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Lagrangian duality and related multiplier methods for variational inequality problems. (English) Zbl 0996.49005

Summary: We consider a new class of multiplier interior point methods for solving variational inequality problems with maximal monotone operators and explicit convex constraint inequalities. Developing a simple Lagrangian duality scheme which is combined with the recent logarithmic-quadratic proximal theory introduced by the authors, we derive three algorithms for solving the variational inequality problem. This provides a natural extension of the methods of multipliers used in convex optimization and leads to smooth interior point multiplier algorithms. We prove primal, dual, and primal-dual convergence under very mild assumptions, eliminating all the usual assumptions used until now in the literature for related algorithms. Applications to complementarity problems are also discussed.

MSC:

49J40 Variational inequalities
90C33 Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming)
49N15 Duality theory (optimization)
65K10 Numerical optimization and variational techniques
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