Search

In this paper, by revisiting coderivative calculus rules for convex multifunctions in finite-dimensional spaces, we derive formulae for estimating/computing the basic subdifferential and the coderivative of the efficient point multifunction of parametric convex vector optimization problems. These results are then applied to a broad class of conventional convex vector optimization problems with the presence of operator constraints and equilibrium ones. Examples are also designed to analyze and illustrate the obtained results. [ABSTRACT FROM AUTHOR]

Can order markets lead participants towards price-taking equilibrium? Viewing market sessions as steps of iterative algorithms, this paper indicates positive prospects for convergence. Mathematical arguments turn on convolution, efficiency and generalized gradients. Economic arguments revolve around reservation costs, derived from indifference or threshold payments for quantities supplied or demanded. [ABSTRACT FROM AUTHOR]

The concept of perfect equilibrium, formulated by Selten (Int J Game Theory 4:25–55, 1975), serves as an effective characterization of rationality in strategy perturbation. In our study, we propose a modified version of perfect equilibrium that incorporates perturbation control parameters. To match the beliefs with the equilibrium choice probabilities, the logistic quantal response equilibrium (logistic QRE) was established by McKelvey and Palfrey (Games Econ Behav 10:6–38, 1995), which is only able to select a Nash equilibrium. By introducing a linear combination between a mixed strategy profile and a given vector with positive elements, this paper develops a variant of the logistic QRE for the selection of the special version of perfect equilibrium. Expanding upon this variant, we construct an equilibrium system that incorporates an exponential function of an extra variable. Through rigorous error-bound analysis, we demonstrate that the solution set of this equilibrium system leads to a perfect equilibrium as the extra variable approaches zero. Consequently, we establish the existence of a smooth path to a perfect equilibrium and employ an exponential transformation of variables to ensure numerical stability. To make a numerical comparison, we capitalize on a variant of the square-root QRE, which yields another smooth path to a perfect equilibrium. Numerical results further verify the effectiveness and efficiency of the proposed differentiable path-following methods. [ABSTRACT FROM AUTHOR]

In this paper, we first propose the model of multi-leader–follower games with fuzzy goals involving fuzzy parameters and introduce its α -FNS equilibrium. Next, we shift our attention to the existence of α -FNS equilibrium and prove it by Kakutani's fixed point theorem. Finally, we illustrate an example to show that the equilibrium existence result is valid. [ABSTRACT FROM AUTHOR]

In this paper is introduced a proposal of resolvent for equilibrium problems in terms of the Busemann's function. A advantage of this new proposal is that, in addition to be a natural extension of its counterpart in the linear setting introduced by Combettes and Hirstoaga (J Nonlinear Convex Anal 6(1): 117–136, 2005), the new term that performs regularization is a convex function in general Hadamard manifolds, being a first step to fully answer to the problem posed by Cruz Neto et al. (J Convex Anal 24(2): 679–684, 2017 Section 5). During our study, some elements of convex analysis are explored in the context of Hadamard manifolds, which are interesting on their own. In particular, we introduce a new definition of convex combination (now commutative) of any finite collection of points and present an associated Jensen-type inequality. [ABSTRACT FROM AUTHOR]

The Gale–Nikaido–Debreu lemma plays an important role in establishing the existence of competitive equilibrium. In this paper, we use Sperner's lemma and basic elements of topology to prove the Gale–Nikaido–Debreu lemma. [ABSTRACT FROM AUTHOR]

In this paper, we use some order-theoretic fixed point theorems to study the upper order-preservation properties of solution mapping for parametric equilibrium problems. In contrast to lots of existing works on the behaviors of solutions to equilibrium problems, the topic of the order-preservation properties of solutions is relatively new for equilibrium problems. It would be useful for us to analyze the changing trends of solutions to equilibrium problems. In order to show the applied value and theoretic value of this subject, we focus on a class of differential variational inequalities, which are currently receiving much attention. By applying the order-preservation properties of solution mapping to variational inequality, we investigate the existence of mild solutions to differential variational inequalities. Since our approaches are order-theoretic and the underlying spaces are Banach lattices, the results obtained in this paper neither require the bifunctions in equilibrium problems to be continuous nor assume the Lipschitz continuity for the involved mapping in ordinary differential equation. [ABSTRACT FROM AUTHOR]

In this paper, we consider the resolvent via Busemann functions introduced by Bento, Cruz Neto, Melo (J Optim Theory Appl 195:1087–1105, 2022), and we present a proximal point method for equilibrium problems on Hadamard manifold. The resolvent in consideration is a natural extension of its counterpart in linear settings, proposed and analyzed by Combettes and Hirstoaga (J Nonlinear Convex Anal 6:117–136, 2005). The advantage of using this resolvent is that the term performing regularization is a convex function in general Hadamard manifolds, allowing us to explore the asymptotic behavior of the proximal point method to solve equilibrium problems. [ABSTRACT FROM AUTHOR]

In this paper, we study the competition of healthcare institutions for medical supplies in emergencies caused by natural disasters. In particular, we develop a two-stage procurement planning model in a random environment. We consider a pre-event policy, in which each healthcare institution seeks to minimize the purchasing cost of medical items and the transportation time from the first stage, and a recourse decision process to optimize the expected overall costs and the penalty for the prior plan, in response to each disaster scenario. Thus, each institution deals with a two-stage stochastic programming model that takes into account the unmet demand at the first stage, and the consequent penalty. Then, the institutions simultaneously solve their own stochastic optimization problems and reach a stable state governed by the stochastic Nash equilibrium concept. Moreover, we formulate the problem as a variational inequality; both the discrete and the general probability distribution cases are described. We also present an alternative formulation using infinite-dimensional duality tools. Finally, we discuss some numerical illustrations applying the progressive hedging algorithm. [ABSTRACT FROM AUTHOR]

To extend the concept of subgame perfect equilibrium to an extensive-form game with imperfect information but perfect recall, Selten (Int J Game Theory 4:25–55, 1975) formulated the notion of perfect equilibrium and attained its existence through the agent normal-form representation of the extensive-form game. As a strict refinement of Nash equilibrium, a perfect equilibrium always yields a sequential equilibrium. The selection of a perfect equilibrium thus plays an essential role in the applications of game theory. Moreover, a different procedure may select a different perfect equilibrium. The existence of Nash equilibrium was proved by Nash (Ann Math 54:289–295, 1951) through the construction of an elegant continuous mapping and an application of Brouwer's fixed point theorem. This paper intends to enhance Nash's mapping to select a perfect equilibrium. By incorporating the complementarity condition into the equilibrium system with Nash's mapping through an artificial game, we successfully eliminate the nonnegativity constraints on a mixed strategy profile imposed by Nash's mapping. In the artificial game, each player solves against a given mixed strategy profile a strictly convex quadratic optimization problem. This enhancement enables us to derive differentiable homotopy systems from Nash's mapping and establish the existence of smooth paths for selecting a perfect equilibrium. The homotopy methods start from an arbitrary totally mixed strategy profile and numerically trace the smooth paths to a perfect equilibrium. Numerical results show that the methods are numerically stable and computationally efficient in search of a perfect equilibrium and outperform the existing differentiable homotopy method. [ABSTRACT FROM AUTHOR]

This paper studies a kind of time-inconsistent linear–quadratic control problem in a more general framework with stochastic coefficients and random jumps. The time inconsistency comes from the dependence of the terminal cost on the current state as well as the presence of a quadratic term of the expected terminal state in the objective functional. Instead of finding a global optimal control, we look for a time-consistent locally optimal equilibrium solution within the class of open-loop controls. A general sufficient and necessary condition for equilibrium controls via a flow of forward–backward stochastic differential equations is derived. This paper further develops a new methodology to cope with the mathematical difficulties arising from the presence of stochastic coefficients and random jumps. As an application, we study a mean-variance portfolio selection problem in a jump-diffusion financial market; an explicit equilibrium investment strategy in a deterministic coefficients case is obtained and proved to be unique. [ABSTRACT FROM AUTHOR]

This paper studies the inverse Stackelberg game with multiple hierarchies under global and local information structures, where the players have discrete strategy spaces. For the classic public goods game, we solve the pure-strategy inverse Stackelberg equilibria under three typical hierarchical structures. The results reveal some counterintuitive characteristics within the systems with hierarchies, such as that the cooperation does not increase with the return rate at the equilibria. Furthermore, by defining a local information structure, we give an estimate of the fewest hierarchies required for full cooperation, which can be a constant multiple of the logarithm or square root of the population size or of the population size itself, according to different information structures and return rates. This paper proposes a novel mechanism to play the game and promote cooperation. Both the formulation and analysis method are different from existing works, and the results can find their ample implications in practice, which might help decision making in hierarchical systems. [ABSTRACT FROM AUTHOR]

In this paper, we introduce some concepts of convexity and semicontinuity for real set-valued mappings similar to those of real single-valued mappings. Then, we obtain different results on the existence of solutions of set-valued equilibrium problems generalizing in a common way several old ones for both single-valued and set-valued equilibrium problems. Applications to Browder variational inclusions, with weakened conditions on the involved set-valued operator, are given. [ABSTRACT FROM AUTHOR]

In this paper, a new concept of equilibrium in dynamic games with incomplete or distorted information is introduced. In the games considered, players have incomplete information about crucial aspects of the game and formulate beliefs about the probabilities of various future scenarios. The concept of belief distorted Nash equilibrium combines optimization based on given beliefs and self-verification of those beliefs. Existence and equivalence theorems are proven, and this concept is compared to existing ones. Theoretical results are illustrated using several examples: extracting a common renewable resource, a large minority game, and a repeated Prisoner's Dilemma. [ABSTRACT FROM AUTHOR]

In this paper, we propose a distribution-free model instead of considering a particular distribution for multiple objective games with incomplete information. We assume that each player does not know the exact value of the uncertain payoff parameters, but only knows that they belong to an uncertainty set. In our model, the players use a robust optimization approach for each of their objective to contend with payoff uncertainty. To formulate such a game, named “robust multiple objective games” here, we introduce three kinds of robust equilibrium under different preference structures. Then, by using a scalarization method and an existing result on the solutions for the generalized quasi-vector equilibrium problems, we obtain the existence of these robust equilibria. Finally, we give an example to illustrate our model and the existence theorems. Our results are new and fill the gap in the game theory literature. [ABSTRACT FROM AUTHOR]

In this paper, we consider a dynamical system for solving equilibrium problems in the framework of Hilbert spaces. First, we prove that under strong pseudo-monotonicity and Lipschitz-type continuity assumptions, the dynamical system has a unique equilibrium solution, which is also globally exponentially stable. Then, we derive the linear rate of convergence of a discrete version of the proposed dynamical system to the unique solution of the problem. Global error bounds are also provided to estimate the distance between any trajectory and this unique solution. Some numerical experiments are reported to confirm the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of solutions for mixed equilibrium problems associated with a set-valued operator in the general setting of vector spaces in duality, and in particular in Banach spaces. We use a Galerkin-type method and the notion of pseudomonotonicity in the sense of Brézis for bifunctions. As application, we study the existence of solutions for quasi-hemivariational inequalities governed by a set-valued mapping and perturbed with a nonlinear term. Our main results can be applied to differential inclusions, evolution equations and evolution hemivariational inequalities. [ABSTRACT FROM AUTHOR]

In this paper, we establish existence theorems for bilevel problems with fixed-point constraints and bilevel problems without fixed-point constraint. The aim of this paper is to investigate under which conditions the existence of a feasible point of a bilevel problem can be assumed in advance and under which conditions there exist minimizers for this type of problems. From this, we establish existence theorems for mathematical programs with equilibrium constraints and semi-infinite problems. [ABSTRACT FROM AUTHOR]

In this paper, we introduce two new constraint qualifications for mathematical programs with equilibrium constraints. One of them is a relaxed version of the No Nonzero Abnormal Multiplier Constraint Qualification, and the other is an adaptation of the Constant Rank of Subspace Component. The new conditions have nice properties. Indeed, they have the local preservation property and imply the error bound property under mild assumptions. Thus, they can be used to extend some known results on stability and sensitivity analysis. Furthermore, they can also be used in the convergence analysis of several methods for solving mathematical programs with equilibrium constraints. [ABSTRACT FROM AUTHOR]

In this paper, we generalize the Markowitz measure of the risk proposed in a stationary setting. We provide an evolutionary Markowitz-type measure of the risk with a memory term and show that this function is effective, namely an existence theorem for the general financial problem can be proved. [ABSTRACT FROM AUTHOR]

In this paper, we study vector quasi-equilibrium problems for the sum of two multivalued bifunctions. The assumptions are required separately on each of these bifunctions. Sufficient conditions for the existence of solutions of such problems are shown in the setting of topological vector spaces. The results in this paper unify, improve and extend some well-known existence theorems from the literature. [ABSTRACT FROM AUTHOR]

Actual implementation of augmented Lagrangian algorithms requires a solution of the subproblem generated at each iterate, which is the most challenging task. In this paper, we propose two approaches to make the augmented Lagrangian algorithms, introduced in Iusem and Nasri (RAIRO Oper Res 44:5-26, ) for equilibrium problems, computer amenable. The first algorithm that we suggest here incorporates the Newton method and the other one benefits from the Shor subgradient method to solve the subproblems that are produced when the augmented Lagrangian algorithms are applied to the equilibrium problem. We also illustrate our findings by numerical results which are obtained when our algorithms are implemented to solve quadratic equilibrium problems and certain generalized Nash equilibrium problem, including the river basin pollution problem, a particular case of the equilibrium problem. Moreover, we compare our numerical results with those presented in Matioli et al. (Comput Optim Appl 52:281-292, ) and Tran et al. (Optimization 57:749-776, ) for the same test problems. [ABSTRACT FROM AUTHOR]

This paper provides formulas for calculating Fréchet and limiting normal cones with respect to a set of sets and the limiting coderivative with respect to a set of set-valued mappings. These calculations are obtained under some qualification constraints and are expressed in the similar forms of the ones of Fréchet and limiting normal cones and the limiting coderivative. By using these new formulas, we state explicit necessary optimality conditions with respect to a set for optimization problems with equilibrium constraints under certain qualification conditions. Some examples are also presented. [ABSTRACT FROM AUTHOR]

In the paper we study, in a Hilbert space setting, a general random traffic equilibrium problem and characterize the random Wardrop equilibrium distribution by means of a random variational inequality. Some existence results are provided and the associated Lagrange function is studied. Some examples illustrate the random variational inequality and the counterintuitive behaviour of the traffic equilibrium problem is focused. [ABSTRACT FROM AUTHOR]

In this paper, we consider two classes of generalized mixed vector equilibrium problems and mixed vector equilibrium problems, and propose some gap functions by using a new method, which is different from the previously known methods used in the literature. Finally, error bounds are obtained for the underlying mixed vector equilibrium problems in terms of regularized gap functions without using projection method. [ABSTRACT FROM AUTHOR]

This paper deals with equilibrium problems with nonlinear constraints. Exploiting a gap function which relies on a polyhedral approximation of the feasible region, we propose two descent methods. They are both based on the minimization of a suitable exact penalty function, but they use different rules for updating the penalization parameter and they rely on different types of line search. The convergence of both algorithms is proved under standard assumptions. [ABSTRACT FROM AUTHOR]

In this paper, we study necessary optimality conditions for nonsmooth mathematical programs with equilibrium constraints. We first show that, unlike the smooth case, the mathematical program with equilibrium constraints linear independent constraint qualification is not a constraint qualification for the strong stationary condition when the objective function is nonsmooth. We then focus on the study of the enhanced version of the Mordukhovich stationary condition, which is a weaker optimality condition than the strong stationary condition. We introduce the quasi-normality and several other new constraint qualifications and show that the enhanced Mordukhovich stationary condition holds under them. Finally, we prove that quasi-normality with regularity implies the existence of a local error bound. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of a solution for a hemivariational inequality problem in a noncoercive framework. The approach adopted is an equilibrium problem formulation associated with a maximal monotone bifunction with pseudomonotone perturbation. We proceed by introducing auxiliary problems that will be studied using a new existence result for equilibrium problems. An example to illustrate the use of the theory is given. [ABSTRACT FROM AUTHOR]

This paper extends the human migration model introduced in previous works to the framework of consistent conjectural variations. First, we introduce the standard multiclass human migration network equilibrium model that describes the movement of migrants between locations. Next, we introduce the concept of conjectural variations, in which migrants conjecture about the (expected) utility of locations after their migration. We define the concept of conjectural variations equilibrium and present results regarding the conditions for its existence and uniqueness. Following that, we define the concept of consistency for the migrants’ conjectures and the consistent conjectural variations equilibrium (CCVE). Finally, we describe the conditions that guarantee the existence of the CCVE. [ABSTRACT FROM AUTHOR]

This paper gives sufficient conditions for the upper and lower semicontinuities of the solution mapping of a parametric mixed generalized Ky Fan inequality problem. We use a new scalarizing approach quite different from traditional linear scalarization approaches which, in the framework of the stability analysis of solution mappings of equilibrium problems, were useful only for weak vector equilibrium problems and only under some convexity and strict monotonicity assumptions. The main tools of our approach are provided by two generalized versions of the nonlinear scalarization function of Gerstewitz. Our stability results are new and are obtained by a unified technique. An example is given to show that our results can be applied, while some corresponding earlier results cannot. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to present a unified approach to study the existence of solutions for two types of variational relation problems, which encompass several generalized equilibrium problems, variational inequalities and variational inclusions investigated in the recent literature. By using two well-known fixed point theorems, we establish several existence criteria for the solutions of these problems. [ABSTRACT FROM AUTHOR]

Multicriteria games describe strategic interactions in which players, having more than one criterion to take into account, don’t have an a-priori opinion on the relative importance of all these criteria. Roemer (Econ. Bull. 3:1–13, ) introduces an organizational interpretation of the concept of equilibrium: each player can be viewed as running a bargaining game among criteria. In this paper, we analyze the bargaining problem within each player by considering the Kalai-Smorodinsky bargaining solution (see Kalai and Smorodinsky in Econometrica 43:513–518, ). We provide existence results for the so called Kalai-Smorodinsky bargaining solution equilibria for a general class of disagreement points which properly includes the one considered by Roemer (Econ. Bull. 3:1–13, ). Moreover we look at the refinement power of this equilibrium concept and show that it is an effective selection device even when combined with classical refinement concepts based on stability with respect to perturbations; in particular, we consider the extension to multicriteria games of the Selten’s trembling hand perfect equilibrium concept (see Selten in Int. J. Game Theory 4:25–55, ) and prove that perfect Kalai-Smorodinsky bargaining solution equilibria exist and properly refine both the perfect equilibria and the Kalai-Smorodinsky bargaining solution equilibria. [ABSTRACT FROM AUTHOR]

This paper presents some results concerning the existence of the Lagrange multipliers for vector optimization problems in the case where the ordering cone in the codomain has an empty interior. The main tool for deriving our assertions is a scalarization by means of a functional introduced by Hiriart-Urruty (Math. Oper. Res. 4:79–97, ) (the so-called oriented distance function). Moreover, we explain some applications of our results to a vector equilibrium problem, to a vector control-approximation problem and to an unconstrainted vector fractional programming problem. [ABSTRACT FROM AUTHOR]

We study a model of intergenerational stochastic game with general state space in which each generation consists of n players. The main objective is to prove the existence of a perfect stationary equilibrium in an infinite-horizon intergenerational game in which cooperation is assumed inside every generation. A suitable change in the terminology used in this paper provides a new equilibrium theorem for stochastic games with so-called “hyperbolic players”. A discussion of perfect equilibria in games of noncooperative generations is also given. Some applications to economic theory are included. [ABSTRACT FROM AUTHOR]

In this paper, necessary and sufficient conditions for null-controllability of a linear system under geometric constraints on the control are given, without the assumption that the origin is an equilibrium point of the system. The criterion for controllability uses the concept of a return condition on an interval which is introduced in the paper. This condition generalizes the existence of an equilibrium point. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the concepts of globally efficient solution and cone-Benson efficient solution for a vector equilibrium problem; we give some scalarization results for Henig efficient solution sets, globally efficient solution sets, weak efficient solution sets, and cone-Benson efficient solution sets in locally convex spaces. Using the scalarization results, we show the connectedness and path connectedness of weak efficient solution sets and various proper efficient solution sets of vector equilibrium problem. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the concept of feasible set for an equilibrium problem with a convex cone and generalize the notion of a Z-function for bifunctions. Under suitable assumptions, we derive some equivalence results of equilibrium problems, least element problems, and nonlinear programming problems. The results presented extend some results of [Riddell, R.C.: Equivalence of nonlinear complementarity problems and least element problems in Banach lattices. Math. Oper. Res. 6, 462—474 (1981)] to equilibrium problems. [ABSTRACT FROM AUTHOR]

This paper studies the optimal tradeoff between the benefits and costs of preventing offenses and treating offenders. Based on a flexible age-structured epidemiological framework, a two-state compartment model is analyzed to reduce the prevalence of offending such as illicit drug consumption or violence. It turns out that, even in this highly simplified model, multiple stationary states exist. In particular, three different kinds of equilibria are identified, i.e., law and order, conservative, and liberal. The optimal mix of the control instruments is calculated providing interesting insight into the structure of the paths minimizing the discounted stream of social costs and expenditures for prevention and treatment. It can happen that a Skiba point exists. This implies that, for an initially small number of offenders, saddle-point convergence to a lawand-order equilibrium (boundary solution with no offenders) or to a conservative equilibrium (with few offenders) occurs, while if the number of offenders is large, the effects of prevention and treatment are too low or too expensive so that a liberal equilibrium (with many offenders) occurs. [ABSTRACT FROM AUTHOR]

In this paper, we prove the existence of the equilibrium in choice for games in choice form. Thus, we add to the research recently appeared in the scientific literature. In fact, our results link the most recent research to the older approaches of the games in normal-form and the qualitative games. [ABSTRACT FROM AUTHOR]

The main goal of this paper is to introduce and study bilevel vector equilibrium problems. We first establish some existence results for solutions of vector equilibrium problems and mixed vector equilibrium problems. We study the existence of solutions of bilevel vector equilibrium problems by considering a vector Thikhonov-type regularization procedure. By using this regularization procedure and existence results for mixed vector equilibrium problems, we establish some existence results for solutions of bilevel vector equilibrium problems. By using the auxiliary principle, we propose an algorithm for finding the approximate solutions of bilevel vector equilibrium problems. The strong convergence of the proposed algorithm is also studied. [ABSTRACT FROM AUTHOR]

In this paper, by an approach, which is based on a notion of sequentially sign property for bifunctions, we establish existence results for equilibrium problems in the setting of Hausdorff locally convex topological vector spaces. The main advantages of this approach are that our conditions are imposed just on a locally segment-dense set, instead of the whole domain. [ABSTRACT FROM AUTHOR]

In this paper, we consider the nonlinear scalarization function in the setting of topological vector spaces and present some properties of it. Moreover, using the nonlinear scalarization function and Fan-Glicksberg-Kakutani's fixed point theorem, we obtain an existence result of a solution for a generalized vector quasi-equilibrium problem without using any monotonicity and upper semi-continuity (or continuity) on the given maps. Our result can be considered as an improvement of the known corresponding result. After that, we introduce a system of generalized vector quasi-equilibrium problem which contains Nash equilibrium and Debreu-type equilibrium problem as well as the system of vector equilibrium problem posed previously. We provide two existence theorems for a solution of a system of generalized vector quasi-equilibrium problem. In the first one, our multi-valued maps have closed graphs and the maps are continuous, while in the second one, we do not use any continuity on the maps. Moreover, the method used for the existence theorem of a solution of a system of generalized vector quasi-equilibrium problem is not based upon a maximal element theorem. Finally, as an application, we apply the main results to study a system of vector optimization problem and vector variational inequality problem. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to introduce the notion of weakly upper semicontinuous set-valued maps and to establish a new fixed-point theorem. The set-valued maps with an approximating upper semicontinuous selection property are also defined. Next, we use our fixed-point result to obtain equilibrium existence in abstract economies with two constraints, which provide a natural scenario for potential applications of our approach to general equilibrium theory. In this regard, we set models of economies with asymmetric informed agents, who are able to improve their initial information through market signals. These economies offer examples in which the informational feasibility requirement represents an additional constraint. [ABSTRACT FROM AUTHOR]

This paper presents a framework of iterative methods for finding a common solution to an equilibrium problem and a countable number of fixed point problems defined in a Hilbert space. A general strong convergence theorem is established under mild conditions. Two hybrid methods are derived from the proposed framework in coupling the fixed point iterations with the iterations of the proximal point method or the extragradient method, which are well-known methods for solving equilibrium problems. The strategy is to obtain the strong convergence from the weak convergence of the iterates without additional assumptions on the problem data. To achieve this aim, the solution set of the problem is outer approximated by a sequence of polyhedral subsets. [ABSTRACT FROM AUTHOR]

In this paper, we prove two strong convergence theorems for finding a common point of the set of zero points of the addition of an inverse-strongly monotone mapping and a maximal monotone operator and the set of zero points of a maximal monotone operator, which is related to an equilibrium problem in a Hilbert space. Such theorems improve and extend the results announced by Y. Liu (Nonlinear Anal. 71:4852-4861, ). As applications of the results, we present well-known and new strong convergence theorems which are connected with the variational inequality, the equilibrium problem and the fixed point problem in a Hilbert space. [ABSTRACT FROM AUTHOR]

In this paper, the image space analysis is employed to study a generalized Ky Fan quasi-inequality with cone constraints. By virtue of a nonlinear scalarization function and a positive linear operator, a nonlinear (regular) weak separation function and a linear regular weak separation function are introduced. Nonlinear and, in particular, linear separations for the generalized Ky Fan quasi-inequality with cone constraints are characterized. Some necessary and sufficient optimality conditions, especially a saddle-point sufficient optimality condition for the generalized Ky Fan quasi-inequality with cone constraints, are obtained. As applications, some sufficient conditions for (weak) vector equilibrium flows of vector traffic equilibrium problems with capacity arc constraints, are derived. [ABSTRACT FROM AUTHOR]

Equilibrium in choice is a solution-concept for noncooperative games defined in a general framework-the game in choice form. There are two leading ideas of the new definition. One is that the players' preferences need not be explicitly represented, but earlier accepted solution concepts should be formally derived as particular cases. Secondly, the choice of a player need not be a best reply to the strategy combination of the others, if the choices of the other players are motivated for themselves and a best reply does not exist. It is shown that in the present framework are included classical models of game theory, and the new concept extends various known noncooperative solutions. The main technical results of the paper concern the existence of the equilibrium in choice. As particular cases, known results on the existence of classical solutions are found. Thus, our approach can be also seen as a general method for proving the existence of different solutions for noncooperative games. [ABSTRACT FROM AUTHOR]

We consider markets in which firms offer supply functions, rather than a quantity or price alone: the most important examples are wholesale electricity markets. The equilibria in such markets can be hard to characterize. In many cases, whole families of supply function equilibria occur so there are difficulties in determining which equilibrium will be chosen. In this paper, we consider supply function equilibria, when firms hold forward contracts, which is common in electricity markets. Under the assumption that contract positions have been fixed in advance, we characterize the families of supply function equilibria in a duopoly. The existence of forward contracts implies a tightening of the conditions for an equilibrium, and a greater likelihood that no equilibrium solution exists. In the case of three firms, there can be at most one supply function equilibrium, provided that the lowest demand be small enough. [ABSTRACT FROM AUTHOR]

We consider a stochastic mathematical program with equilibrium constraints (SMPEC) and show that, under certain assumptions, global optima and stationary solutions are robust with respect to changes in the underlying probability distribution. In particular, the discretization scheme sample average approximation (SAA), which is convergent for both global optima and stationary solutions, can be combined with the robustness results to motivate the use of SMPECs in practice. We then study two new and natural extensions of the SMPEC model. First, we establish the robustness of global optima and stationary solutions to an SMPEC model where the upper-level objective is the risk measure known as conditional value-at-risk (CVaR). Second, we analyze a multiobjective SMPEC model, establishing the robustness of weakly Pareto optimal and weakly Pareto stationary solutions. In the accompanying paper (Cromvik and Patriksson, Part 2, J. Optim. Theory Appl., , to appear) we present applications of these results to robust traffic network design and robust intensity modulated radiation therapy. [ABSTRACT FROM AUTHOR]

In this paper, we introduce some iterative algorithms for finding a common element of the set of solutions of a mixed equilibrium problem, the set of fixed points of a strict pseudocontraction and the set of solutions of a variational inequality for a monotone, Lipschitz continuous mapping. We obtain both weak and strong convergence theorems for the sequences generated by these processes in Hilbert spaces. [ABSTRACT FROM AUTHOR]