Gap Function Approach to the Generalized Nash Equilibrium Problem |
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Authors: | K Kubota M Fukushima |
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Institution: | (3) Institute Mat. Pura e Apl., Rio de Janeiro, Brazil; |
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Abstract: | We consider an optimization reformulation approach for the generalized Nash equilibrium problem (GNEP) that uses the regularized
gap function of a quasi-variational inequality (QVI). The regularized gap function for QVI is in general not differentiable,
but only directionally differentiable. Moreover, a simple condition has yet to be established, under which any stationary
point of the regularized gap function solves the QVI. We tackle these issues for the GNEP in which the shared constraints
are given by linear equalities, while the individual constraints are given by convex inequalities. First, we formulate the
minimization problem involving the regularized gap function and show the equivalence to GNEP. Next, we establish the differentiability
of the regularized gap function and show that any stationary point of the minimization problem solves the original GNEP under
some suitable assumptions. Then, by using a barrier technique, we propose an algorithm that sequentially solves minimization
problems obtained from GNEPs with the shared equality constraints only. Further, we discuss the case of shared inequality
constraints and present an algorithm that utilizes the transformation of the inequality constraints to equality constraints
by means of slack variables. We present some results of numerical experiments to illustrate the proposed approach. |
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