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Solving a class of variational inequalities with inexact oracle operators

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  • Deren Han
  • Wei Xu
  • Hai Yang

Abstract

Consider a class of variational inequality problems of finding $${x^*\in S}$$ , such that $$f(x^*)^\top (z-x^*)\geq 0,\quad \forall z\in S,$$ where the underlying mapping f is hard to evaluate (sometimes its explicit form is unknown), and S has the following structure $$S=\{x\in R^n\; | \; Ax\le b, x\in K\}.$$ For any given Lagrangian multiplier y associated with the linear inequality constraint in S, a solution of the relaxed variational inequality problem of finding $${\hat x\in K}$$ , such that $$(x^\prime-\hat x)^\top (f(\hat x)+A^\top y)\geq 0 \quad\forall x^\prime \in K \quad\quad\quad\quad (1)$$ can be given by an oracle. This class of problems arises frequently in economics and engineering. In this paper, we focus on considering the above problems where the underlying mapping f, though is unknown, is strongly monotone. We propose an iterative method for solving this class of variational inequality problems. At each iteration, the method consists of two steps: predictor and corrector. At the predictor step, a trial multiplier is given and the oracle is called for a solution of the relaxed variational inequality problem (1); then at the corrector step, the multiplier y is updated, using the information from the predictor step. We allow the oracle to give just an inexact solution of the relaxed variational inequality problem at the predictor step, which makes the method very efficient and practical. Under some suitable conditions, the global convergence of the method is proved. Some numerical examples are presented to illustrate the method. Copyright Springer-Verlag 2010

Suggested Citation

  • Deren Han & Wei Xu & Hai Yang, 2010. "Solving a class of variational inequalities with inexact oracle operators," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 71(3), pages 427-452, June.
  • Handle: RePEc:spr:mathme:v:71:y:2010:i:3:p:427-452
    DOI: 10.1007/s00186-009-0299-0
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    References listed on IDEAS

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    1. B. S. He & L. Z. Liao, 2002. "Improvements of Some Projection Methods for Monotone Nonlinear Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 112(1), pages 111-128, January.
    2. D. Han, 2007. "Inexact Operator Splitting Methods with Selfadaptive Strategy for Variational Inequality Problems," Journal of Optimization Theory and Applications, Springer, vol. 132(2), pages 227-243, February.
    3. Li, Michael Z. F., 2002. "The role of speed-flow relationship in congestion pricing implementation with an application to Singapore," Transportation Research Part B: Methodological, Elsevier, vol. 36(8), pages 731-754, September.
    4. Anna Nagurney & Padma Ramanujam, 1996. "Transportation Network Policy Modeling with Goal Targets and Generalized Penalty Functions," Transportation Science, INFORMS, vol. 30(1), pages 3-13, February.
    5. Yang, Hai & Bell, Michael G. H., 1997. "Traffic restraint, road pricing and network equilibrium," Transportation Research Part B: Methodological, Elsevier, vol. 31(4), pages 303-314, August.
    6. Liqun Qi, 1999. "Regular Pseudo-Smooth NCP and BVIP Functions and Globally and Quadratically Convergent Generalized Newton Methods for Complementarity and Variational Inequality Problems," Mathematics of Operations Research, INFORMS, vol. 24(2), pages 440-471, May.
    7. Yang, Hai & Meng, Qiang & Lee, Der-Horng, 2004. "Trial-and-error implementation of marginal-cost pricing on networks in the absence of demand functions," Transportation Research Part B: Methodological, Elsevier, vol. 38(6), pages 477-493, July.
    8. Stella Dafermos, 1980. "Traffic Equilibrium and Variational Inequalities," Transportation Science, INFORMS, vol. 14(1), pages 42-54, February.
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