# Affine Variational Inequalities (AVI)¶

## Problem statement¶

The Affine Variational Inequality (AVI) is defined by

Given $$q\in\mathbb{R}^n$$ , $$M\in\mathbb{R}^{n\times n}$$ and a set $$K\in\mathbb{R}^n$$ , find $$z\in\mathbb{R}^n$$ such that:

$\begin{equation*}(Mz+q)^T(y -z) \geq 0,\quad \text{ for all } y \in K,\end{equation*}$

or equivalently,

$\begin{equation*}-(Mz + q) \in \mathcal{N}_K(z)\end{equation*}$

where $$\mathcal{N}_K$$ is the normal cone to $$K$$ at $$z$$ .

The AVI is a special case of a Variational Inequality (VI), where the function $$F$$ is affine. For VI solvers, see Variational Inequality (VI).

From more details on theory and analysis of AVI (and VI in general), we refer to [6]

## Implementation in numerics¶

Structure to define the problem: AffineVariationalInequalities.

The generic driver for all VI is avi_driver().

solvers list AVI_SOLVER

## AVI Available solvers¶

### Cao Ferris (SICONOS_AVI_CAOFERRIS)¶

Direct solver for AVI based on pivoting method principle for degenerate problem. Choice of pivot variable is performed via lexicographic ordering

driver: avi_caoferris()

parameters:

• iparam[SICONOS_IPARAM_MAX_ITER] = 10000

• dparam[SICONOS_DPARAM_TOL] = 1e-12

### Pathvi (SICONOS_AVI_PATHAVI)¶

Direct solver for VI based on pivoting method principle for degenerate problem.

Ref: “A structure-preserving Pivotal Method for Affine Variational Inequalities” Y. Kim, O. Huber, M.C. Ferris, Math Prog B (2017).

driver: avi_pathvi()

parameters:

• iparam[SICONOS_IPARAM_MAX_ITER] = 10000

• dparam[SICONOS_DPARAM_TOL] = 1e-12