The subdifferential and the directional derivatives of the maximum of a family of convex functions. II (Q2759281)

The paper deals with calculating the directional derivative and the subdifferential of the maximum of convex functions without compactness conditions on the index set.NEWLINENEWLINENEWLINEIn detail, let \(X\) be a linear space (or a locally convex linear topological space, respectively), \(f_s: X\to\overline{\mathbb{R}}\) \((s\in S)\) be (extended real-valued) convex functions and NEWLINE\[NEWLINEf(x):= \sup_{s\in S} f_s(x).NEWLINE\]NEWLINE Further, let \(x_0\in\text{dom}(f)\) be a fixed point, let \(y\in X\) be a direction such that \(f\) is finite and let \(S(x)= \{s\in S\mid f_s(x)= f(x)\}\) be nonempty on the segment \([x_0, x_1]\) with \(x_1= x_0+ \delta y\) \((\delta> 0)\).NEWLINENEWLINENEWLINEIn the first part of the paper, sufficient conditions are presented such that the well-known relations NEWLINE\[NEWLINEf'(x_0, y)= \sup_{s\in S(x_0)} f_s'(x_0, y),NEWLINE\]NEWLINE NEWLINE\[NEWLINE\partial f(x_0)= \overline{\text{conv}} \bigcup_{s\in S(x_0)}\partial f_s(x_0)NEWLINE\]NEWLINE hold. The author gives also further characterizations which are equivalent to these formulas. All the used conditions are essentially (upper) semicontinuity conditions to the functions \(f(x)\) and \(f'(x,y)\) with respect to \(x\) on the segment \([x_0, x_1]\).NEWLINENEWLINENEWLINEIn the second part of the paper, the results are applied to problems of minimax theory, to Lagrange duality for non-convex optimization problems and especially (as illustration) to a special problem of optimal design.NEWLINENEWLINENEWLINE(See also the review of Part I [Izv. Math. 62, No. 4, 807-832 (1998); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 62, No. 4, 173-200 (1998)] in Zbl 0921.49009).