Regularization Algorithms for Solving Monotone Ky Fan Inequalities with Application to a Nash-Cournot Equilibrium Model

Iterative methods for solving monotone equilibrium problems via dual gap functions

“…which proves the first inequality of (13). From the choice of ρ and since λ k+1 > 0, we have λ k+1 ( ρ 2 − L 2 2(ρ+τ ) ) ≥ 0.…”

“…We prove that the rate of convergence of the algorithm is linear. It is noticeable that the global contraction rate is better than the projection method proposed in [9,13] when the condition number of (PEP) is greater than (2 + √ 5) 1/2 that is often the case in practice. However, as a compensation, in each iteration of the algorithm, two convex programming problems need to be solved instead of one as in the projection method.…”

“…This problem is also referred as Ky-Fan's inequality due to his results in this field [13]. Associated with the primal form (PEP), its dual form is defined as follows:…”

“…Problem (PEP) on one hand covers many practical problems in optimization and nonlinear analysis such as optimization problems, variational inequalities, complementarity problems, fixed point problems and Nash equilibrium models [1,13,16,18,19]. On the other hand, it is resulted from many practical problems in economics, transportation, mechanics and engineering.…”

“…Other solution methods have been well developed in mathematical programming and variational inequalities such as gap function-based, extragradient and bundle methods [5,[14][15][16] recently have been extended to equilibrium problems [13,19,22].…”