Solving variational inequalities with a quadratic cut method: a primal-dual, Jacobian-free approach

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Abstract

We extend in two directions the Analytic Center, Cutting Plane Method for Variational Inequalities with quadratic cuts, ACCPM-VI(quadratic cuts), introduced by Denault and Goffin in 1998. First, we define a primal–dual method to find the analytic center at each iteration. Second, the Broyden–Fletcher–Goldfarb–Shanno Jacobian approximation, of quasi-Newton fame, is used in the definition of the cuts, making the algorithm applicable to problems without tractable Jacobians. The algorithm is tested on a variety of variational inequality problems, including one challenging problem of pricing the pollution permits put forward in the Kyoto Protocol.

Introduction

Finite-dimensional variational inequalities, and the closely related complementarity problems, have been a focus of renewed attention in the mathematical programming and operations research communities. The natural ability of variational inequalities to model equilibrium problems has led to numerous applications in the areas of transportation, economics, finance, environment, engineering, etc. The development of algorithms to solve variational inequalities has also been intense, and has assimilated many advances of other branches of mathematical programming, e.g. interior-point methods.

In this paper, the ACCPM-VI (quadratic cuts) approach of [1] is implemented in a primal-dual, infeasible setting; primal-dual methods, while more difficult to analyze, are robust and have a built-in centrality measure. Of great practical interest, we give promising results on the use of approximations of the Jacobian information that is required by the quadratic cut. The approximation is done with the BFGS matrix, in the spirit of quasi-Newton methods. The algorithm is admittedly heuristic, in that we do not provide a convergence proof; algorithms that drop cuts, as this one, are notoriously difficult to analyze.

The use of “quasi-Jacobian”, temporary, quadratic cuts allows us to solve efficiently a particularly difficult variational inequality that arises in CO2 emission permits trading (see [2]); this VI is such that the Jacobian of the associated mapping is not defined.

Following this introduction, Section 2 presents the generic ACCPM-VI (quadratic cuts) algorithm introduced in [1], while recalling some definitions and results on variational inequalities and analytic center. In Section 3, we derive our primal-dual method to locate an analytic center. We discuss in Section 4 the use of Jacobian proxies, and provide in Section 5 the results of our numerical experiments. We conclude in Section 6.

Section snippets

Solution of variational inequalities as convex feasibility problems: a quadratic cut approach

We begin, in Section 2.1, with a succinct overview of the literature on cutting plane approaches for variational inequalities. Then in Section 2.2, we introduce more formally the variational inequality, the convex feasibility problem, and the theorem that links them. Section 2.3 defines and motivates the quadratic cuts. Section 2.4 introduces the analytic center and other related concepts. Finally, in Section 2.5, we bring together linear cuts, quadratic cuts, and analytic centers in an

Primal–dual updating and recentering steps to the next analytic center

We describe here a primal–dual method for computing the next analytic center yk+1, that of YkQ(yk), when the quadratic cut Q(yk) is introduced at the current point yk. More than one step may be necessary to reach yk+1 from yk, and it is useful to distinguish the first step out of yk, called the updating step, from the following ones, called centering steps. The distinction is needed because of a special difficulty with the updating step: since yk lies on the cut, the corresponding slack value

Jacobian matrix approximations

Faced with an algorithm that uses derivative information (the Jacobian ∇F), one asks if it is not possible to keep the spirit of the method while avoiding the derivatives evaluations. The answer, often, is yes; in optimization, the idea led to the quasi-Newton methods. In our case, the use of Jacobian approximations based on mapping evaluations can also be fruitful.

Drawing from well-established optimization theory, we use a Broyden–Fletcher–Goldfarb–Shanno (BFGS) matrix for the Jacobian

Notes on the implementation

Here are some details on our implementation of the primal-dual algorithm described above. We used the matrix computation software MATLAB for most computations; the linear programming software CPLEX was used for the gap evaluations.

Sparsity is exploited whenever possible. Within the matrix A, there are columns from the definition of the feasible set Y of the VI(F,Y), usually sparse, and columns from the linear cuts of the algorithm, always dense in our applications. The two categories are

Conclusion

In this paper, we have introduced a primal–dual algorithm to find the analytic centers used in the ACCPM-VI (quadratic cut) approach of [1]. Primal–dual techniques are robust, and in opposition to dual methods, no supplementary effort is required to obtain a centrality measure. We also present numerical evidence that Jacobian-proxied cuts, based on first-order information, can perform as well as cuts using the Jacobian itself.

The algorithm is successfully used to solve a number of problems from

Acknowledgements

We would like to thank B. Büeler, H.-J. Lüthi, and J.-Ph. Vial for the discussions we had with them about this paper. We specifically want to express our gratitude to Büeler and Lüthi for making both their MMMR model and their computer available to us for testing.

Finally, we sincerely thank the two anonymous referees for their insightful comments on the first version of the paper.

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1

Research supported by NSERC grants and McGill University Fellowships.

2

Research supported by NSERC and FCAR grants.

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