A Projected Subgradient Algorithm for Bilevel Equilibrium Problems and Applications

A self-adaptive extragradient–CQ method for a class of bilevel split equilibrium problem with application to Nash Cournot oligopolistic electricity market models

“…In this section, some numerical results are presented to compare the convergence behavior of the proposed Algorithm 1 with that of Van Quy's Algorithm (6). Example 3.9 This example focuses on applying the previous results to a real-world problem of the Nash-Cournot oligopolistic electricity market equilibrium model involving three electricity companies j, ( j = 1, 2, 3) (see Quoc et al 2012;Thuy and Hai 2017;Yen et al 2016) in which each company j has its own, independent generating units with index set I j . Suppose that I 1 = {1}, I 2 = {2, 3} and I 3 = {4, 5, 6}.…”

“…Then, the Nash-Cournot equilibrium models of electricity markets can be reformulated as an equilibrium problem (see Thuy and Hai 2017;Van Quy 2018):…”

“…Here, f and g are bifunctions from C × C to R satisfying f (x, x) = 0 = g(x, x), for all x ∈ C. In the last few years, bilevel problems have aroused considerable research interests in nonlinear analysis due to its wide variety of applications in different fields ranging from transportation engineering (optimal design, optimal chemical equilibria, and network design, see, e.g., Iiduka 2012;Dempe 2003;Luo et al 1996) to management economics (network facility location, Stackelberg games, principal-agent problem, taxation, optimal pricing, policy decisions, and coordination of multi-divisional firms, see, e.g., Mastroeni 2003;Thuy and Hai 2017;Luo et al 1996;Dempe 2003…”

Extragradient subgradient methods for solving bilevel equilibrium problems

“…In 2014, Quy [23] introduced the algorithm by combining the proximal method with the Halpern method for solving bilevel monotone equilibrium and fixed point problem. For more details and most recent works on the methods for solving bilevel equilibrium problems, we refer the reader to [2, 5, 24]. The authors considered the method for monotone and pseudoparamonotone equilibrium problem.…”

Contraction-Mapping Algorithm for the Equilibrium Problem Over the Fixed Point Set of a Nonexpansive Semigroup

“…Let C be a nonempty, closed, convex subset of a real Hilbert space H and f : C × C → R be a bifunction such that f (x, x) = 0 for all x ∈ C. Such a bifunction is called an equilibrium bifunction. The equilibrium problem for f on C can be formulated as others; see, for instance, [1,9,10,11,12,14,19,20,25] and references therein. Equilibrium problems have been generalized and extensively studied in many directions due to its importance.…”