Interior Proximal Method Without the Cutting Plane Property

Nils Langenberg ()
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Nils Langenberg: Universität Trier

Journal of Optimization Theory and Applications, 2015, vol. 166, issue 2, No 9, 529-557

Abstract: 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.

Keywords: Variational inequalities; Bregman distances; Proximal Point Algorithm; Extragradient step; Generalized monotonicity; Interior point effect; 47H05; 47J20; 65J20; 65K10; 90C26 (search for similar items in EconPapers)
Date: 2015
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DOI: 10.1007/s10957-014-0605-8

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