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Interior Proximal Method Without the Cutting Plane Property

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  • Nils Langenberg

    (Universität Trier)

Abstract

A new interior proximal method for variational inequalities with generalized monotone operators is developed. It transforms a given variational inequality (which, maybe, is constrained and ill-posed) into unconstrained and well-posed equations as well as, at each iteration, one single additional extragradient step with rather small numerical efforts. Convergence is established under mild assumptions: The frequently assumed maximal monotonicity is weakened to pseudo- and quasimonotonicity with respect to the solution set, and a wide class of even nonlinearly constrained feasible sets is allowed for. In this general setting, the presented scheme constitutes the first interior proximal method that works without the so-called cutting plane property. Such a demanding assumption is completely left out, which allows to solve, e.g., wide classes of saddle point and equilibrium problems by means of an interior proximal method for the first time. As another application, we study variational inequalities derived from quasiconvex optimization problems.

Suggested Citation

  • Nils Langenberg, 2015. "Interior Proximal Method Without the Cutting Plane Property," Journal of Optimization Theory and Applications, Springer, vol. 166(2), pages 529-557, August.
    • Handle: RePEc:spr:joptap:v:166:y:2015:i:2:d:10.1007_s10957-014-0605-8
    DOI: 10.1007/s10957-014-0605-8
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    References listed on IDEAS

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    1. Jonathan Eckstein & Paulo Silva, 2010. "Proximal methods for nonlinear programming: double regularization and inexact subproblems," Computational Optimization and Applications, Springer, vol. 46(2), pages 279-304, June.
    2. Nils Langenberg, 2012. "An Interior Proximal Method for a Class of Quasimonotone Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 155(3), pages 902-922, December.
    3. Nils Langenberg, 2010. "Pseudomonotone operators and the Bregman Proximal Point Algorithm," Journal of Global Optimization, Springer, vol. 47(4), pages 537-555, August.
    4. D. Aussel & N. Hadjisavvas, 2004. "On Quasimonotone Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 121(2), pages 445-450, May.
    5. M. V. Solodov & B. F. Svaiter, 2000. "An Inexact Hybrid Generalized Proximal Point Algorithm and Some New Results on the Theory of Bregman Functions," Mathematics of Operations Research, INFORMS, vol. 25(2), pages 214-230, May.
    6. Regina S. Burachik & Alfredo N. Iusem, 2008. "Set-Valued Mappings and Enlargements of Monotone Operators," Springer Optimization and Its Applications, Springer, number 978-0-387-69757-4, June.
    7. Regina S. Burachik & Alfredo N. Iusem, 2008. "Enlargements of Monotone Operators," Springer Optimization and Its Applications, in: Set-Valued Mappings and Enlargements of Monotone Operators, chapter 0, pages 161-220, Springer.
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