A numerical algorithm for finding solutions of a generalized Nash equilibrium problem

A family of nonempty closed convex sets is built by using the data of the Generalized Nash equilibrium problem (GNEP). The sets are selected iteratively such that the intersection of the selected sets contains solutions of the GNEP. The algorithm introduced by Iusem-Sosa (2003) is adapted to obtain solutions of the GNEP. Finally some numerical experiments are given to illustrate the numerical behavior of the algorithm.

Implicit vector equilibrium problems with applications

Let X and Y be Hausdorff topological vector spaces, K a nonempty, closed, and convex subset of X, C: K--> 2(Y) a point-to-set mapping such that for any x is an element of K, C(x) is a pointed, closed, and convex cone in Y and int C(x) not equal 0. Given a mapping g : K --> K and a vector valued bifunction f : K x K - Y, we consider the implicit vector equilibrium problem (IVEP) of finding x* is an element of K such that f (g(x*), y) is not an element of - int C(x) for all y is an element of K. This problem generalizes the (scalar) implicit equilibrium problem and implicit variational inequality problem. We propose the dual of the implicit vector equilibrium problem (DIVEP) and establish the equivalence between (IVEP) and (DIVEP) under certain assumptions. Also, we give characterizations of the set of solutions for (IVP) in case of nonmonotonicity, weak C-pseudomonotonicity, C-pseudomonotonicity, and strict C-pseudomonotonicity, respectively. Under these assumptions, we conclude that the sets of solutions are nonempty, closed, and convex. Finally, we give some applications of (IVEP) to vector variational inequality problems and vector optimization problems. (C) 2003 Elsevier Science Ltd. All rights reserved.

Coincidence theorems and its applications to equilibrium problems

Given two maps h : X x K -> R and g : X -> K such that, for all x is an element of X, h(x, g(x)) = 0, we consider the equilibrium problem of finding (x) over tilde is an element of X such that h((x) over tilde, g(x)) >= 0 for every x is an element of X. This question is related to a coincidence problem.

Existence of solutions to some equilibrium problems

The concept of a monotone family of functions, which need not be countable, and the solution of an equilibrium problem associated with the family are introduced. A fixed-point theorem is applied to prove the existence of solutions to the problem.

Uso de redes neurais recorrentes na determinação das constantes de acidez para a 7-epiclusianona em misturas etanol-água

This work propose a recursive neural network to solve inverse equilibrium problem. The acidity constants of 7-epiclusianone in ethanol-water binary mixtures were determined from multiwavelength spectrophotmetric data. A linear relationship between acidity constants and the %w/v of ethanol in the solvent mixture was observed. The proposed method efficiency is compared with the Simplex method, commonly used in nonlinear optimization techniques. The neural network method is simple, numerically stable and has a broad range of applicability.

Location-price competition with externality: an application to the Tenerife tram

[EN] This paper presents a location–price equilibrium problem on a tree. A sufficient condition for having a Nash equilibrium in a spatial competition model that incorporates price, transport, and externality costs is given. This condition implies both competitors are located at the same point, a vertex that is the unique median of the tree. However, this is not an equilibrium necessary condition. Some examples show that not all medians are equilibria. Finally, an application to the Tenerife tram is presented.

Variational Principles for Monotone and Maximal Bifunctions

2000 Mathematics Subject Classification: 49J40, 49J35, 58E30, 47H05

A physical packing sequence algorithm for the container loading problem with static mechanical equilibrium conditions

The container loading problem (CLP) is a combinatorial optimization problem for the spatial arrangement of cargo inside containers so as to maximize the usage of space. The algorithms for this problem are of limited practical applicability if real-world constraints are not considered, one of the most important of which is deemed to be stability. This paper addresses static stability, as opposed to dynamic stability, looking at the stability of the cargo during container loading. This paper proposes two algorithms. The first is a static stability algorithm based on static mechanical equilibrium conditions that can be used as a stability evaluation function embedded in CLP algorithms (e.g. constructive heuristics, metaheuristics). The second proposed algorithm is a physical packing sequence algorithm that, given a container loading arrangement, generates the actual sequence by which each box is placed inside the container, considering static stability and loading operation efficiency constraints.

The land assembly problem revisited

As in the standard land assembly problem, a developer wants to buy two adjacent blocks of land belonging to two different owners. The value of the two blocks of land to the developer is greater than the sum of the individual values of the blocks for each owner. Unlike the land assembly literature, however, our focus is on the incentive that each lot owner has to delay the start of negotiations, rather than on the public goods nature of the problem. An incentive for delay exists, for example, when owners perceive that being last to sell will allow them to capture a larger share of the joint surplus from the development. We show that competition at point of sale can cause equilibrium delay, and that cooperation at point of sale will eliminate delay. This suggests that strategic delay is another source for the inefficient allocation of land, in addition to the public-good type externality pointed out by Grossman and Hart [Bell Journal of Economics 11 (1980) 42] and O'Flaherty [Regional Science and Urban Economics 24 (1994) 287]. (C) 2004 Elsevier B.V. All rights reserved.

The stability of the equilibrium outcomes in the admission games induced by stable matching rules

A stable matching rule is used as the outcome function for the Admission game where colleges behave straightforwardly and the students` strategies are given by their preferences over the colleges. We show that the college-optimal stable matching rule implements the set of stable matchings via the Nash equilibrium (NE) concept. For any other stable matching rule the strategic behavior of the students may lead to outcomes that are not stable under the true preferences. We then introduce uncertainty about the matching selected and prove that the natural solution concept is that of NE in the strong sense. A general result shows that the random stable matching rule, as well as any stable matching rule, implements the set of stable matchings via NE in the strong sense. Precise answers are given to the strategic questions raised.

The equilibrium penetration of monolayers. Is equilibrium really established?

In studying the penetration of water-soluble surfactants into water-insoluble monolayers the main theoretical problem is to find a relationship that would enable the amount of surfactant that has entered the monolayer to be calculated from a set of equilibrium surface pressure-area isotherms. Despite many attempts, no current theory gives satisfactory results when applied to experimental data (Langmuir 14 (1998) 2148). One possible reason is that equilibrium had not been established when the surface pressure-area curves were measured. The three experiments reported here suggest that equilibrium is extremely difficult to establish in such systems when the area is low or the surface pressure is high. The essence of these experiments is to try to reach the same final condition by two different routes. In the first route, the one nearly always used in equilibrium penetration measurements, the surfactant is injected under the expanded monolayer, which is then slowly compressed in steps, with time allowed at each step for a steady surface pressure to be attained. In the second procedure, the monolayer is first compressed to a high surface pressure and the surfactant then injected. A stepped expansion isotherm may then be observed. Surface pressure-area per monolayer molecule isotherms, reflection spectra, and slow neutron reflectivity data all show the same pattern: if the surfactant was allowed to penetrate while the monolayer was in an expanded state, it was not completely removed when the monolayer was compressed; but if the monolayer was in a highly compressed state when exposed to the surfactant little penetration took place until the film was expanded. There thus appear to be very large energy barriers to the ejection of surfactant from a compressed monolayer and to the penetration of surfactant into a compressed monolayer. Although these experiments have some limitations, it now seems likely that at least some of the penetration data used in evaluating the various thermodynamic treatments of equilibrium penetration were not equilibrium data. (C) 2001 Elsevier Science B.V. All rights reserved.

Quasiequilibrium problem in H-spaces

In the present paper, we study the quasiequilibrium problem and generalized quasiequilibrium problem of generalized quasi-variational inequality in H-spaces by a new method. Some new equilibrium existence theorems are given. Our results are different from corresponding given results or contain some recent results as their special cases. (C) 2003 Elsevier Science Ltd. All rights reserved.

Dynamic equilibrium through reinforcement learning

Reinforcement Learning is an area of Machine Learning that deals with how an agent should take actions in an environment such as to maximize the notion of accumulated reward. This type of learning is inspired by the way humans learn and has led to the creation of various algorithms for reinforcement learning. These algorithms focus on the way in which an agent’s behaviour can be improved, assuming independence as to their surroundings. The current work studies the application of reinforcement learning methods to solve the inverted pendulum problem. The importance of the variability of the environment (factors that are external to the agent) on the execution of reinforcement learning agents is studied by using a model that seeks to obtain equilibrium (stability) through dynamism – a Cart-Pole system or inverted pendulum. We sought to improve the behaviour of the autonomous agents by changing the information passed to them, while maintaining the agent’s internal parameters constant (learning rate, discount factors, decay rate, etc.), instead of the classical approach of tuning the agent’s internal parameters. The influence of changes on the state set and the action set on an agent’s capability to solve the Cart-pole problem was studied. We have studied typical behaviour of reinforcement learning agents applied to the classic BOXES model and a new form of characterizing the environment was proposed using the notion of convergence towards a reference value. We demonstrate the gain in performance of this new method applied to a Q-Learning agent.

Solving a signalized traffic intersection problem with an hyperbolic penalty function

Mathematical Program with Complementarity Constraints (MPCC) finds many applications in fields such as engineering design, economic equilibrium and mathematical programming theory itself. A queueing system model resulting from a single signalized intersection regulated by pre-timed control in traffic network is considered. The model is formulated as an MPCC problem. A MATLAB implementation based on an hyperbolic penalty function is used to solve this practical problem, computing the total average waiting time of the vehicles in all queues and the green split allocation. The problem was codified in AMPL.

Efficient Nash Equilibrium under Adverse Selection

This paper revisits the problem of adverse selection in the insurance market of Rothschild and Stiglitz [28]. We propose a simple extension of the game-theoretic structure in Hellwig [14] under which Nash-type strategic interaction between the informed customers and the uninformed firms results always in a particular separating equilibrium. The equilibrium allocation is unique and Pareto-efficient in the interim sense subject to incentive-compatibility and individual rationality. In fact, it is the unique neutral optimum in the sense of Myerson [22].