concave on \(X\). Then, what can be said on the differentiability of the function \(g(x)\) on \(X\)? Some examples of functions \(f(x,y)\) for the which function \(g(x)\) is concave are first presented. In these examples, \(X\) is a certain nonempty open convex set from \(\mathbb{R}^m\) and \(Y\) is a nonempty compact subset of \(\mathbb{R}^n\). Now, let us study the differential properties of the function \(g(x)\) under the same assumptions that have been made in the formulation of the above mentioned problem and the following theorems are proved. Theorem 1. Under the assumptions made above, the function \(g(x)\) is differentiable on \(X\). Theorem 2. The function \(g_x'(\xi)\) is continuous on \(X\). References consists of 10 names.