A new error bound result for Generalized Nash Equilibrium Problems and its algorithmic application

Abstract

We present a new algorithm for the solution of Generalized Nash Equilibrium Problems. This hybrid method combines the robustness of a potential reduction algorithm and the local quadratic convergence rate of the LP-Newton method. We base our local convergence theory on a local error bound and provide a new sufficient condition for it to hold that is weaker than known ones. In particular, this condition implies neither local uniqueness of a solution nor strict complementarity. We also report promising numerical results. © 2013 Springer Science+Business Media New York

On error bounds and Newton-type methods for generalized Nash equilibrium problems

“…For instance, our approach does not require any auxiliary variables and any additional constraints. On the other hand, unlike our analysis of Newton methods above, local quadratic convergence of the method in [7] does not require strict complementarity. However, in the case of strict complementarity our assumptions for local quadratic convergence of Newtonian methods become weaker than those in [7].…”

“…Remark 4 Subsequently to the original version of this paper, there appeared an independent technical report [7] concerned with similar issues, namely, primal-dual error bounds and Newton-type methods for GNEP.…”

“…In our setting (joint constraints only) Assumption 1 in [7] needed to obtain an error bound consists of saying that there exists a partition (I 1 , I 2 ) of A (i.e., I 1 ∪ I 2 = A, I 1 ∩ I 2 = ∅) such that I 1 ⊂ A …”

“…Finally, the two error bounds are of different nature, as explained next. The error bound in [7] is valid only on the set including all nonnegativity conditions, and on this set the residual used in the error bound is smooth. Our residual is generally nonsmooth, and the error bound is on the entire space.…”

“…As for the Newtonian methods in this paper and in [7], we emphasize that they are different. Each may have some advantages (as well as disadvantages, of course).…”