1,721,042 research outputs found
Gap functions and penalization for solving equilibrium problems with nonlinear constraints
No full textThe paper deals with equilibrium problems (EPs) with nonlinear convex constraints. First, EP is reformulated as a global optimization problem introducing a class of gap functions, in which the feasible set of EP is replaced by a polyhedral approximation. Then, an algorithm is given for solving EP through a descent type procedure related to exact penalties of the gap functions and its global convergence is proved. Finally, the algorithm is tested on a network oligopoly problem with nonlinear congestion constraints<br /
Stability for equilibrium problems: from variational inequalities to dynamical systems
No full textWe study the connections between solutions of variational inequalities and equilibrium points of a generalized dynamical system. Furthermore, we analyze some stability questions arising in this field
Optimal road maintenance investment in traffic networks with random demands
Full text linkThe paper deals with a traffic network with random demands in which some of the roads need maintenance jobs. Due to budget constraints, a central authority has to choose which of them are to be maintained in order to decrease as much as possible the average total travel time spent by all the users, assuming that the network flows are distributed according to the Wardrop equilibrium principle. This optimal road maintenance problem is modeled as an integer nonlinear program, where the objective function evaluation is based on the solution of a stochastic variational inequality. We propose a regularization and approximation procedure for its computation and prove its convergence. Finally, the results of preliminary numerical experiments on some test networks are reported
Descent and penalization techniques for equilibrium problems with nonlinear constraints
Full text linkThis paper deals with equilibrium problems with nonlinear constraints. Exploiting a gap function recently introduced, which rely 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
Auxiliary problem principles for equilibria
Full text linkThe auxiliary problem principle allows solving a given equilibrium problem (EP) through an equivalent auxiliary problem with better properties. The paper investigates two families of auxiliary EPs: the classical auxiliary problems, in which a regularizing term is added to the equilibrium bifunction, and the regularized Minty EPs. The conditions that ensure the equivalence of a given EP with each of these auxiliary problems are investigated exploiting parametric definitions of different kinds of convexity and monotonicity. This analysis leads to extending some known results for variational inequalities and linear EPs to the general case together with new equivalences. Stationarity and convexity properties of gap functions are investigated as well in this framework. Moreover, both new results on the existence of a unique solution and new error bounds based on gap functions with good convexity properties are obtained under weak quasimonotonicity or weak concavity assumptions
A two-stage game theoretical model of social interactions and location choice in city areas
No full textIn this article we further investigate a previous game theoretical model of social and economic interactions where players are considered according to their position in a social network and also to their geographic location. Our model allows for intermediate geographic areas, between the city center and the periphery. Moreover, an analysis of some simple but paradigmatic networks shows that the game whose solution yields the players’ location can have multiple equilibria
Gap functions for quasi-equilibria
Full text linkAn approach for solving quasi-equilibrium problems (QEPs) is proposed relying on gap functions, which allow reformulating QEPs as global optimization problems. The (generalized) smoothness properties of a gap function are analysed and an upper estimate of its Clarke directional derivative is given. Monotonicity assumptions on both the equilibrium and constraining bifunctions are a key tool to guarantee that all the stationary points of a gap function actually solve QEP. A few classes of constraints satisfying such assumptions are identified covering a wide range of situations. Relying on these results, a descent method for solving QEP is devised and its convergence proved. Finally, error bounds are given in order to guarantee the boundedness of the sequence generated by the algorithm
Gap functions and Lyapunov functions
No full textEquilibrium problems play a central role in the study of complex and competitive systems. Many variational formulations of these problems have been presented in these years. So, variational inequalities are very useful tools for the study of equilibrium solutions and their stability. More recently a dynamical model of equilibrium problems based on projection operators was proposed. It is designated as globally projected dynamical system (GPDS). The equilibrium points of this system are the solutions to the associated variational inequality (VI) problem. A very popular approach for finding solution of these VI and for studying its stability consists in introducing the so-called "gap-functions", while stability analysis of an equilibrium point of dynamical systems can be made by means of Lyapunov functions. In this paper we show strict relationships between gap functions and Lyapunov functions
D-gap functions and descent techniques for solving equilibrium problems
Full text linkA new algorithm for solving equilibrium problems with differentiable bifunctions is provided. The algorithm is based on descent directions of a suitable family of D-gap functions. Its convergence is proved under assumptions which do not guarantee the equivalence between the stationary points of the D-gap functions and the solutions of the equilibrium problem. Moreover, the algorithm does not require to set parameters according to thresholds which depend on regularity properties of the equilibrium bifunction. The results of preliminary numerical tests on Nash equilibrium problems with quadratic payoffs are reported. Finally, some numerical comparisons with other D-gap algorithms are drawn relying on some further tests on linear equilibrium problems
Stability of equilibrium points of projected dynamical systems
No full textWe present a survey of the main results about asymptotic stability, exponential stability and monotone attractors of locally and globally projected dynamical systems, whose stationary points coincide with the solutions of a corresponding variational inequality. In particular, we show that the global monotone attractors of locally projected dynamical systems are characterized by the solutions of a corresponding Minty variational inequality. Finally, we discuss two special cases: when the domain is a polyhedron, the stability analysis for a locally projected dynamical system, at regular solutions to the associated variational inequality, is reduced to one of a standard dynamical system of lower dimension; when the vector field is linear, some global stability results, for locally and globally projected dynamical systems, are proved if the matrix is positive definite (or strictly copositive when the domain is a convex cone)
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