Introduction

Abstract

Let X ⊆ Rⁿ be a nonempty, closed and convex set, u : Rⁿ R∪ {+∞} a lower semicontinuous (1.s.c.), proper¹ and convex function, and F : dom u ∩ X ↦ R^u a vector-valued and continuous mapping on dom u ∩ X.² The problem under study is defined by three operators: the normal cone operator for X,

$${{N}_{X}}\left( x \right): = \left\{ \begin{gathered} \left\{ {z \in {{\Re }^{n}}\left| {{{z}^{T}}\left( {y - x} \right) \leqslant 0,\quad \forall y \in } \right.} \right\},\quad x \in X, \hfill \\ \phi \quad \in \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad x \notin X; \hfill \\ \end{gathered} \right.$$

(1.1)

the subdifferential operator for u,

$$\partial u(x): = {\left\{ \zeta \right._u} \in {\Re ^n}/u(y) \geqslant u(x) + \zeta _u^T(y - x),\forall y \in \left. {{\Re ^n}} \right\};$$

and the mapping F. Consider the problem of finding a vector x ^* ∈ Rⁿ such that

$$\begin{array}{*{20}{c}} {[GVIP(F,u,X)]} \hfill \\ {F({{x}^{*}}) + \partial u({{x}^{*}}) + {{N}_{X}}({{x}^{*}}){{0}^{n}}} \hfill \\ \end{array}$$

(1.2)