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Zbl 0647.90071
Pelant, Jan; Poljak, Svatopluk
Extensions of cyclically monotone mappings. (English)
[J] Rend. Circ. Mat. Palermo, II. Ser. Suppl. 11, 81-88 (1985).

The authors study cyclically monotone mappings and the subclass of them called strongly cyclically monotone (s.c.m.) mappings (the mapping $f: S\subset R$ $n\to R\sp n$ is said to be s.c.m. on S if $\sum\sp{k}\sb{i=1}(x\sb i-x\sb{i-1})f(x\sb i)\ge 0$ for all $k\ge 2$ and $x\sb 1,...,x\sb k=x\sb 0\in S$, and $\sum\sp{k}\sb{i=1}(x\sb i-x\sb{i- 1})f(x\sb i)=0$ implies that $f(x\sb 1)=...=f(x\sb k)=0$ for all $k\ge 2$ and every $x\sb 1,...,x\sb k=x\sb 0\in S)$. They prove the following main result: Let $f: S\subset R$ $n\to R\sp n$ be s.c.m. Then f can be extended to some s.c.m. mapping $\bar f$ defined on conv S (the convex hull of S).

MSC 2000:: *90C25 Convex programming
49J35 Minimax problems (existence)
90C31 Sensitivity, etc.

Keywords: subgradients; multivalued mapping; cyclically monotone mappings

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