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Quasiconvex duality for the max of two functions. (English) Zbl 0921.49027

Gritzmann, Peter (ed.) et al., Recent advances in optimization. Proceedings of the 8th French-German conference on Optimization. Trier, Germany, July 21–26, 1996. Berlin: Springer. Lect. Notes Econ. Math. Syst. 452, 365-379 (1997).
The author gives two duality formulas on the infimum of the maximum of two functions. The first one, \[ \inf_{x\in X}\bigl\{f(x)\vee g(x) \bigr\}= \max_{\substack{ y\in X^* \setminus\{0\}\\ r\in R}} \Bigl\{\inf_{\langle x,y \rangle<r} f(x)\wedge\inf_{\langle x,y\rangle>r}g(x)\Bigr\}, \] is valid, under mild assumptions, for quasiconvex functions \(f,g:X\to \overline R\) on a real topological vector space \(X\) \((X^*\) is its topological dual, \(\langle \cdot, \cdot \rangle\) is the canonical linear pairing between \(X\) and \(X^*\), and \(\vee\) and \(\wedge\) denote maximum and minimum, respectively). The second one, \[ \inf_{ z \in Z}\biggl\{f(z)\vee-g\bigl(u(z)\bigr)\biggr\}=\inf_{\substack{ y\in X^*\\ r \in R}} \Bigl\{\inf_{\langle u(x),y\rangle>r}f(z)\vee-\inf_{\langle x,y\rangle> r} g(x) \Bigr\}, \] involves a function \(f:Z\to\overline R\) on an arbitrary set \(Z\), a mapping \(u:Z\to X\) and a lower semicontinuous quasiconvex function \(g:X\to \overline R\). As a particular case of the first result, a duality formula for quasiconvex minimization on a convex set follows; from the second one, duality formulas for quasiconvex maximization on an arbitrary subset and for minimization of an arbitrary function on the complement of a closed convex set are obtained.
For the entire collection see [Zbl 0868.00068].

MSC:

49N15 Duality theory (optimization)
90C26 Nonconvex programming, global optimization
90C25 Convex programming
90C48 Programming in abstract spaces
49J45 Methods involving semicontinuity and convergence; relaxation
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