Existence and solution methods for equilibria

Abstract

Equilibrium problems provide a mathematical framework which includes optimization, variational inequalities, fixed-point and saddle point problems, and noncooperative games as particular cases. This general format received an increasing interest in the last decade mainly because many theoretical and algorithmic results developed for one of these models can be often extended to the others through the unifying language provided by this common format. This survey paper aims at covering the main results concerning the existence of equilibria and the solution methods for finding them

Generalized Nash equilibria for SaaS/PaaS Clouds

“…Constraint family (15) states that X k cannot be larger than the actual arrival rate Λ k and smaller than the minimum admission rate λ k . Constraint family (16) entails that the response time for application k negotiated with the PaaS is smaller than R S k . As in the previous section, constraints (17) and (18) guarantee that the application response time satisfies the SLA and that physical servers are not saturated, respectively.…”

“…[16,19,23,26,37]), which is an extension of the classical Nash equilibrium problem [35], in which both the objective function and the feasible region of each player depend on the strategies chosen by the other players. We then use results from game theory to develop efficient algorithms for the run-time management and allocation of PaaS resources to competing SaaSs.…”

Gap functions for quasi-equilibria

“…Though descent type methods based on gap functions have been extensively developed for EPs (see, for instance, [49,44,12,43,45,6,8,9] and Section 3.2 in the survey paper [7]), the analysis of gap functions for QVIs is focused on smoothness properties [19,30,59,18,35] and error bounds [2,33] while no algorithm is developed. A descent method has been developed in [38] for jointly convex GNEPs; anyway, the choice of restricting to the computation of normalized equilibria makes the problem actually fall within the EP (and not the QEP) framework.…”

“…It is also worth noting that (QEP ) is a natural generalization of the so-called abstract equilibrium problem (shortly EP), i.e., the case in which the set-valued map C is constant. As EP subsumes optimization, multiobjective optimization, variational inequalities, fixed point and complementarity problems, Nash equilibria in noncooperative games and inverse optimization in a unique mathematical model (see, for instance, [10,7]), further "quasi" type models could be analysed through the QEP format beyond QVIs and GNEPs.…”

Service Provisioning Problem in Cloud and Multi-Cloud Systems

“…To capture the behavior of SaaSs and IaaSs in this conflicting situation, in which the best choice for one depends on the choices of the others, we recur to the Generalized Nash Equilibrium (GNE) concept (see e.g., (Bigi et al 2013, Cavazzuti et al 2002, Debreu 1952, Facchinei and Kanzow 2010a, Rosen 1965). GNE is an extension of the Nash equilibrium, in which not only the objective function but also the feasible region of each player depend on the strategies chosen by the other players.…”