A structure-preserving pivotal method for affine variational inequalities

Abstract

Affine variational inequalities (AVI) are an important problem class that subsumes systems of linear equations, linear complementarity problems and optimality conditions for quadratic programs. This paper describes PathAVI, a structure-preserving pivotal approach, that can efficiently process (solve or determine infeasible) large-scale sparse instances of the problem with theoretical guarantees and at high accuracy. PathAVI implements a strategy known to process models with good theoretical properties without reducing the problem to specialized forms, since such reductions may destroy sparsity in the models and can lead to very long computational times. We demonstrate formally that PathAVI implicitly follows the theoretically sound iteration paths, and can be implemented in a large scale setting using existing sparse linear algebra and linear programming techniques without employing a reduction. We also extend the class of problems that PathAVI can process. The paper illustrates the effectiveness of our approach by comparison to the Path solver used on a complementarity reformulation of the AVI in the context of applications in friction contact and Nash Equilibria. PathAVI is a general purpose solver, and freely available under the same conditions as Path.

Appendix

Lemma 5

(Theorem 4.4 [6]) Consider an $\textit{AVI}(C,q,M)$ with ${\text {lin}}\,C = \{0\}$ and let M be semimonotone with respect to ${\text {rec}}\,C$. Suppose that an unbounded ray occurs. Then the value of the auxiliary variable t is constant on that ray and $\varDelta z$, the variation in z, is nonzero and satisfies

$$\begin{aligned} \varDelta z \in {\text {rec}}\,C,\qquad M\varDelta z \in {({\text {rec}}\,C)}^{D}, \qquad \text {and}\qquad \varDelta z^TM\varDelta z = 0. \end{aligned}$$

(14)

Proof

The fact that t is constant and that $\varDelta z$ is a solution to (14) follows from the first part of the proof of Theorem 4.4 in [6]. To see that the direction $\varDelta z$ is nonzero, we proceed by contradiction: at the current iterate $(x^{k}, t^{k})$ we have

$$\begin{aligned} G_{C}(x^{k}, t^{k})&= Mz^{k} + q + x^{k} - z^{k} - t^{k} r = 0. \end{aligned}$$

(15)

Let $x^{k+1}$ belong to the unbounded ray and suppose that $\varDelta z = 0$:

$$\begin{aligned} G_{C}(x^{k+1}, t^{k})&= Mz^{k} + q + x^{k+1} - z^{k} - t^{k} r = 0. \end{aligned}$$

It immediately follows that $x^{k+1} = x^{k}$. $\square$