Asymptotic Behavior of Regularized OptimizationProblems with Quasi-variational Inequality Constraints
The great interest into hierarchical optimization problems and the increasing use of game theory in many economic or engineering applications led to investigate optimization problems with constraints described by the solutions to a quasi-variational inequality (variational problems having constraint sets depending on their own solutions, present in many applications as social and economic networks, financial derivative models, transportation network congestion and traffic equilibrium). These problems are bilevel problems such that at the lower level a parametric quasi-variational inequality is solved (by one or more followers) meanwhile at the upper level the leader solves a scalar optimization problem with constraints determined by the solutions set to the lower level problem. In this paper, mainly motivated by the use of approximation methods in infinite dimensional spaces (penalization, discretization, Moreau-Yosida regularization ...), we are interested in the asymptotic behavior of the sequence of the infimal values and of the sequence of the minimum points of the upper level when a general scheme of perturbations is considered. Unfortunately, we show that the global convergence of exact values and exact solutions of the perturbed bilevel problems cannot generally be achieved. Thus, we introduce suitable concepts of regularized optimization problems with quasi-variational inequality constraints and we investigate, in Banach spaces, the behavior of the approximate infimal values and of the approximate solutions under and without perturbations.
|Date of creation:||28 Dec 2013|
|Date of revision:|
|Publication status:||Forthcoming in Journal of Mathematical Analysis and Applications|
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- M. Lignola & Jacqueline Morgan, 2012. "Approximate values for mathematical programs with variational inequality constraints," Computational Optimization and Applications, Springer, vol. 53(2), pages 485-503, October.
- Jong-Shi Pang & Masao Fukushima, 2005. "Quasi-variational inequalities, generalized Nash equilibria, and multi-leader-follower games," Computational Management Science, Springer, vol. 2(1), pages 21-56, 01.
- Laurent Drouet & Alain Haurie & Francesco Moresino & Jean-Philippe Vial & Marc Vielle & Laurent Viguier, 2008. "An oracle based method to compute a coupled equilibrium in a model of international climate policy," Computational Management Science, Springer, vol. 5(1), pages 119-140, February.
- Jacqueline Morgan & Fabien Prieur, 2011. "Global emission ceiling versus international cap and trade: What is the most efficient system when countries act non-cooperatively?," CSEF Working Papers 275, Centre for Studies in Economics and Finance (CSEF), University of Naples, Italy.
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