approximating saddle points as equilibria of differential inclusions abstract motivated by nonlinear programming we study a process involving continuous adjustments of both primal and dual variables nonsmoothness in problem data or simple restrictions on variables make the velocity of the process possibly nonunique the process obtained is shown to converge locally under strict monotonicity conditions examples are given in terms of augmented lagrangians the results are believed to provide a basis for the construction of algorithms

infinite horizon optimization we consider the general problem of choosing a discounted cost minimizing infinite sequence of decisions from a closed subset of the product space formed by a sequence of arbitrary compact metric spaces examples include equipment replacement production planning and more generally infinite stage mathematical programs it is shown that the optimal costs for finite horizon approximating problems converge to the optimal infinite horizon cost as the horizons diverge to infinity moreover the existence of a unique algorithmically optimal i e accumulation point solution is shown to be a necessary and sufficient condition for convergence in the product topology i e policy convergence of all finite horizon optima under the weaker condition of hausdorff convergence of the sets of finite horizon optima to the set of infinite horizon optima we show how to force policy convergence through a natural tie breaking rule finally a forward algorithm is presented which in the presence of a unique infinite horizon optimum is guaranteed to converge

finite dimensional approximation in infinite dimensional mathematical programming we consider the problem of approximating an optimal solution to a separable doubly infinite mathematical program p with lower staircase structure by solutions to the programs p n obtained by truncating after the first n variables and n constraints of p viewing the surplus vector variable associated with the nth constraint as a state and assuming that all feasible states are eventually reachable from any feasible state we show that the efficient set of all solutions optimal to all possible feasible surplus states for p n converges to the set of optimal solutions to p a tie breaking algorithm which selects a nearest point efficient solution for p n is shown for convex programs to converge to an optimal solution to p a stopping rule is provided for discovering a value of n sufficiently large to guarantee any prespecified level of accuracy the theory is illustrated by an application to production planning

a convolution test equation for double delay integral equations in this paper we consider volterra integral equations with two constant delays and we carry out the stability analysis of direct quadrature methods with respect to the linear convolution test equation y t email protected t t 2 t t 1 email protected t s y s ds email protected t 2 t we investigate the analytical behavior of the solution of the test equation and derive the qualitative and quantitative properties of the numerical solution the numerical experiments show that the stability conditions obtained represent sufficient conditions for the stability of the numerical method applied to a more general equation

non constant discounting and differential games with random time horizon previous results on non constant discounting in continuous time are extended to the field of deterministic differential games with a stochastic terminal time a dynamic programming equation is derived for problems with general time inconsistent preferences and random duration different cooperative and non cooperative solution concepts for differential games with random duration are analyzed the results are illustrated by solving the cake eating problem describing the classical model of management of a nonrenewable resource

approximation to optimization problems an elementary review during the last two decades the concept of epi convergence was introduced and then was used in various investigations in optimization and related areas the aim of this review is to show in an elementary way how closely the arguments in the epi convergence approach are related to those of the classical theory of convergence of functions

a stable numerical method for volterra integral equations with discontinuous kernel numerical methods for volterra integral equations with discontinuous kernel need to be tuned to their peculiar form here we propose a version of the trapezoidal direct quadrature method adapted to such a type of equations in order to delineate its stability properties we first investigate about the behavior of the solution of a suitable basic test equation and then we find out under which hypotheses the trapezoidal direct quadrature method provides numerical solutions which inherit the properties of the continuous problem

aggregation and discretization in multistage stochastic programming multistage stochastic programs have applications in many areas and support policy makers in finding rational decisions that hedge against unforeseen negative events in order to ensure computational tractability continuous state stochastic programs are usually discretized and frequently the curse of dimensionality dictates that decision stages must be aggregated in this article we construct two discrete stage aggregated stochastic programs which provide upper and lower bounds on the optimal value of the original problem the approximate problems involve finitely many decisions and constraints thus principally allowing for numerical solution

some clarifications on the transversality condition in this paper the authors study a general concave discrete time infinite horizon optimal control problem they establish necessary and sufficient conditions for optimality in the weak sense of w a brock and for optimality in the strong sense of d gale the corresponding transversality conditions are general exhaustion properties of the limit value of the optimal state variables these properties cover and extend the well known results obtained in special cases copyright 1990 by the econometric society

discrete time pontryagin principles with infinite horizon we establish necessary conditions of optimality for problems of optimal control theory in the discrete time framework with infinite horizon our necessary conditions are in the form of pontryagin principles we treat smooth and partially nonsmooth settings without concavity a strong motivation to the study of such problems comes from dynamical macroeconomic theory and there exist also some motivations provided by physics

existence of stationary correlated equilibria with symmetric information for discounted stochastic games the following theorem is proved every nonzero sum discounted stochastic game in countably generated measurable state space with compact metric action spaces admits a stationary correlated equilibrium point with symmetric public information whenever the immediate rewards and transition densities are measurable with respect to the state variable and continuous with respect to joint actions

numerical analysis of a high order method for nonlinear delay integral equations abstract in this paper we introduce a new class of multistep collocation method mcm for solving the non linear volterra functional integral equations vfies including two types of linear and non linear lag functions we divide the definition domain into several subintervals according to the primary discontinuous points associated with the delay function in each subinterval where the solution is smooth enough we can apply mcm to approximate the solution convergence analysis of the method for both types is also analyzed and illustrative five examples are provided to confirm the applicability of this method

on the stability of runge kutta methods for delay integral equations we present a class of runge kutta methods for the numerical solution of a class of delay integral equations dies described by two different kernels and with a fixed delay the stability properties of these methods are investigated with respect to a test equation with linear kernels depending on complex parameters the results are then applied to collocation methods in particular we obtain that any collocation method for dies resulting from ana stable collocation method for odes with a stepsize which is submultiple of the delay preserves the asymptotic stability properties of the analytic solutions

transversality conditions for some infinite horizon discrete time optimization problems this paper considers a class of discrete time infinite horizon optimization problems arising in economics necessary optimality conditions for such problems consist of an euler equation and a transversality condition at infinity our concern here is with the latter condition previous results for this type of problem have required such restrictions as concavity and boundedness of the utility functions whose sum is being maximized here we derive a transversality condition that remains valid under considerably more general hypotheses in doing so we introduce methods that may also be applicable to other situations such as optimization in continuous time

the existence of subgame perfect equilibrium in continuous games with almost perfect information a case for public randomization the starting point of this paper is a simple regular dynamic game in which subgame perfect equilibrium fails to exist examination of this example shows that existence would be restored if players were allowed to observe the output of a public randomization device the main result of the paper shows that the introduction of public randomization yields existence not only in the example but also in a large class of dynamic games it is also argued that the introduction of public randomization is the minimal robust extension of subgame perfect equilibrium in this class of games

convergence of approximate saddle points abstract relationships between convergence properties of a family of extended real valued functions and convergence properties of the family of the sets of their approximate or exact saddle points are investigated the use of limits on convergence spaces enables us to refine several theorems and to get entirely new results i e on the convergence of approximate saddle points to an exact saddle point the fundamental role of marginal functions in the convergence of approximate saddle points is observed the corresponding results on convergence of minima are recovered as a special case

convex infinite horizon programs we establish conditions under which a sequence of finite horizon convex programs monotonically increases in value to the value of the infinite program a subsequence of optimal solutions converges to the optimal solution of the infinite problem if the conditions we impose fail then roughtly the optimal value of the infinite horizon problem is an improper convex function under more restrictive conditions we establish the necessary and sufficient conditions for optimality this constructive procedure gives us a way to solve the infinite long range problem by solving a finite short range problem it appears to work well in practice

convex optimal control problems with smooth hamiltonians optimal control problems with convex costs for which hamiltonians have lipschitz continuous gradients are considered examples of such problems including extensions of the linear quadratic regulator with hard and possibly state dependent control constraints and piecewise linear quadratic penalties are given lipschitz continuous differentiability and strong convexity of the terminal cost are shown to be inherited by the value function leading to lipschitz continuity of the optimal feedback with no regularity assumptions on the limiting problem epi convergence of costs which can be equivalently described by pointwise convergence of hamiltonians is shown to guarantee epi convergence of value functions resulting schemes of approximating any concave convex hamiltonian by continuously differentiable ones are displayed auxiliary results about existence and stability of saddle points of quadratic functions over polyhedral sets are also proved tools used are based on duality theory of convex and saddle functions

dynamic choices of hyperbolic consumers laboratory and field studies of time preference find that discount rates are much greater in the short run than in the long run hyperbolic discount functions capture this property this paper solves the decision problem of a hyperbolic consumer who faces stochastic income and a borrowing constraint the paper uses the bounded variation calculus to derive the hyperbolic euler relation a natural generalization of the standard exponential euler relation the hyperbolic euler relation implies that consumers act as if they have endogenous rates of time preference that rise and fall with the future marginal propensity to consume e g discount rates that endogenously range from 5 to 41 for the example discussed in the paper

approximations of nash equilibria inspired by previous works on approximations of optimization problems and recent papers on the approximation of walrasian and nash equilibria and on stochastic variational inequalities the present paper investigates the approximation of nash equilibria and clarifies the conditions required for the convergence of the approximate equilibria via a direct approach a variational approach and an optimization approach besides directly addressing the issue of convergence of nash equilibria via approximation our investigation leads to a deeper understanding of various notions of functional convergence and their interconnections more importantly the investigation yields improved conditions for convergence of the approximate nash equilibria via the variational approach an illustrative application of our results to the approximation of a nash equilibrium in a competitive capacity expansion model under uncertainty is presented

lopsided convergence an extension and its quantification much of the development of lopsided convergence for bifunctions defined on product spaces was in response to motivating applications a wider class of applications requires an extension that would allow for arbitrary domains not only product spaces this leads to an extension of the definition and its implications that include the convergence of solutions and optimal values of a broad class of minsup problems in the process we relax the definition of lopsided convergence even for the classical situation of product spaces we now capture applications in optimization under stochastic ambiguity generalized nash games and many others we also introduce the lop distance between bifunctions which leads to the first quantification of lopsided convergence this quantification facilitates the study of convergence rates of methods for solving a series of problems including minsup problems generalized nash games and various equilibrium problems

an information based approximation scheme for stochastic optimization problems in continuous time dynamic stochastic optimization problems with a large possibly infinite number of decision stages and high dimensional state vectors are inherently difficult to solve in fact scenario tree based algorithms are unsuitable for problems with many stages while dynamic programming type techniques are unsuitable for problems with many state variables this paper proposes a stage aggregation scheme for stochastic optimization problems in continuous time thus having an extremely large i e uncountable number of decision stages by perturbing the underlying data and information processes we construct two approximate problems that provide bounds on the optimal value of the original problem moreover we prove that the gap between the bounds converges to zero as the stage aggregation is refined if massive aggregation of stages is possible without sacrificing too much accuracy the aggregate approximate problems can be addressed by means of scenario tree based methods the suggested approach applies to problems that exhibit randomness in the objective and the constraints while the constraint functions are required to be additively separable in the decision variables and random parameters

shadow prices in infinite dimensional linear programming we consider the class of linear programs that can be formulated with infinitely many variables and constraints but where each constraint has only finitely many variables this class includes virtually all infinite horizon planning problems modeled as infinite stage linear programs examples include infinite horizon production planning under time varying demands and equipment replacement under technological change we provide under a regularity condition conditions that are both necessary and sufficient for strong duality to hold moreover we show that under these conditions the lagrangean function corresponding to any pair of primal and dual optimal solutions forms a linear support to the optimal value function thus extending the shadow price interpretation of an optimal dual solution to the infinite dimensional case we illustrate the theory through an application to production planning under time varying demands and costs where strong duality is established

stationary markov equilibria conditions are established under which there are equilibria in various settings described by ergodic markov processes with a borel state space s let rho s denote the probability measures on s and let s approaching g s is a subset of rho s be a possibly empty valued correspondence with closed graph a non empty measurable set j is a subset of s is self justified if g s intersects rho j is not empty for all s an element of j a time homogeneous markov equilibrium thme for g is a self justified set j and a measurable selection ii j approaching rho j from the restriction of g to j the paper gives sufficient conditions for existence of compact self justified sets and applies the theorem if g is convex valued and has a compact self justified set then g has an thme with an ergodic measure coauthors are j geanakoplos a mas colell and a mclennan copyright 1994 by the econometric society

discrete time finite horizon approximation of infinite horizon optimization problems with steady state invariance the computation of large scale nonlinear intertemporal optimization problems requires time aggregation a procedure generally adopted is shown to introduce a dependency of the solution steady state to a specific choice of sequence of time intervals the authors establish necessary and sufficient conditions to avoid this dependency the result is a considerable improvement in the numerical accuracy of the time aggregated approximation the conditions apply to a broad class of models and prove useful in large scale applied general equilibrium modeling this conclusion is highlighted by a comparison with a spectral projection method using optimal orthogonal collocation as in k l judd 1992 copyright 1994 by the econometric society

duality gaps in semi infinite linear programming an approximation problem this paper builds upon the relationship between the objective function of a semi infinite linear program and its constraints to identify a class of semi infinite linear programs which do not have a duality gap the key idea is to guarantee the approximation of the primal program by a sequence of linear programs where thenth approximating program is to minimize the objective function subject to the firstn constraints the paper goes on to show that any program not in the identified class can be linearly perturbed into it with the optimal value of the perturbed program converging to the optimal value of the original program the results are then extended to the case when an uncountable number of constraints are present by reducing this case to the countable case

asymptotic behavior of statistical estimators and of optimal solutions of stochastic optimization problems we study the asymptotic behavior of the statistical estimators that maximize a not necessarily dieren tiable criterion function possibly subject to side constraints equalities and inequalities the consistency results generalize those of wald and huber conditions are also given under which one is still able to obtain asymptotic normality the analysis brings to the fore the relationship between the problem of nding statistical estimators and that of nding the optimal solutions of stochastic optimization problems with partial information the last section is devoted to the properties of the saddle points of the associated lagrangians

variational convergence of bifunctions motivating applications it is shown that a number of variational and equilibrium problems can be cast as finding the maxinf points or minsup points of bivariate functions for short bifunctions these problems include linear and nonlinear complementarity problems fixed points variational inequalities inclusions noncooperative games and walras and nash equilibrium problems one appeals to the theory of lopsided convergence for bifunctions to derive stability results for each one of these problems

stochastic games of resource extraction we study stochastic games of resource extraction in which the transition probability is a convex combination of stochastic kernels with coefficients depending on the joint investments of the players our approach covers the unbounded utility case which was not examined in this class of games beforehand we give two theorems on the existence of pure stationary markov perfect equilibria for the models of games under consideration a detailed discussion with illustrative examples explains the meaning of our assumptions and their relation to the conditions used earlier in the literature

isometries for the legendre fenchel transform it is shown that on the space of lower semicontinuous convex functions defined on r the conjugation map the legendre fenchel transform is an isometry with respect to some metrics consistent with the epi topology we also obtain isometries for the infinite dimensional case hilbert space and reflexive banach space but this time they correspond to topologies finer than the mosco epi topology

model building techniques for the correction of end effects in multistage convex programs this paper considers model building procedures that can be used to reduce end effects in multistage planning models in particular it emphasizes some problems that are peculiar to energy planning applications the paper presents and compares four methods for mitigating end effects it concludes that one of these four methods the dual equilibrium procedure is best the major drawback of the dual equilibrium procedure is its lack of simplicity if it is to be widely used then it must be widely understood

stability criteria for certain delay integral equations of volterra type abstract in this paper we study the asymptotic stability of the solution of the following delay integral equation of volterra type o x a 0 a 1 x s y s ds 1 alpha o x a 0 a 1 x s y s ds y x x x 0 is constant and 0 1 stability criteria are provided for certain s and the parameters a 0 a 1 and the aim of this study is to understand the effect of the delay on the asymptotic stability of the solution of volterra integral equations as such the parameters and 1 appear with the same kernel in both integrals of the equation we also provide four algorithmic stability tests and include several examples and stability regions for certain values of the parameters a 0 a 1 and

on univariate function identification problems applications in the areas of data fitting forecasting and estimation naturally lead to a rich class of constrained infinite dimensional optimization problems over extended real valued semicontinuous functions we discuss a framework for dealing with such applications even in the presence of nearly arbitrary constraints on the functions we formulate computationally tractable approximating problems relying on piecewise polynomial semicontinuous approximations of the actual functions the approximations enable the study of evolving problems caused by incrementally arriving data and other information in contrast to an earlier more general treatment we focus on optimization over functions defined on a compact interval of the real line which still addresses a large number of applications the paper provides an introduction to the subject through simplified derivations and illustrative examples

robust markov perfect equilibria abstract in this paper we study a markov decision model with quasi hyperbolic discounting and transition probability function depending on an unknown parameter assuming that the set of parameters is finite the sets of states and actions are borel and the transition probabilities satisfy some additivity conditions and are atomless we prove the existence of a non randomised robust markov perfect equilibrium

discounted value representations of temporal preferences this paper examines the proposition that temporal preferences may be represented in terms of discount functions an analytical framework is developed within which preferences are defined over a set of programs or prospects over time necessary and sufficient conditions for generalized discounting discounting as well as exponential discounting are established for a class of program sets over discrete and continuous time

circumventing the slater conundrum in countably infinite linear programs abstract duality results on countably infinite linear programs are scarce subspaces that admit an interior point which is a sufficient condition for a zero duality gap yield a dual where the constraints cannot be expressed using the ordinary transpose of the primal constraint matrix subspaces that permit a dual with this transpose do not admit an interior point this difficulty has stumped researchers for a few decades it has recently been called the slater conundrum we find a way around this hurdle we propose a pair of primal dual spaces with three properties the series in the primal and dual objective functions converge the series defined by the rows and columns of the primal constraint matrix converge and the order of sums in a particular iterated series of a double sequence defined by the primal constraint matrix can be interchanged so that the dual is defined by the ordinary transpose weak duality and complementary slackness are then immediate instead of using interior point conditions to establish a zero duality gap we call upon the planning horizon method when the series in the primal and dual constraints are continuous we prove that strong duality holds if a sequence of optimal solutions to finite dimensional truncations of the primal and dual cilps has an accumulation point we show by counterexample that the requirement that such an accumulation point exist cannot be relaxed our results are illustrated using several examples and are applied to countable state markov decision processes and to a problem in robust optimization

a shadow simplex method for infinite linear programs we present a simplex type algorithm that is an algorithm that moves from one extreme point of the infinite dimensional feasible region to another not necessarily adjacent extreme point for solving a class of linear programs with countably infinite variables and constraints each iteration of this method can be implemented in finite time whereas the solution values converge to the optimal value as the number of iterations increases this simplex type algorithm moves to an adjacent extreme point and hence reduces to a true infinite dimensional simplex method for the important special cases of nonstationary infinite horizon deterministic and stochastic dynamic programs

another isometry for the legendre fenchel transform abstract a new distance is introduced on the space of extended real valued lower semicontinuous convex functions defined on reflexive banach spaces it is shown that with this notion of distance the legendre fenchel is an isometry

duality in infinite dimensional linear programming we consider the class of linear programs with infinitely many variables and constraints having the property that every constraint contains at most finitely many variables while every variable appears in at most finitely many constraints examples include production planning and equipment replacement over an infinite horizon we form the natural dual linear programming problem and prove strong duality under a transversality condition that dual prices are asymptotically zero that is we show under this transversality condition that optimal solutions are attained in both primal and dual problems and their optimal values are equal the transversality condition and hence strong duality is established for an infinite horizon production planning problem

existence results and finite horizon approximates for infinite horizon optimization problems the paper deals with infinite horizon optimization problems the existence of optimal solutions is obtained as a consequence of an asymptotic growth condition we also exhibit finite horizon approximates that yield upper and lower bounds for the optimal values and whose optimal solutions converge to the long term optimal trajectories

when uniform weak convergence fails empirical processes for dependence functions via epi and hypographs for copulas whose partial derivatives are not continuous everywhere on the interior of the unit cube the empirical copula process does not converge weakly with respect to the supremum distance this makes it hard to verify asymptotic properties of inference procedures for such copulas to resolve the issue a new metric for locally bounded functions is introduced and the corresponding weak convergence theory is developed convergence with respect to the new metric is related to epi and hypoconvergence and is weaker than uniform convergence still for continuous limits it is equivalent to locally uniform convergence whereas under mild side conditions it implies lp convergence even in cases where uniform convergence fails weak convergence with respect to the new metric is established for empirical copula and tail dependence processes no additional assumptions are needed for tail dependence functions and for copulas the assumptions reduce to existence and continuity of the partial derivatives almost everywhere on the unit cube the results are applied to obtain asymptotic properties of minimum distance estimators goodness of fit tests and resampling procedures

convergence of functions equi semicontinuity the ever increasing complexity of the systems to be modeled and analyzed taxes the existing mathematical and numerical techniques far beyond our present day capabilities by their intrinsic nature some problems are so difficult to solve that at best we may hope to find a solution to an approximation of the original problem stochastic optimization problems except in a few special cases are typical examples of this class this however raises the question of what is a valid approximate to the original problem the design of the approximation must be such that i the solution to the approximate provides approximate solutions to the original problem and ii a refinement of the approximation yields a better approximate solution the classical techniques for approximating functions are of little use in this setting in fact very simple examples show that classical approximation techniques dramatically fail in meeting the objectives laid out above what is needed at least at a theoretical level is to design the approximates to the original problem in such a way that they satisfy an epi convergence criterion the convergence of the functions defining the problem is to be replaced by the convergence of the sets defined by these functions that type of convergence has many properties but for our purpose the main one is that it implies the convergence of the optimal solutions this article is devoted to the relationship between the epi convergence and the classical notion of pointwise convergence a strong semicontinuity condition is introduced and it is shown to be the link between these two types of convergences it provides a number of useful criteria which can be used in the design of approximates to difficult problems

what shape is your conjugate a survey of computational convex analysis and its applications computational convex analysis algorithms have been rediscovered several times in the past by researchers from different fields to further communications between practitioners we review the field of computational convex analysis which focuses on the numerical computation of fundamental transforms arising from convex analysis current models use symbolic numeric and hybrid symbolic numeric algorithms our objective is to disseminate widely the most efficient numerical algorithms useful for applications in image processing computing the distance transform the generalized distance transform and mathematical morphology operators partial differential equations solving hamilton jacobi equations and using differential equations numerical schemes to compute the convex envelope max plus algebra computing the equivalent of the fast fourier transform multifractal analysis etc the fields of applications include among others computer vision robot navigation thermodynamics electrical networks medical imaging and network communication

optimal control problems involving optimal economic growth with infinite horizon abstract we examine the existence of optimal solutions for a very general optimal economic growth problem with unbounded planning time horizon a convexity hypothesis about a certain orientor field plays a crucial role in the proof then we conduct sensitivity analysis and determine the relation between optimal pairs and the initial capital accumulation finally we see what happens if the convexity hypothesis we have made no longer exists this leads us to the introduction of the relaxed system and its comparison with the original one

variational convergence of bivariate functions lopsided convergence we explore convergence notions for bivariate functions that yield convergence and stability results for their maxinf or minsup points this lays the foundations for the study of the stability of solutions to variational inequalities the solutions of inclusions of nash equilibrium points of non cooperative games and walras economic equilibrium points of fixed points of solutions to inclusions the primal and dual solutions of convex optimization problems and of zero sum games these applications will be dealt with in a couple of accompanying papers

existence of perfect equilibria in a class of multigenerational stochastic games of capital accumulation in this paper we introduce a model of multigenerational stochastic games of capital accumulation where each generation consists of m different players the main objective is to prove the existence of a perfect stationary equilibrium in an infinite horizon game a suitable change in the terminology used in this paper provides in the case of perfect altruism between generations a new nash equilibrium theorem for standard stochastic games with uncountable state space

a stability result for linear markovian stochastic optimization problems in this paper we propose a semi metric for markov processes that allows to bound optimal values of linear markovian stochastic optimization problems similar to existing notions of distance for general stochastic processes our distance is based on transportation metrics as opposed to the extant literature the proposed distance is problem specific i e dependent on the data of the problem whose objective value we want to bound as a result we are able to consider problems with randomness in the constraints as well as in the objective function and therefore relax an assumption in the extant literature we derive several properties of the proposed semi metric and demonstrate its use in a stylized numerical example

an existence result for optimal economic growth problems abstract an existence result for optimal control problems of lagrange type with unbounded time domain is derived very directly from a corresponding result for problems with bounded time domain this subsumes the main existence result of r f baum j optim theory appl 19 1976 89 116 and has the existence results for optimal economic growth problems of s i takekuma j math econom 7 1980 193 208 and m j p magill econometrica 49 1981 679 711 j math anal appl 82 1981 66 74 as simple corollaries in addition a new notion of uniform integrability is used which coincides with the classical notion if the time domain is bounded