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

singularly perturbed difference equations in optimal control problems a linear regular problem involving singularly perturbed difference equations is considered the basic features of singularly perturbed systems order reduction separation of time scales boundary layers etc are highlighted application to the numerical treatment of control problems involving stiff differential equations is discussed

elastic wave propagation in unsaturated porous media using hybrid trefftz stress elements summary the stress model of the hybrid trefftz finite element is formulated for the analysis of elastodynamic problems defined on unsaturated porous media the supporting mathematical model is the theory of mixtures with interfaces and considers the full coupling between the solid fluid and gas phases including the effect of seepage acceleration hybrid trefftz stress elements use the free field regular solutions of the homogeneous navier or beltrami equation to construct the approximation of the generalized stresses in the domain of the element the influence of non homogeneous terms in the navier equation is modelled using solutions of the corresponding static problem the resulting elements are highly convergent under p refinement and robust to both low and high excitation frequencies as the trial functions embody relevant physical information on the modelled phenomenon copyright 2013 john wiley sons ltd

on the matrix riccati equation for a singularly perturbed linear discrete control system abstract a difference approximation to a singularly perturbed linear time varying continuous time control system is presented and is called a singularly perturbed linear discrete time control system under several assumptions it is shown that the riccati equation for the reduced system in the sense of singular perturbation agrees with that for the associated subsystem of the full non reduced system when a small parameter in the equation is set zero on the basis of this result it is also shown that the optimal controls and the corresponding trajectories for the two systems are consistent with each other when a small parameter for the full system is set zero

distributed computation of nash equilibria in linear quadratic stochastic differential games in this paper a class of n person stochastic linear quadratic differential games under multiple probabilistic modeling is studied with each player acquiring a noisy measurement of the initial state conditions for the existence and uniqueness of the nash equilibrium is obtained and a method is provided for an iterative distributed computation of the solution the distributed algorithm involves learning in the policy space and it does not require that each player knows the others perceptions of the probabilistic model underlying the decision process for the finite horizon problem such an iteration converges whenever the length of the time horizon is sufficiently small and the limit in this case is an affine policy for all players if the underlying distributions are jointly gaussian when the horizon is infinite and a discount factor is used in the cost functionals the iteration converges under conditions depending on the magnitude of the discount factor the limiting policies again being affine in

singular perturbation methods for a class of initial and boundary value problems in discrete systems a class of singularly perturbed initial and boundary value problems occurring in discrete systems is formulated singular perturbation methods are developed separately for initial value problems and boundary value problems the methods essentially consist of seeking an approximate solution in terms of an outer series based on the degenerate problem and a correction series to recover the auxiliary conditions sacrificed in the process of degeneration some interesting features are explored and the asymptotic correctness of the formal series expansions is clearly established illustrative examples are provided in support of the proposed methods

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

a variational inequality approach for the determination of oligopolistic market equilibrium this paper presents an alternative approach to that by murphy sherali and soyster 13 for computing market equilibria with mathematical programming methods this approach is based upon a variational inequality representation of the problem and the use of a diagonalization relaxation algorithm

decentralized method for computing pareto solutions in multiparty negotiations this paper presents a decentralized method for computing pareto optimal solutions in multiparty negotiations over continuous issues the method is based on the well known weighting method which is decomposed by introducing an own decision variable for each decision maker and by applying the dual decomposition method to the resulting problem the method offers a systematic way for generating some or all pareto optimal solutions so that decision makers do not have to know each others value functions under the assumption of quasilinear value function the requirement that a decision maker knows the explicit form for his value function can be relaxed in that case the decision maker is asked to solve a series of multiobjective programming problems where an additional artificial decision variable is introduced

analysis of the constraint proposal method for two party negotiations abstract in the constraint proposal method a mediator locates points at which the two decision makers have joint tangent hyperplanes we give conditions under which these points are pareto optimal and we prove that under these conditions the mediator s problem has a solution in practice the mediator adjusts a hyperplane going through a reference point until the decision makers most preferred alternatives on the hyperplane coincide we give local convergence conditions for fixed point iteration as an adjustment process we also discuss the relationship of exchange economies and the constraint proposal method and the possible ways of using the method

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

approximate representation of the pareto frontier in multiparty negotiations decentralized methods and privacy preservation abstract multiparty negotiations have drawn much research attention in recent years and an important problem is how to find a pareto optimal solution or the entire pareto frontier in a decentralized way privacy preservation is also important in negotiation analysis the main aim of this paper is to find an approximate representation of the pareto frontier in a decentralized manner and meanwhile all parties privacy can be effectively protected in this paper we propose a decentralized discrete time algorithm based on a weight sum method and the well known subgradient optimization algorithm where a mediator works as a coordinator to help negotiators the proposed algorithm is easily executable and it only requires the mediator to compute a weighted average of the noisy estimates received from negotiators and negotiators to follow a subgradient optimization iteration at this weighted average the proposed algorithm can generate an approximate pareto optimal solution for one particular weight vector and an approximate representation of the pareto frontier by varying appropriately weight vectors the approximation error between the obtained approximate representation and the pareto frontier can be controlled by the number of iterations and the step size moreover it also reveals that the proposed algorithm is privacy preserving as a result of the random disturbance technique and the weighted average scheme used in this algorithm

hybrid trefftz stress element for bounded and unbounded poroelastic media the equations that govern the dynamic response of saturated porous media are first discretized in time to define the boundary value problem that supports the formulation of the hybrid trefftz stress element the total stress and pore pressure fields are directly approximated under the condition of locally satisfying the domain conditions of the problem the solid displacement and the outward normal component of the seepage displacement are approximated independently on the boundary of the element unbounded domains are modelled using either unbounded elements that locally satisfy the sommerfeld condition or absorbing boundary elements that enforce that condition in weak form as the finite element equations are derived from first principles the associated energy statements are recovered and the sufficient conditions for the existence and uniqueness of the solutions are stated the performance of the element is illustrated with the time domain response of a biphasic unbounded domain to show the quality of the modelling that can be attained for the stress pressure displacement and seepage fields using a high order wavelet based time integration procedure copyright 2010 john wiley sons ltd

hybrid finite element formulations for elastodynamic analysis in the frequency domain abstract three alternative sets of hybrid formulations to solve linear elastodynamic problems by the finite element method are presented they are termed hybrid mixed hybrid and hybrid trefftz and differ essentially on the field conditions that the approximation functions are constrained to satisfy locally two models namely the displacement and the stress models are obtained for each formulation depending on whether the tractions or the boundary displacements are the field chosen to implement interelement continuity a fourier time discretization is used to uncouple the solving system in the frequency domain the basic space discretization criterion is implemented directly on the fundamental relations of elastodynamics and used to derive the stress and displacement models of the hybrid mixed formulation the hybrid and hybrid trefftz formulations are presented in sequence as the variants of the hybrid mixed formulation obtained by progressively increasing the constraints on the approximation bases numerical implementation aspects are briefly discussed and the performance of the finite element models is illustrated with numerical applications

a distributed computation algorithm for solving portfolio problems with integer variables a portfolio problem with integer variables can facilitate the use of complex models including models containing discrete asset values transaction costs and logical constraints this study proposes a distributed algorithm for solving a portfolio program to obtain a global optimum for a portfolio problem with n integer variables the objective function first is converted into an ellipse function containing n separated quadratic terms next the problem is decomposed into m equal size separable programming problems solvable by a distributed computation system composed of m personal computers linked via the internet the numerical examples illustrate that the proposed method can obtain the global optimum effectively for large scale portfolio problems involving integral variables

singular perturbational approximations for discrete time balanced systems a new zero order singular perturbational type approximation is developed for model reduction of discrete time linear systems this approximation is also suitable for model reduction of systems which have fast subsystems and which are represented in the internally balanced format subsystem elimination as suggested by moore for continuous time systems does not generalize for the discrete time case and the singular perturbational approximation gives a particular solution to the discrete time internally balanced model reduction problem

order reducing approximation of two time scale discrete linear time varying systems abstract in this paper we study a two time scale discrete time linear time varying system we heuristically find a reduced order approximation to its asymptotic behaviour as the time scale separation tends to infinity this approximation results in a white noise representation of the fast state vector and a corresponding approximation error in the slow state vector after introducing the approximate system we define concepts of continuity and rate of variation which are needed in the discrete time analysis we then prove that as the time scale separation increases the state of the reduced order system asymptotically coincides with the slow state of the original system in the mean square sense on compact time intervals we also find the order of the slow state approximation error covariance

distributed computation of pareto solutions in n player games the problem of computing pareto optimal solutions with distributed algorithms is considered inn player games we shall first formulate a new geometric problem for finding pareto solutions it involves solving joint tangents for the players objective functions this problem can then be solved with distributed iterative methods and two such methods are presented the principal results are related to the analysis of the geometric problem we give conditions under which its solutions are pareto optimal characterize the solutions and prove an existence theorem there are two important reasons for the interest in distributed algorithms first they can carry computational advantages over centralized schemes second they can be used in situations where the players do not know each others objective functions

generating pareto solutions in a two party setting constraint proposal methods this paper presents a class of methods called constraint proposal methods for generating pareto optimal solutions in two party negotiations in these methods joint tangents of the decision makers value functions are searched by adjusting an artificial plane constraint the problem of generating pareto optimal solutions decomposes into ordinary multiple criteria decision making problems for the individual decision makers and into a coordination problem for an assisting mediator depending on the numerical iteration scheme used to solve the coordination problem different constraint proposal methods are obtained we analyze and illustrate the behaviour of some iteration schemes by numerical examples using both precise and imprecise answers from decision makers an example of a method belonging to the class under study is the ramona method that has been previously described from a practical point of view we present the underlying theory for it by describing it as a constraint proposal method and include some applications

constraint proposal method for computing pareto solutions in multi party negotiations abstract the constraint proposal method for computing pareto optimal solutions is extended to multi party negotiations in the method a neutral coordinator assists decision makers in finding pareto optimal solutions so that the elicitation of the decision makers value functions is not required during the procedure the decision makers have to indicate their most preferred points on different sets of linear constraints the method can be used to generate either one pareto optimal solution dominating the status quo solution of the negotiation or an approximation to the pareto frontier in the latter case a distributive negotiation among the efficient agreements can be carried out afterwards

mixed finite element solution of time dependent problems a mixed formulation of the finite element method is used to establish a higher order incremental method for the solution of second order hyperbolic problems the displacement the velocity and optionally the acceleration fields are approximated independently in time using hierarchical bases the time approximation criterion preserves hyperbolicity in the sense that it replaces the solution of hyperbolic problems by the solution of uncoupled helmholtz type elliptic problems which can be subsequently solved using the alternative methods currently in use for discretization of the space dimension the development of the time integration procedure the characterization of its performance in terms of stability accuracy and convergence are illustrated using a polynomial time basis in order to stress the fact that the procedure can be implemented using alternative time bases a wavelet system is used in the solution of nonlinear parabolic and hyperbolic problems the method is well suited to parallel processing and to large time stepping the extension of its application to the solution of other than linear second order hyperbolic problems is discussed