Extensions of cyclically monotone mappings. (English) Zbl 0647.90071

Pelant, Jan; Poljak, Svatopluk

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 \(f: S\subset R\) \(n\to R^ n\) is said to be s.c.m. on S if \(\sum^{k}_{i=1}(x_ i-x_{i-1})f(x_ i)\geq 0\) for all \(k\geq 2\) and \(x_ 1,...,x_ k=x_ 0\in S\), and \(\sum^{k}_{i=1}(x_ i-x_{i- 1})f(x_ i)=0\) implies that \(f(x_ 1)=...=f(x_ k)=0\) for all \(k\geq 2\) and every \(x_ 1,...,x_ k=x_ 0\in S)\). They prove the following main result: Let \(f: S\subset R\) \(n\to R^ n\) 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:

90C25	Convex programming
49J35	Existence of solutions for minimax problems
90C31	Sensitivity, stability, parametric optimization

Keywords:

subgradients; multivalued mapping; cyclically monotone mappings