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BIRS Workshop Lecture Videos

Sharp Penalty Mapping Approach to Approximate Solution of Monotone Variational Inequalities Nurminski, Evgeni

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We consider here a generalization of exact penalty functions approach to solution of variational inequalities. Instead of more common potential mappings, associated with the gradient field of a penalty function the oriented field of a sharp penalty mapping is considered. Linearly combined with the variational inequality operator it directs the iteration method toward approximate feasibility and optimality,which is unavoidably only approximate as the exact balance between these two acting forces can not being attained beforehand.
Nevertheless the accuracy of the limit points can be controlled by the parameters of the scheme and can be made arbitrary high and in the dynamic variants of this idea can even guarantee the exact solution of a monotone variational inequality. If one is content with the finite accuracy, the notion of sharp penalty mappings gives an additional freedom in construction of iteration schemes for approximate solution of variational inequalities with controllable accuracy. Due to some advantages of iteration methods, such as low memory requirements, data splitting, parallel execution, etc. it may be of interest for solution of variational inequalities of high dimension in particular for transportation equilibrium problems.
Some part of the talk is devoted to a new approach for studying convergence properties of iteration processes arising from the above.Typically convergence proofs are based on Lyapunov-like statements about relaxation properties of a related sequences of values of convergence indicators. There are several general nontrivial methods for establishing such properties, but the analysis of algorithms for obtaining approximate solutions of optimization and equilibrium problems excludes the very idea of such convergence so new techniques used toward the validation of iteration methods were developed and will be discussed in this report.

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