on relaxation algorithms in computation of noncooperative equilibria this paper considers a special class of numerical algorithms the so called relaxation algorithm for nash equilibrium points in noncooperative games the relaxation algorithms have been studied by various authors for the deterministic case convergence conditions of this algorithm are based on fixed point theorems for example basar 1987 and li and basar 1987 have proved its convergence for a two player game via the contraction mapping theorem for the quadratic case these conditions can be easily checked for other nonlinear payoff functions it is sometimes difficult to check these convergence conditions in this paper the authors propose an alternative approach using the residual terms of the nikaido isoda function the convergence theorem is proved for nonsmooth weakly convex concave nikaido isoda functions the family of weakly convex concave functions is broad enough for applications since if includes the family of smooth functions when the payoff functions are twice continuously differentiable the condition for the residual terms is reduced to strict positiveness of a matrix representing the difference of the hessians of the nikaido isoda function with respect to the first and second groups of variables an analogous condition was used by uryas ev 1988 to prove convergence of the gradient type algorithm for the nash equilibrium problem in this paper the authors discuss only the deterministic case nevertheless this approach can be generalized for the stochastic nash equilibrium problems with uncertainties in parameters

numerical aspects of the sweeping process the sweeping process introduced some time ago by the author with motivation in plasticity theory today remains an object of mathematical research it is considered in this paper as the prototype of an evolution conditioned by inequality constraints since the governing differential requirements are only of order one with respect to time this provides a simplified setting for analysing some numerical and theoretical features also present in unilateral dynamics the latter is governed by differential inclusions of order two for the numerical handling of which the existing literature proposes diverse strategies briefly discussed the paper is especially intended to offer an introduction to the numerical approach called contact dynamics

adaptive timestep control for the contact stabilized newmark method the aim of this paper is to devise an adaptive timestep control in the contact stabilized newmark method contacx for dynamical contact problems between two viscoelastic bodies in the framework of signorini s condition in order to construct a comparative scheme of higher order accuracy we extend extrapolation techniques this approach demands a subtle theoretical investigation of an asymptotic error expansion of the contact stabilized newmark scheme on the basis of theoretical insight and numerical observations we suggest an error estimator and a timestep selection which also cover the presence of contact finally we give a numerical example

time integration schemes for the finite element dynamic signorini problem a large variety of discretizations have been proposed in the literature for the numerical solution of the dynamic signorini problem we classify the different discretizations into four groups the first three groups correspond to different ways of enforcing the contact condition exact enforcement enforcement with penalty and enforcement with contact condition in velocity the fourth approach is based on a modification of the mass matrix numerical simulations on two one dimensional benchmark problems with analytical solutions illustrate the properties of representative methods within each class focusing first on spurious oscillations triggered by contact and then on energy behavior after multiple impacts selected schemes are also tested on a two dimensional benchmark

force stepping integrators in lagrangian mechanics we present a class of integration schemes for lagrangian mechanics referred to as energy stepping integrators that are momentum and energy conserving symplectic and convergent in order to achieve these properties we replace the original potential energy by a piecewise constant or terraced approximation at steps of uniform height by taking steps of diminishing height an approximating sequence of energies is generated the trajectories of the resulting approximating lagrangians can be characterized explicitly and consist of intervals of piecewise rectilinear motion we show that the energy stepping trajectories are symplectic exactly conserve all the momentum maps of the original system and subject to a transversality condition converge to trajectories of the original system when the energy step is decreased to zero these properties the excellent long term behavior of energy stepping and its automatic time step selection property are born out by selected examples of application including the dynamics of a frozen argon cluster the spinning of an elastic cube and the collision of two elastic spheres

design of energy conserving algorithms for frictionless dynamic contact problems this paper proposes a formulation of dynamic contact problems which enables exact algorithmic conservation of linear momentum angular momentum and energy in finite element simulations it is seen that a lagrange multiplier enforcement of an appropriate contact rate constraint produces these conservation properties a related method is presented in which a penalty regularization of the aforementioned rate constraint is utilized this penalty method sacrifices the energy conservation property but is dissipative under all conditions of changing contact so that the global algorithm remains stable notably it is also shown that augmented lagrangian iteration utilizing this penalty kernel reproduces the energy conserving i e lagrange multiplier solution to any desired degree of accuracy the result is a robust stable method even in the context of large deformations as is shown by some representative numerical examples in particular the ability of the formulation to produce accurate results where more traditional integration schemes fail is emphasized by the numerical simulations 1997 by john wiley sons ltd

distributed algorithms for the computation of noncooperative equilibria abstract in this paper a general class of nonquadratic convex nash games is studied from the points of view of existence stability and iterative computation of noncooperative equilibria conditions for contraction of general nonlinear operators are obtained which are then used in the stability study of such games these lead to existence and uniqueness conditions for stable nash equilibrium solutions under both global and local analysis also convergence of an algorithm which employs inaccurate search techniques is verified it is shown in the context of a fish war example that the algorithm given is in some aspects superior to various algorithms found in the literature and is furthermore more meaningful for real world implementation

variational integrators and the newmark algorithm for conservative and dissipative mechanical systems the purpose of this work is twofold first we demonstrate analytically that the classical newmark family as well as related integration algorithms are variational in the sense of the veselov formulation of discrete mechanics such variational algorithms are well known to be symplectic and momentum preserving and to often have excellent global energy behavior this analytical result is veried through numerical examples and is believed to be one of the primary reasons that this class of algorithms performs so well second we develop algorithms for mechanical systems with forcing and in particular for dissipative systems in this case we develop integrators that are based on a discretization of the lagrange d alembert principle as well as on a variational formulation of dissipation it is demonstrated that these types of structured integrators have good numerical behavior in terms of obtaining the correct amounts by which the energy changes over the integration run

a nonsmooth multiscale method for solving frictional two body contact problems in 2d and 3d with multigrid efficiency we present a nonsmooth multiscale method for the numerical solution of frictional contact problems in 2d and 3d the computational effort is comparable to that of solving linear problems our method does not require any regularization neither of the nonpenetration condition nor of the friction law and can be applied to contact problems with tresca friction as well as to contact problems with coulomb friction for the case of tresca friction the global convergence of the method is shown for the more complicated case of coulomb friction we develop a nonsmooth multiscale method which can be directly applied to the corresponding variational quasi inequality no outer iteration is required moreover our multiscale approach is general in the sense that it can be used in the context of geometric as well as algebraic multigrid methods nonconforming domain decomposition techniques or mortar methods are employed in order to enforce the transmission conditions at the interface between different bodies with nonmatching meshes numerical examples illustrate the high robustness and efficiency of our method

partially b regular optimization and equilibrium problems this paper introduces a concept termed partial b regularity for a feasible solution to a bivariate constraint system and shows that this condition leads to the equivalence between the b stationarity of a pair of lifted and unlifted programs in particular for an optimization problem with a univariate pseudoconvex objective function constrained by such a nonconvex bivariate system partial b regularity provides a sufficient condition for a b stationary point to be globally optimal applications of partial b regularity to several classes of optimization and equilibrium problems are presented these include a lexicographic optimization problem a nonconvex mathematical program with equilibrium constraints mpec that arises from a convex implicit value function optimization problem and a nash equilibrium program with equilibrium constraints

newton s method for computing a normalized equilibrium in the generalized nash game through fixed point formulation we consider the generalized nash equilibrium problem gnep where not only the players cost functions but also their strategy spaces depend on the rivals decision variables existence results for gneps are typically shown by using a fixed point argument for a certain set valued function here we use a regularization of this set valued function in order to obtain a single valued function that is easier to deal with from a numerical point of view we show that the fixed points of the latter function constitute an important subclass of the generalized equilibria called normalized equilibria this fixed point formulation is then used to develop a nonsmooth newton method for computing a normalized equilibrium the method uses a so called computable generalized jacobian that is much easier to compute than clarke generalized jacobian or b subdifferential we establish local superlinear quadratic convergence of the method under the constant rank constraint qualification which is weaker than the frequently used linear independence constraint qualification and a suitable second order condition some numerical results are presented to illustrate the performance of the method

a stable energy conserving approach for frictional contact problems based on quadrature formulas a common approach for the numerical simulation of non linear multi body contact problems is the use of lagrange multipliers to model the contact conditions the stability of standard algorithms is improved by introducing a modified mass matrix which assigns no mass to the potential contact nodes by this the spurious algorithmic oscillations in the multiplier do not occur any more which facilitates the application of the primal dual active set strategy to dynamical contact problems the new mass matrix is calculated via a modified quadrature formula that needs no extra computational cost in addition the conservation properties of the underlying algorithm are transferred to the modified mass version different numerical examples for frictional two body contact problems illustrate the improvement in the results for the contact stresses copyright 2007 john wiley sons ltd

improved implicit integrators for transient impact problems geometric admissibility within the conserving framework the value of energy and momentum conserving algorithms has been well established for the analysis of highly non linear systems including those characterized by the nonsmooth non linearities of an impact event this work proposes an improved integration scheme for frictionless dynamic contact seeking to preserve the stability properties of exact energy and momentum conservation without the heretofore unavoidable compromise of violating geometric admissibility as established by the contact constraints the physically motivated introduction of a discrete contact velocity provides an algorithmic framework that ensures exact conservation locally while remaining independent of the choice of constraint treatment thus making full conservation equally possible in conjunction with a penalty regularization as with an exact lagrange multiplier enforcement the discrete velocity effects are incorporated as a post convergence update to the system velocities and thus have no direct effect on the non linear solution of the displacement equilibrium equation the result is a robust implicit algorithmic treatment of dynamic frictionless impact appropriate for large deformations and fully conservative for a range of geometric constraints copyright 2001 john wiley sons ltd

iterative techniques for the nash solution in quadratic games with unknown parameters we study adaptive schemes for repeated quadratic nash games in a deterministic and a stochastic framework the convergence of the schemes is demonstrated under certain conditions

a time discretization scheme based on rothe s method for dynamical contact problems with friction abstract we present a new dissipative and contact stabilized time discretization scheme for dynamic frictional contact problems which is based on rothe s method especially for the case of coulomb friction the stability of the contact stresses is of crucial importance as they directly influence the frictional behavior in our approach we obtain an accurate description of the frictional stresses by a time discretized friction law allowing for an implicit treatment of the contact forces in the framework of the newmark scheme moreover undesirable oscillations at the contact interface are removed by employing an additional l 2 projection within the predictor step since the implicit treatment of the material behavior and the frictional response requires a quasi variational inequality to be solved in each time step we derive a non smooth multiscale method which allows for the efficient and robust solution of these highly non linear problems the convergence of this multiscale method is proven for the case of tresca friction for the case of coulomb friction an inexact fixed point iteration in the normal stresses is used we furthermore consider the case of two body contact here the information transfer at the contact interface is realized by means of mortar methods which provide a stable discretization of the relative displacements and the stresses at the contact boundary numerical results for the resulting fully discrete scheme in 3d are presented showing the high accuracy of the proposed method

a family of space time connecting discretization schemes with local impact detection for elastodynamic contact problems abstract we derive formulate and analyze a new family of discretization schemes for elastodynamic contact problems which implicitly resolve the individual impact times for each node on the contact interface within our approach information from the space discrete setting is incorporated into the time discretization by means of pointwise chosen parameters for the time discretization scheme the members of this family can be interpreted as modified newmark schemes thus making them easily understandable and implementable we prove that for certain parameter choices the algorithms are dissipative methods further as our analysis and our numerical experiments show a special solution dependent choice of parameters leads to a new space time connecting discretization with a highly stable behavior of displacements velocities and boundary stresses at the contact interface

finite element analysis of nonsmooth contact this work develops robust contact algorithms capable of dealing with complex contact situations involving several bodies with corners amongst the mathematical tools we bring to bear on the problem is nonsmooth analysis following clarke f h clarke optimization and nonsmooth analysis john wiley and sons new york 1983 we specifically address contact geometries for which both the use of normals and gap functions have difficulties and therefore precludes the application of most contact algorithms proposed to date such situations arise in applications such as fragmentation where angular fragments undergo complex collision sequences before they scatter we demonstrate the robustness and versatility of the nonsmooth contact algorithms developed in this paper with the aid of selected two and three dimensional applications

a note on the time compression approximation in this paper the work by sivapalan and milly 1989 is extended beyond the time of ponding to estimate the errors of the time compression approximation tca when an exact solution exists the diffusion coefficient is set to be constant and gravitational effects are not considered here the largest error of standard tca in calculating cumulative infiltration is about 2 5 while its underestimate of the time to ponding is about 19 a modified tca is also proposed in which the exact time of ponding is used directly instead of the flux to improve the estimate of postponding infiltration the error for the modified tca is about half of the error for the standard tca in predicting cumulative infiltration and thus the modified tca should be preferred wherever ponding time is or can be measured

real and complex monotone communication games noncooperative game theoretic tools have been increasingly used to study many important resource allocation problems in communications networking smart grids and portfolio optimization in this paper we consider a general class of convex nash equilibrium problems neps where each player aims at solving an arbitrary smooth convex optimization problem differently from most of current works we do not assume any specific structure for the players problems and we allow the optimization variables of the players to be matrices in the complex domain our main contribution is the design of a novel class of distributed asynchronous best response algorithms suitable for solving the proposed neps even in the presence of multiple solutions the new methods whose convergence analysis is based on variational inequality vi techniques can select among all the equilibria of a game those that optimize a given performance criterion at the cost of limited signaling among the players this is a major departure from existing best response algorithms whose convergence conditions imply the uniqueness of the ne some of our results hinge on the use of vi problems directly in the complex domain the study of these new kind of vis also represents a noteworthy innovative contribution we then apply the developed methods to solve some new generalizations of single input single output siso and multiple input multiple output mimo games in cognitive radio systems showing a considerable performance improvement over classical pure noncooperative schemes

penalty methods for the solution of generalized nash equilibrium problems the generalized nash equilibrium problem gnep is an extension of the classical nash equilibrium problem where both the objective functions and the constraints of each player may depend on the rivals strategies this class of problems has a multitude of important engineering applications and yet solution algorithms are extremely scarce in this paper we analyze in detail a globally convergent penalty method that has favorable theoretical properties we also consider strengthened results for a particular subclass of problems very often considered in the literature basically our method reduces the gnep to a single penalized and nonsmooth nash equilibrium problem we suggest a suitable method for the solution of the latter penalized problem and present extensive numerical results

three modeling paradigms in mathematical programming celebrating the sixtieth anniversary since the zeroth international symposium on mathematical programming was held in 1949 this paper discusses several promising paradigms in mathematical programming that have gained momentum in recent years but have yet to reach the main stream of the field these are competition dynamics and hierarchy the discussion emphasizes the interplay between these paradigms and their connections with existing subfields including disjunctive equilibrium and nonlinear programming and variational inequalities we will describe the modeling approaches mathematical formulations and recent results of these paradigms and sketch some open mathematical and computational challenges arising from the resulting optimization and equilibrium problems our goal is to elucidate the need for a systematic study of these problems and to inspire new research in the field