Directional derivates in minimax problems. (English) Zbl 0566.49005

The authors study the existence and characterization of the (one-sided) directional derivative \(h'(x_ 0;d)\) of the function \(h(x)=\inf_{u\in U_ 0}\sup_{v\in V_ 0}L(x,u,v)\) defined on a Hausdorff topological vector space. It is assumed that L(x,\(\cdot,\cdot)\) is defined on the product of two Hausdorff spaces U and V, and satisfies the equality \[ \inf_{u\in U_ 0}\sup_{v\in V_ 0}L(x,u,v)=\sup_{v\in V_ 0}\inf_{u\in U_ 0}L(x,u,v) \] for all \(x\in [x_ 0,x_ 0+\alpha d]\) (where \(\alpha >0\), \(U_ 0\subset U\), \(V_ 0\subset V)\). The conclusion of the main result is the same as in a previous paper of the authors [Nonlinear Anal., Theory Methods Appl. 9, 13-22 (1985; Zbl 0556.49007)] but the assumptions are somewhat different.
Reviewer: M.Studniarski


49J50 Fréchet and Gateaux differentiability in optimization
49K35 Optimality conditions for minimax problems
46G05 Derivatives of functions in infinite-dimensional spaces
49K27 Optimality conditions for problems in abstract spaces
46A03 General theory of locally convex spaces


Zbl 0556.49007
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