Lagrangian duality and related multiplier methods for variational inequality problems

Alfred Auslender*, Marc Teboulle

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


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 (LQP) theory introduced by the authors, we derive three algorithms for solving the variational inequality (VI) 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.

Original languageEnglish
Pages (from-to)1097-1115
Number of pages19
JournalSIAM Journal on Optimization
Issue number4
StatePublished - 2000


  • Complementarity problems
  • Global convergence
  • Interior proximal methods
  • Lagrangian duality
  • Lagrangian multiplier methods
  • Variational inequalities


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