Abstract
Let X ⊆ R n be a nonempty, closed and convex set, u : R n 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,
the subdifferential operator for u, and the mapping F. Consider the problem of finding a vector x * ∈ Rn such that
$${{N}_{X}}\left( x \right): = \left\{ \begin{gathered} \left\{ {z \in {{\Re }^{n}}\left| {{{z}^{T}}\left( {y - x} \right) \leqslant 0,\quad \forall y \in } \right.} \right\},\quad x \in X, \hfill \\ \phi \quad \in \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad x \notin X; \hfill \\ \end{gathered} \right.$$
(1.1)
$$\partial u(x): = {\left\{ \zeta \right._u} \in {\Re ^n}/u(y) \geqslant u(x) + \zeta _u^T(y - x),\forall y \in \left. {{\Re ^n}} \right\};$$
$$\begin{array}{*{20}{c}} {[GVIP(F,u,X)]} \hfill \\ {F({{x}^{*}}) + \partial u({{x}^{*}}) + {{N}_{X}}({{x}^{*}}){{0}^{n}}} \hfill \\ \end{array}$$
(1.2)
Keywords
Variational Inequality Step Length Line Search Variational Inequality Problem Merit Function
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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Copyright information
© Springer Science+Business Media Dordrecht 1999