Monotone and maximal monotone affine subspaces

Abstract

Affine monotone and maximal monotone subspaces are characterized.

Introduction

Variational inequality problems and equilibrium problems occur in a large field of applications arising from various domains such as physics, mechanics, economics and operations research. They encompass optimization problems and related problems such as complementarity problems and saddle-point problems.

With a subset $G$ of ${\mathbb{R}}^n\times {\mathbb{R}}^n$ we associate the multivalued maps $\Gamma :{\mathbb{R}}^n\stackrel{⟶}{⟶}{\mathbb{R}}^n$ and ${\Gamma }^{-1}:{\mathbb{R}}^n\stackrel{⟶}{⟶}{\mathbb{R}}^n$ such that

$$G=\{(x,x^{\ast }):x^{\ast }\in \Gamma \left(x\right)\}=\{(x,x^{\ast }):x\in {\Gamma }^{-1}\left(x^{\ast }\right)\}\text{.}$$

The set $G$ and the maps $\Gamma$ and ${\Gamma }^{-1}$ are said to be monotone if

$$〈x^{\ast }-y^{\ast },x-y〉\ge 0\text{for all }(x,x^{\ast }),(y,y^{\ast })\in G\text{.}$$

They are said to be maximal monotone if they are monotone and for all monotone subsets $S\supseteq G$ of ${\mathbb{R}}^n\times {\mathbb{R}}^n$ one has $S=G$.

(Maximal) Monotone maps play in variational inequality problems the role of (lower semicontinuous) convex functions in optimization problems. We say that a multivalued map $\Gamma$ is affine if its graph $G$ is an affine subspace, i.e., 

$$G=\{(x,x^{\ast })\in {\mathbb{R}}^n\times {\mathbb{R}}^n:Ax+B{x}^{\ast }=c\}\text{,}$$

where $A$ and $B$ are two $p\times n$ matrices and $c\in {\mathbb{R}}^p$. Affine (monotone) maps play in variational inequality problems the role of quadratic (convex) functions in optimization problems. An important class of affine variational inequality problems is the class of linear complementarity problems. A slightly more general form of these problems is as follows:

$$\text{Find }x\ge 0,x^{\ast }\ge 0\text{ such that }〈x,x^{\ast }〉=0\text{ and }Ax+B{x}^{\ast }=c\text{.}$$

Linear (monotone) complementarity problems encompass quadratic (convex) optimization problems and Nash bimatrix equilibrium problems. They can be solved in a finite number of steps by an algorithm due to Lemke [6] which uses, as the simplex algorithm, pivotal rules. For linear complementarity problems see the text book by Cottle, Pang and Stone [3] and for quadratic programming see the seminal paper [5] of Frank and Wolfe.

The purpose of this paper is the analysis of monotone affine maps. In Section 2 we characterize monotone and maximal monotone affine subsets of ${\mathbb{R}}^n\times {\mathbb{R}}^n$. Section 3 analyses affine monotone subspaces under linear transformations. Such transformations occur along the completion of Lemke’s algorithm. In Section 4 we provide an algorithm that builds a maximal monotone affine extension of a monotone affine subspace.