Subdifferential characterization of quasiconvex and convex vector functions (Q1277219)

Let \(C \subset R^m\) be a cone generated by \(m\) linearly independent vectors \(c_1,\cdots,c_m\) and let \(f\) be a C-lower semicontinuous function defined on a Banach space \(X\) and taking values on \(R^m \cup \{ \infty^+ \}\), where \(\infty^+ = c_1(+\infty) + \cdots + c_n (+\infty)\). In this paper suitable concepts of directional subderivative \(f^\uparrow (x;v)\) and subdifferential \(\partial^\uparrow f (x)\) are introduced in order to characterize C-convex or C-quasiconvex functions, that is such that for \(x,y \in X\), \(t\in (0,1)\) we, respectively, have \(f(tx+(1-t)y) \preceq tf(x)+(1-t)f(y)\) or \(f(tx+(1-t)y) \preceq \sup\{f(x), f(y)\}\), where ''\(\preceq\)'' is the partial order induced by the cone \(C\). The following equivalence is proved: \(f\) is C-convex iff \(\partial^\uparrow f \) is C-monotone and \(f\) is C-quasiconvex iff \(\partial^\uparrow f\) is C-quasimonotone. A set-valued map \(F\) from \(X\) to \(L(X,R^m)\) is said to be C-monotone if \(x,y \in \text{dom } F \), \(A\in F(x), B\in F(y) \) imply \((B-A)(y-x)\in C\), whereas it is said to be C-quasimonotone if \(x,y \in \text{dom } F \), \(A\in F(x), B\in F(y), A(y-x)\in \text{int}(C)\) imply \(B(y-x)\in C\). Analogous results for scalar functions (\(m=1\)) have been established by \textit{D. T. Luc} [ Bull. Aust. Math. Soc. 48, No. 3, 393-406 (1993; Zbl 0790.49015)]. As regards the properties of \(\partial^\uparrow f\), it is proved that it contains the gradient of \(f\) when \(f\) is differentiable (but it does not coincide with it). Moreover, in the case \(X= R^n\), \(\partial^\uparrow f\) contains the subdifferential in the sense of Clarke and it coincides with the subgradient in the sense of convex analysis when \(f\) is convex.