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Abstract: The generalized Nash equilibrium problem (GNEP) is a kind of game to find strategies for a group of players such that each player’s objective function is optimized. Solutions for GNEPs are called generalized Nash equilibria (GNEs). In this paper, we propose a numerical method for finding GNEs of GNEPs of polynomials based on the polyhedral homotopy continuation and the Moment-SOS hierarchy of semidefinite relaxations. We show that our method can find all GNEs if they exist, or detect the nonexistence of GNEs, under some genericity assumptions. Some numerical experiments are made to demonstrate the efficiency of our method.

We consider the constrained piecewise Levenberg--Marquardt method globalized by linesearch, and apply it to "min" reformulations of the optimality systems for generalized Nash equilibrium problems. Numerical comparison of the performance of this method with some relevant existing alternatives is provided. [ABSTRACT FROM AUTHOR]

Electric vehicle (EV) has emerged as a crucial component in addressing both energy and environmental problems, and has become an essential part of nowadays' intelligent transportation systems. However, the charging demands of large amounts of EVs can put substantial pressure on power grid systems and cause potential grid congestion problems. In this article, we propose a hierarchical EV charging control strategy that considers the network‐wide communication overheads, computational complexity, total energy cost, EV user preferences, and data privacy protection. The hierarchical charging structure contains two phases, that is, a centralized control for EV aggregators and a distributed control for EVs within an aggregator. We prove that the centralized control with the objective to minimize the total energy cost of the power system constitutes a convex quadratic programming problem. Then a unique global optimum for the energy consumption profiles of EV aggregators can be achieved. The distributed charging control for EVs within an aggregator is studied using a generalized Nash equilibrium problem (GNEP). We show that the solution of the GNEP can be obtained via a variational inequality. Then the Solodov and Svaiter hyperplane projection method is employed to iteratively approach the variational equilibrium while ensuring the protection of EV users' data privacy. Extensive simulation studies are conducted to verify the correctness and effectiveness of our proposed hierarchical charging control algorithm for EVs. [ABSTRACT FROM AUTHOR]

Generation expansion planning (GEP) is a pivotal problem for power system planners due to increasing electric power consumption. This article aims to construct a methodology for GEP in power networks while incorporating the prohibited operating zones (POZ) and multi-fuel option (MFO) into the problem. The POZ and MFO are not considered in the existing GEP models, whereas these factors might considerably change the optimal planning scheme. A decentralized framework that relies on a generalized Nash equilibrium problem (GNEP) is considered in this paper, where several rival generation companies (Gencos) compete with each other to specify their own optimal decisions. Due to modeling the POZ and MFO in the GEP problem, integer variables are introduced, leading to a discretely constrained GNEP (DC-GNEP). Employing Karush–Kuhn–Tucker (KKT) optimality conditions, the DC-GNEP is recast as mixed-integer linear programming, which commercial branch-and-cut solvers can efficiently solve. Numerical experiments are performed on the IEEE 118-bus system to analyze and compare optimal GEP solutions. The numerical results point to the effectiveness of the proposed model considering POZ and MFO. [ABSTRACT FROM AUTHOR]

This paper studies convex generalized Nash equilibrium problems that are given by polynomials. We use rational and parametric expressions for Lagrange multipliers to formulate efficient polynomial optimization for computing generalized Nash equilibria (GNEs). The Moment-SOS hierarchy of semidefinite relaxations are used to solve the polynomial optimization. Under some general assumptions, we prove the method can find a GNE if there exists one, or detect nonexistence of GNEs. Numerical experiments are presented to show the efficiency of the method. [ABSTRACT FROM AUTHOR]

In recent years, shared energy storage has gained significant attention for mitigating the supply and demand imbalance caused by the intermittency of distributed renewable energy. Considering the subjective perception of prosumers when facing uncertainty, this paper proposes a new dynamic competitive on-demand renting framework for energy storage capacity (ESC) sharing to increase energy storage utilization, increase energy storage operator (ESO) profits, and reduce prosumer costs. In this framework, a demand-based dynamic capacity pricing mechanism is introduced, modeling the relationship between ESO and prosumers as a Stackelberg game while establishing a generalized Nash equilibrium (GNE) problem among prosumers. ESO determines the dynamic capacity pricing mechanism, while prosumers determine the hourly renting capacity based on demand. In capacity sharing, prospect theory is introduced for the first time to describe the subjective perceptions of prosumers when facing the uncertainty of renewable energy. Moreover, the existence of SE and the uniqueness of GNE are analyzed, followed by a summary and proposal of a method to determine the existence of equilibrium in a nested generalized non-cooperative Stackelberg game. Simulations show the effectiveness of the proposed framework on improving the ESC utilization rate, the impact of subjective perceptions on prosumers' decision-making, and the profit favorability of the correct estimation of subjective perceptions on ESO. Specifically, the framework increases ESO utilization by 24.07% and profit by 13.73%. • A new framework for dynamic on-demand energy storage sharing is proposed. • Prospect theory is introduced to describe the subjective uncertainty of prosumers. • A nested generalized non-cooperative Stackelberg game model is constructed. • Proven: Stackelberg equilibrium exists; Generalized Nash equilibrium is unique. [ABSTRACT FROM AUTHOR]

This paper provides a sufficient condition for the existence of solutions for generalized Nash equilibrium problems in the infinite-dimensional setting and with a countable (possibly infinite) number of players. The result has been achieved as a consequence of a modified version of Michael's selection theorem that works even when the range space is not metrizable and the set-valued map has not closed values. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the generalized Ekeland’s variational principle in several forms. The general setting of our results includes a graphical metric structure and also employs a generalized ພ-distance. We then applied the proposed variational principles to obtain existence theorems for a class of quasi-equilibrium problems whose constraint maps are induced from the graphical structure. The conditions used in our existence results are based on a very general concept called a convergence class. Finally, we deduce the existence of a generalized Nash equilibrium via its quasi-equilibrium reformulation. A validating example is also presented. [ABSTRACT FROM AUTHOR]

The concept of projected solution has been introduced in Aussel et al. (J Optim Theory Appl 170:818–837, 2016) for studying quasivariational problems where the constraint map may not be a self-map. Aim of this paper is to establish a new result on the existence of projected solutions for finite-dimensional quasiequilibrium problems without any monotonicity assumptions and without assuming the compactness of the feasible set. These two facts allow us to improve some recent results. Additionally, we deduce the existence of projected solutions for quasivariational inequalities, quasioptimization problems and generalized Nash equilibrium problems. Also, a comparison with similar results is provided. [ABSTRACT FROM AUTHOR]

Quasi-equilibrium problems (QEPs) allow a change of the feasible region together with the considered point, in contrast to standard equilibrium problems (EPs). In this work, we extend the Newton-type method for EPs considered by Santos et al. to the setting of QEPs. We prove locally the convergence of the method and we establish the superlinear/quadratic convergence of the algorithm to a solution of the problem under suitable assumptions. Applications to generalized Nash equilibrium problems (GNEPs) are considered. Some preliminary numerical results are reported showing the performance of the algorithm. [ABSTRACT FROM AUTHOR]

A generalized Nash equilibrium problem corresponds to a noncooperative interaction between a finite set of players in which the cost function and the feasible set of each player depend on the decisions of the others. The classical existence result for generalized equilibria due to Arrow and Debreu requires continuity of the cost functions. In this work, we provide an existence of solutions transferring this hypothesis to a "continuity-like" condition over the sublevel sets of the aforementioned functions. Comparison with Reny's approach for discontinuous games is also considered. [ABSTRACT FROM AUTHOR]

In this paper, we study a numerical approach to compute a solution of the generalized Nash equilibrium problem (GNEP). The GNEP is a potent modeling tool that has been increasingly developing in recent decades. Much of this development has centered around applying variational methods to the so-called GNSC, a useful but restricted subset of GNEP where each player has the same constraint set. One popular approach to solve the GNSC is to use the apparent separability of each player to build a decomposition method. This method has the benefit of being easily implementable and can be parallelized. Our aim in this paper is to show an extension of the decomposition method to a class of convex GNEP. We prove convergence of the proposed algorithm under a full convexity assumption. Then, we show numerical results on some examples to validate our approach and discuss the assumptions. [ABSTRACT FROM AUTHOR]

The "finite intersection property" for bifunctions is introduced and its relationship with generalized monotonicity properties is studied. Some characterizations are considered involving the Minty equilibrium problem. Also, some results concerning existence of equilibria and quasi-equilibria are established recovering several results in the literature. Furthermore, we give an existence result for generalized Nash equilibrium problems and variational inequality problems. [ABSTRACT FROM AUTHOR]

This paper concerns the generalized Nash equilibrium problem of polynomials (GNEPP). We apply the Gauss–Seidel method and Moment-SOS relaxations to solve GNEPPs. The convergence of the Gauss–Seidel method is known for some special GNEPPs, such as generalized potential games (GPGs). We give a sufficient condition for GPGs and propose a numerical certificate, based on Putinar's Positivstellensatz. Numerical examples for both convex and nonconvex GNEPPs are given for demonstrating the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

Pareto-based evolutionary multiobjective approaches are methods that use the Pareto dominance concept to guide the search of evolutionary algorithms towards the Pareto frontier of a problem. To address the challenge of providing an entire set of optimal solutions they use specially designed mechanisms for preserving search diversity and maintaining the non-dominated solutions set. The limitation of the Pareto dominance relation in high-dimensional spaces has rendered these methods inefficient for many-objective optimization. In this paper we aim to exploit existing Pareto-based methods to compute the generalized Nash equilibrium for multi-player games by replacing the Pareto dominance relation with an equilibrium generative relation. The generalized Nash equilibrium extends the Nash equilibrium concept by considering constraints over players' strategies. Numerical experiments indicate that the selected methods can be employed for equilibria computation even for games with up to twenty players. [ABSTRACT FROM AUTHOR]

The optimality system of a quasi-variational inequality can be reformulated as a non-smooth equation or a constrained equation with a smooth function. Both reformulations can be exploited by algorithms, and their convergence to solutions usually relies on the nonsingularity of the Jacobian, or the fact that the merit function has no nonoptimal stationary points. We prove new sufficient conditions for the absence of nonoptimal constrained or unconstrained stationary points that are weaker than some known ones. All these conditions exploit some properties of a certain matrix, but do not require the nonsingularity of the Jacobian. Further, we present new necessary and sufficient conditions for the nonsingularity of the Jacobian that are based on the signs of certain determinants. Additionally, we consider generalized Nash equilibrium problems that are a special class of quasi-variational inequalities. Exploiting their structure, we also prove some new sufficient conditions for stationarity and nonsingularity results. [ABSTRACT FROM AUTHOR]

Nash Equilibrium (NE) plays a crucial role in game theory. The relaxation method in conjunction with the Nikaido–Isoda (NI) function, namely the NI-based relaxation method, has been widely applied to the determination of NE. Genetic Algorithm (GA) with adaptive penalty is introduced and incorporated in the original NI-based relaxation method. The GA enhances the capability in the optimization step for computing the optimum response function. The optimization of the non-convex and non-concave NI function is made possible by GA. The proposed method thus combines the advantageous feature of the GA in its optimization capability and that of the relaxation method in its implementation simplicity together. The applicability of the method is shown through the illustrative examples, including the generalized Nash Equilibrium problem with nonlinear payoff functions and coupled constraints, the game with multiple strategic variables for individual players, and the non-differentiable payoff functions. All test example results suggest the appropriate crossover and mutation rate to be 0.05 and 0.002 for use in GA. These numbers are closed to the recommended values by DeJong. The proposed method shows its capability of finding correct NEs in all test examples.

In this paper, we propose a best-response approach to select an equilibrium in a two-player generalized Nash equilibrium problem. In our model we solve, at each of a finite number of time steps, two independent optimization problems. We prove that convergence of our Jacobi-type method, for the number of time steps going to infinity, implies the selection of the same equilibrium as in a recently introduced continuous equilibrium selection theory. Thus the presented approach is a different motivation for the existing equilibrium selection theory, and it can also be seen as a numerical method. We show convergence of our numerical scheme for some special cases of generalized Nash equilibrium problems with linear constraints and linear or quadratic cost functions. [ABSTRACT FROM AUTHOR]

In this paper, we utilize a new homotopy method to solve generalized Nash equilibrium problem with equality and inequality constraints on unbounded sets. Based on the existing homotopy method, we establish a new homotopy equation by introducing a suitable perturbation on the equality constraint, the existence and the global convergence of homotopy path under certain assumptions have also been proved. In the proposed method, the initial point only needs to satisfy the inequality constraints. Compared with the existing homotopy method, this method expands the scope of the initial points and provides the convenience for solving generalized Nash equilibrium problem. The numerical results illustrate the effectiveness of this method. [ABSTRACT FROM AUTHOR]

As the main air pollution control pattern used in China, the independent emission reduction (IER) model mandated that each province reduce its air pollutant independently. Under this model, provinces paid huge pollutant removal costs because they could not select the optimal pollutant removal rate based on their emission reduction capacity and their marginal cost of pollutant reduction. To overcome the ineffectiveness and high cost of the IER model, a generalized Nash equilibrium game (GNEG) model was constructed to achieve the State-set regional emission reduction target while minimizing the pollutant removal cost for each sub-region. Next, the case of Yangtze River Delta region was investigated using the GNEG model. Further, an incentive mechanism was suggested to improve the willingness of all participants. Result shows that the optimal SO 2 removal solution in the GNEG model saved US$4.8 × 107 compared with the IER model, amounting to 3.1% of the total pollutant removal cost. Finally, a sensitivity analysis was performed to simulate the effects of changes in the State-set SO 2 reduction targets and the upper and lower bounds on the optimal pollutant removal rate and the removal cost of each province. The proposed model will help policy makers develop pollutant-reduction strategies that improve the effectiveness and economic benefits of pollution control. One province's emission-reduction strategy is determined not only by its own decision variable but also by the strategy of the rivals. The generalized Nash equilibrium game (GNEG) model was constructed to achieve the State-set regional emission reduction target while minimizing the pollutant removal cost for each sub-region. The case of Yangtze River Delta region was investigated. Results showed that the SO 2 removal cost for the whole region in the proposed GNEG model is significantly lower than that in the traditional IER model. An incentive mechanism was established to improve the willingness of all participants. Image 106668 • The model helps the region achieve the emission reduction target at minimum cost. • The model saved 48 million dollars, amounting to 3.1% of the total cost. • An incentive mechanism to improve the willingness was suggested. • The sensitivity analysis shows what measures offer an attractive return. [ABSTRACT FROM AUTHOR]

Recently a new solution concept for generalized Nash equilibrium problems was published by the author. This concept selects a reasonable equilibrium out of the typically infinitely many. The idea is to model the process of finding a compromise by solving parametrized generalized Nash equilibrium problems. In this paper we propose an algorithmic realization of the concept. The model produces a solution path, which is under suitable assumptions unique. The algorithm is a homotopy method that tries to follow this path. We use semismooth Newton steps as corrector steps in our algorithm in order to approximately solve the generalized Nash equilibrium problems for each given parameter. If we have a unique solution path, we need three additional theoretical assumptions: a stationarity result for the merit function, a coercivity condition for the constraints, and an extended Mangasarian–Fromowitz constraint qualification. Then we can prove convergence of our semismooth tracing algorithm to the unique equilibrium to be selected. We also present convincing numerical results on a test library of problems from literature. The algorithm also performs well on a number of problems that do not satisfy all the theoretical assumptions. [ABSTRACT FROM AUTHOR]

Pessimistic bilevel optimization problems, as do optimistic ones, possess a structure involving three interrelated optimization problems. Moreover, their finite infima are only attained under strong conditions. We address these difficulties within a framework of moderate assumptions and a perturbation approach which allows us to approximate such finite infima arbitrarily well by minimal values of a sequence of solvable single-level problems. To this end, as has already been done for optimistic problems, for the first time in the literature we introduce the standard version of the pessimistic bilevel problem. For its algorithmic treatment, we reformulate it as a standard optimistic bilevel program with a two follower Nash game in the lower level. The latter lower-level game, in turn, is replaced by its Karush--Kuhn--Tucker conditions, resulting in a single-level mathematical program with complementarity constraints. We prove that the perturbed pessimistic bilevel problem, its standard version, the two follower game, as well as the mathematical program with complementarity constraints are equivalent with respect to their global minimal points. We study the more intricate connections between their local minimal points in detail. As an illustration, we numerically solve a regulator problem from economics for different values of the perturbation parameters. [ABSTRACT FROM AUTHOR]

This paper deals with generalized Nash equilibrium problems (GNEPs) in Banach spaces. We give an existence result for normalized equilibria of jointly convex GNEPs and then propose an augmented Lagrangian-type algorithm for their computation. A thorough convergence analysis is conducted which considers the existence of subproblem solutions as well as the feasibility and optimality of limit points. We then apply our investigations to differential economic games and multiobjective optimal control problems governed by linear partial differential equations. Numerical results are provided to demonstrate the practical performance of the method. [ABSTRACT FROM AUTHOR]

We define and discuss different enumerative methods to compute solutions of generalized Nash equilibrium problems with linear coupling constraints and mixed-integer variables. We propose both branch-and-bound methods based on merit functions for the mixed-integer game, and branch-and-prune methods that exploit the concept of dominance to make effective cuts. We show that under mild assumptions the equilibrium set of the game is finite and we define an enumerative method to compute the whole of it. We show that our branch-and-prune method can be suitably modified in order to make a general equilibrium selection over the solution set of the mixed-integer game. We define an application in economics that can be modelled as a Nash game with linear coupling constraints and mixed-integer variables, and we adapt the branch-and-prune method to efficiently solve it. [ABSTRACT FROM AUTHOR]

We prove an existence result for the time-dependent generalized Nash equilibrium problem under generalized convexity without neither a quasi-variational inequality reformulation nor a quasi-equilibrium problem reformulation. Furthermore, an application to the time-dependent abstract economy is considered. [ABSTRACT FROM AUTHOR]

We consider the generalized Nash equilibrium problem in a Hilbert space setting. The joint constraints are eliminated by an augmented Lagrangian-type approach, and we present a fully distributed version by using ideas from alternating direction methods of multipliers (ADMM methods). Convergence follows, under a cocoercivity condition, from the fact that this method can be interpreted as a suitable splitting approach in our Hilbert space endowed with a modified scalar product. This observation also leads to a second algorithmic approach, which yields convergence under a Lipschitz assumption and monotonicity. Numerical results are presented for some examples arising in both finite- and infinite-dimensional Hilbert spaces. [ABSTRACT FROM AUTHOR]

We propose a new solution concept for generalized Nash equilibrium problems. This concept leads, under suitable assumptions, to unique solutions, which are generalized Nash equilibria and the result of a mathematical procedure modeling the process of finding a compromise. We first compute the favorite strategy for each player, if he could dictate the game, and use the best response on the others’ favorite strategies as starting point. Then, we perform a tracing procedure, where we solve parametrized generalized Nash equilibrium problems, in which the players reduce the weight on the best possible and increase the weight on the current strategies of the others. Finally, we define the limiting points of this tracing procedure as solutions. Under our assumptions, the new concept selects one reasonable out of typically infinitely many generalized Nash equilibria. [ABSTRACT FROM AUTHOR]

We present a new covering theorem for a nonlinear mapping on a convex cone, under the assumptions weaker than the classical Robinson’s regularity condition. When the latter is violated, one cannot expect to cover the entire neighborhood of zero in the image space. Nevertheless, our covering theorem gives rise to natural conditions guaranteeing stability of a solution of a cone-constrained equation subject to wide classes of perturbations, and allowing for nonisolated solutions, and for systems with the same number of equations and variables. These features make these results applicable to various classes of variational problems, like nonlinear complementarity problems. We also consider the related stability issues for generalized Nash equilibrium problems. [ABSTRACT FROM AUTHOR]

The generalized Nash equilibrium problem (GNEP) is often difficult to solve by Newton-type methods since the problem tends to have locally nonunique solutions. Here we take an existing trust-region method which is known to be locally fast convergent under a relatively mild error bound condition, and modify this method by a nonmonotone strategy in order to obtain a more reliable and efficient solver. The nonmonotone trust-region method inherits the nice local convergence properties of its monotone counterpart and is also shown to have the same global convergence properties. Numerical results indicate that the nonmonotone trust-region method is significantly better than the monotone version, and is at least competitive to an existing software applied to the same reformulation used within our trust-region framework. Additional tests on quasi-variational inequalities (QVI) are also presented to validate efficiency of the proposed extension. [ABSTRACT FROM AUTHOR]

We consider a model of pay-as-clear electricity market based on a Equilibrium Problem with Complementarity Constraints approach where the producers are playing a noncooperative game parameterized by the decisions of regulator of the market (ISO). In the proposed approach the bids are assumed to be convex quadratic functions of the production quantity. The demand is endogenously determined. The ISO problem aims to maximize the total welfare of the market. The demand being elastic, this total welfare take into account at the same time the willingness to pay of the aggregated consumer, as well as the cost of transactions. The market clearing will determine the market price in a pay-as-clear way. An explicit formula for the optimal solution of the ISO problem is obtained and the optimal price is proved to be unique. We also state some conditions for the existence of equilibria for this electricity market with elastic demand. Some numerical experiments on a simplified market model are also provided. [ABSTRACT FROM AUTHOR]

Summary: We consider optimal control problems with ordinary differential equations that are coupled by shared, possibly nonconvex, constraints. For these problems, we use the generalized Nash equilibrium approach and provide a reformulation of normalized Nash equilibria as solutions to a single optimal control problem. By this reformulation, we are able to prove existence, and in some settings, exploiting convexity properties, we also get a limited number or even uniqueness of the normalized Nash equilibria. Then, we use our approach to discuss traffic scenarios with several autonomous vehicles, whose dynamics is described through differential equations, and the avoidance of collisions couples the optimal control problems of the vehicles. For the solution to the discretized problems, we prove strong convergence of the states and weak convergence of the controls. Finally, using existing optimal control software, we show that the generalized Nash equilibrium approach leads to reasonable results for a crossing scenario with different vehicle models. [ABSTRACT FROM AUTHOR]

We consider generalized potential games, that constitute a fundamental subclass of generalized Nash equilibrium problems. We propose different methods to compute solutions of generalized potential games with mixed-integer variables, i.e., games in which some variables are continuous while the others are discrete. We investigate which types of equilibria of the game can be computed by minimizing a potential function over the common feasible set. In particular, for a wide class of generalized potential games, we characterize those equilibria that can be computed by minimizing potential functions as Pareto solutions of a particular multi-objective problem, and we show how different potential functions can be used to select equilibria. We propose a new Gauss-Southwell algorithm to compute approximate equilibria of any generalized potential game with mixed-integer variables. We show that this method converges in a finite number of steps and we also give an upper bound on this number of steps. Moreover, we make a thorough analysis on the behaviour of approximate equilibria with respect to exact ones. Finally, we make many numerical experiments to show the viability of the proposed approaches. [ABSTRACT FROM AUTHOR]

The European-Union Emissions Trading System (EU-ETS) is a cap and trade scheme that requires the industries participating in the program to obtain allowances to cover their carbon emissions. Energy Intensive Industries claim that this system puts their European plants at an economics disadvantage compared to facilities located outside the EU. As a direct consequence, industries may relocate their production activities in unregulated countries, leading to the so-called carbon leakage effect. In order to curb this effect, several policies have been devised, including grandfathering of CO 2 allowances and border tax adjustment. This paper investigates the impact of these two policies on the cement sector, with a particular focus on the Italian market, particularly prone to carbon leakage. The analysis is based on an oligopolistic partial equilibrium model with a detailed technological representation of the market. The model is a Generalized Nash Equilibrium Problem that accounts for the interactions of cement companies. Simulations show that neither the grandfathering nor the border tax adjustment fully solve the carbon leakage problem because cement companies modify their cement and clinker trade strategies according to the measure applied in order to avoid or reduce their carbon costs. [ABSTRACT FROM AUTHOR]

With the development and deployment of new technologies, the oligopolistic taxi industry is transforming into a shared market with coexistence of both traditional taxi service (TTS) and app-based third-party taxi service (ATTS). The ATTS is different from TTS in both entry policy and fare setting, and brings competition into the market. To account for the revolution of the taxi industry, in this study, we analyze the characteristics of the TTS and ATTS, model the taxi market as a multiple-leader-follower game at the network level, and investigate the equilibrium of taxi market with competition (TMC Equilibrium). In particular, passengers are modeled as the leaders who seek to minimize their travel cost associated with taxi rides. Followers involve TTS and ATTS drivers, who compete for passengers to maximize their revenue. The network model captures selfish behavior of passengers and drivers in the taxi market, and we prove the existence of TMC Equilibrium for the proposed model using variational inequality formulations. An iterative algorithm is further developed to find the TMC Equilibrium, which corresponds to the strongly stationary point of the multi-leader-follower game. Based on numerical results, it is observed that fleet size and pricing policy are closely associated with the level of competition in the market and may have significant impact on total passengers cost, average waiting time, and fleet utilization. [ABSTRACT FROM AUTHOR]

The generalized Nash equilibrium problem is an extension of the Nash equilibrium problem by assuming that each player’s feasible set depends on the rival player’s strategies. By the Nikaido-Isoda function, we reformulate the generalized Nash equilibrium problem as a constrained optimization problem. This reformulation allows us to apply optimization techniques to the constrained optimization problem in order to solve the generalized Nash equilibrium problem. Conditions for the stationary point to be the global minimum of the constrained optimization problem are also given. [ABSTRACT FROM PUBLISHER]

We define the concept of reproducible map and show that, whenever the constraint map defining the quasivariational inequality (QVI) is reproducible then one can characterize the whole solution set of the QVI as a union of solution sets of some variational inequalities (VI). By exploiting this property, we give sufficient conditions to compute any solution of a generalized Nash equilibrium problem (GNEP) by solving a suitable VI. Finally, we define the class of pseudo-Nash equilibrium problems, which are (not necessarily convex) GNEPs whose solutions can be computed by solving suitable Nash equilibrium problems. [ABSTRACT FROM AUTHOR]

Deployment of small cell base stations (SBSs) overlaying the coverage area of a macrocell BS (MBS) results in a two-tier hierarchical small cell network. Cross-tier and inter-tier interference not only jeopardize primary macrocell communication but also limit the spectral efficiency of small cell communication. This paper focuses on distributed interference management for downlink small cell networks. We address the optimization of transmit strategies from both the game theoretical and the network utility maximization (NUM) perspectives and show that they can be unified in a generalized Nash equilibrium problem (GNEP) framework. Specifically, the small cell network design is first formulated as a GNEP, where the SBSs and MBS compete for the spectral resources by maximizing their own rates while satisfying global quality of service (QoS) constraints. We analyze the GNEP via variational inequality theory and propose distributed algorithms, which only require the broadcasting of some pricing information, to achieve a generalized Nash equilibrium (GNE). Then, we also consider a nonconvex NUM problem that aims to maximize the sum rate of all BSs subject to global QoS constraints. We establish the connection between the NUM problem and a penalized GNEP and show that its stationary solution can be obtained via a fixed point iteration of the GNE. We propose GNEP-based distributed algorithms that achieve a stationary solution of the NUM problem at the expense of additional signaling overhead and complexity. The convergence of the proposed algorithms is proved and guaranteed for properly chosen algorithm parameters. The proposed GNEP framework can scale from a QoS constrained game to an NUM design for small cell networks by trading off signaling overhead and complexity. [ABSTRACT FROM AUTHOR]