# Introduction

• Michael Patriksson
Part of the Applied Optimization book series (APOP, volume 23)

## 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 uX ↦ R u a vector-valued and continuous mapping on dom uX.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)

## 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.