Auslender, Alfred; Teboulle, Marc Lagrangian duality and related multiplier methods for variational inequality problems. (English) Zbl 0996.49005 SIAM J. Optim. 10, No. 4, 1097-1115 (2000). 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. Cited in 2 ReviewsCited in 55 Documents 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 Keywords:Lagrangian duality; variational inequalities; interior proximal methods; complementarity problems; global convergence; Lagrangian multiplier methods; logarithmic-quadratic proximal theory PDFBibTeX XMLCite \textit{A. Auslender} and \textit{M. Teboulle}, SIAM J. Optim. 10, No. 4, 1097--1115 (2000; Zbl 0996.49005) Full Text: DOI