computationally feasible bounds for partially observed markov decision processes a partially observed markov decision process pomdp is a sequential decision problem where information concerning parameters of interest is incomplete and possible actions include sampling surveying or otherwise collecting additional information such problems can theoretically be solved as dynamic programs but the relevant state space is infinite which inhibits algorithmic solution this paper explains how to approximate the state space by a finite grid of points and use that grid to construct upper and lower value function bounds generate approximate nonstationary and stationary policies and bound the value loss relative to optimal for using these policies in the decision problem a numerical example illustrates the methodology

an adaptive finite element method for viscoplastic fluid flows in pipes the numerical modelling of the fully developed poiseuille flow of a yield stress fluid in a square section is presented the dead regions in outer corners and the plug region in the center are exhibited numerical computations cover the complete range of the dimensionless number describing the yield stress effect from a newtonian flow to a fully stopped flow the resolution of variational inequalities describing the flow is based on the augmented lagrangian method and a specific mixed finite element method the localization of yield surfaces is approximated by an anisotropic auto adaptive mesh procedure the limit load analysis and the associated limit yield surface are obtained by an extrapolation procedure

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

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

typical kernel size and number of sparse random matrices over galois fields a statistical physics approach using methods of statistical physics we study the average number and kernel size of general sparse random matrices over gf q with a given connectivity profile in the thermodynamical limit of large matrices we introduce a mapping of gf q matrices onto spin systems using the representation of the cyclic group of order q as the q th complex roots of unity this representation facilitates the derivation of the average kernel size of random matrices using the replica approach under the replica symmetric ansatz resulting in saddle point equations for general connectivity distributions numerical solutions are then obtained for particular cases by population dynamics similar techniques also allow us to obtain an expression for the exact and average number of random matrices for any general connectivity profile we present numerical results for particular distributions

intersection of hadamard codes for two binary codes c1 c2 define i c1 c2 c1capc2 to be their intersection number this correspondence establishes that there exist hadamard codes of length 2t for all tges3 with intersection number i if and only if iisin 0 2 4 2t 1 12 2t 1 8 2t 1 also it is proved that for all tges4 if there exists a hadamard matrix of order 4s then there exist hadamard codes of length 2t 2 s with intersection number i if and only if iisin 0 2 4 2 t 3s 12 2t 3s 8 2t 3s

on the additive spl zopf sub 4 linear and non spl zopf sub 4 linear hadamard codes rank and kernel all the possible nonisomorphic additive spl zopf sub 4 linear and non spl zopf sub 4 linear hadamard codes are characterized and the rank and the dimension of the kernel are computed for each one

approximating an infinite stage search problem with a finite horizon model a target is moving randomly on a lattice a searcher starting at the origin wants to follow a search plan that has minimum expected cost very general cost functions are allowed the cost function depends on the time to capture and the search procedure followed at each stage the searcher can move only to adjacent positions or remain at his current position the target s motion may depend on the search procedure thus allowing for the possibility that the target may take evasive action it is shown that only a finite number of plans need to be considered in order to approximate the optimal cost it is also shown that the leading elements of optimal finite horizon plans eventually become the leading elements of optimal infinite horizon plans the case of search on the real line where the cost is o k where k 1 and is the time to capture is considered in detail the results of this paper also generalise the important planning horizon theorems of bean and smith bean j c smith r l 1984 conditions for the existence of planning horizons math oper res 9 391 401

fast solution and detection of minimal forecast horizons in dynamic programs with a single indicator of the future applications to dynamic lot sizing models in most dynamic planning problems one observes that an optimal decision at any given stage depends on limited information i e information pertaining to a limited set of adjacent or nearby stages this holds in particular for planning problems over time where an optimal decision in a given period depends on information related to a limited future time horizon a so called forecast horizon only in this paper we identify a general class of dynamic programs in which an efficient forward algorithm can be designed to solve the problem and to identify minimal forecast horizons such a procedure specifies necessary and sufficient conditions for a stage to arise as a forecast horizon this class of dynamic programs includes the single item dynamic lot sizing model with general concave costs both with and without backlogging to which special attention is given

solution and forecast horizons for infinite horizon nonhomogeneous markov decision processes we consider a nonhomogeneous infinite horizon markov decision process mdp problem with multiple optimal first period policies we seek an algorithm that given finite data delivers an optimal first period policy such an algorithm can thus recursively generate within a rolling horizon procedure an infinite horizon optimal solution to the original problem however it can happen that no such algorithm exists i e the mdp is not well posed equivalently it is impossible to solve the problem with a finite amount of data assuming increasing marginal returns in actions with respect to states and stochastically increasing state transitions with respect to actions we provide an algorithm that is guaranteed to solve the given mdp whenever it is well posed this algorithm determines in finite time a forecast horizon for which an optimal solution delivers an optimal first period policy as an application we solve all well posed instances of the time varying version of the classic asset selling problem

an absolutely stabilized finite element method for the stokes problem on presente une formulation a elements finis absolument stabilisee pour le probleme de stokes on etablit des estimations d erreurs optimales dans la norme l 2 pour l approximation des champs de vitesses et de pressions

technical note identifying forecast horizons in nonhomogeneous markov decision processes a procedure for identifying forecast horizons in nonhomogeneous markov decision processes based on convergence results for relative value functions is developed two different algorithmic implementations of this procedure are discussed and a closed form expression for computing sufficiently long horizons to guarantee epsilon optimality is presented

deterministic equivalence in stochastic infinite horizon problems we consider a general infinite horizon problem with stochastic data we establish conditions under which the stochastic problem may be correctly solved using an equivalent deterministic problem in particular we establish conditions under which there exists an equivalent interest rate which permits replacement of the stochastic data by their expectations this equivalent interest rate can be approximated by a linear function of the ratio of the variance to the mean of the stochastic data applications to production planning are included

a stochastic programming approach for planning horizons of infinite horizon capacity planning problems planning horizon is a key issue in production planning different from previous approaches based on markov decision processes we study the planning horizon of capacity planning problems within the framework of stochastic programming we first consider an infinite horizon stochastic capacity planning model involving a single resource linear cost structure and discrete distributions for general stochastic cost and demand data non markovian and non stationary we give sufficient conditions for the existence of an optimal solution furthermore we study the monotonicity property of the finite horizon approximation of the original problem we show that the optimal objective value and solution of the finite horizon approximation problem will converge to the optimal objective value and solution of the infinite horizon problem when the time horizon goes to infinity these convergence results together with the integrality of decision variables imply the existence of a planning horizon we also develop a useful formula to calculate an upper bound on the planning horizon then by decomposition we show the existence of a planning horizon for a class of very general stochastic capacity planning problems which have complicated decision structure

the z sub 4 linearity of kerdock preparata goethals and related codes certain notorious nonlinear binary codes contain more codewords than any known linear code these include the codes constructed by nordstrom robinson 1967 kerdock 1972 preparata 1968 goethals 1974 and delsarte goethals 1975 it is shown here that all these codes can be very simply constructed as binary images under the gray map of linear codes over z sub 4 the integers mod 4 although this requires a slight modification of the preparata and goethals codes the construction implies that all these binary codes are distance invariant duality in the z sub 4 domain implies that the binary images have dual weight distributions the kerdock and preparata codes are duals over z sub 4 and the nordstrom robinson code is self dual which explains why their weight distributions are dual to each other the kerdock and preparata codes are z sub 4 analogues of first order reed muller and extended hamming codes respectively all these codes are extended cyclic codes over z sub 4 which greatly simplifies encoding and decoding an algebraic hard decision decoding algorithm is given for the preparata code and a hadamard transform soft decision decoding algorithm for the i kerdock code binary first and second order reed muller codes are also linear over z sub 4 but extended hamming codes of length n spl ges 32 and the golay code are not using z sub 4 linearity a new family of distance regular graphs are constructed on the cosets of the preparata code

existence and discovery of average optimal solutions in deterministic infinite horizon optimization we consider the problem of making a sequence of decisions each chosen from a finite action set over an infinite horizon so as to minimize its associated average cost both the feasibility and cost of a decision are allowed to depend upon all of the decisions made prior to that decision moreover time varying costs and constraints are allowed a feasible solution is said to be efficient if it reaches each of the states through which it passes at minimum cost we show that efficient solutions exist and that under a state reachability condition efficient solutions are also average optimal exploiting the characterization of efficiency via a solution s short run as opposed to long run behavior a forward algorithm is constructed which recursively discovers the first second and subsequent decisions of an efficient and hence average optimal infinite horizon solution

conditions for the discovery of solution horizons we present necessary and sufficient conditions for discrete infinite horizon optimization problems with unique solutions to be solvable these problems can be equivalently viewed as the task of finding a shortest path in an infinite directed network we provide general forward algorithms with stopping rules for their solution the key condition required is that of weak reachability which roughly requires that for any sequence of nodes or states it must be possible from optimal states to reach states close in cost to states along this sequence moreover the costs to reach these states must converge to zero applications are considered in optimal search undiscounted markov decision processes and deterministic infinite horizon optimization

on stable equal order finite element formulations for incompressible flow problems in this work we discuss stable equal order finite element formulations for incompressible flow problems based on petrov galerkin methods constructed by adding to the classical galerkin formulation least squares of the governing equations continuous and discontinuous pressure interpolations are considered numerical results are presented reinforcing the numerical analysis

degeneracy in infinite horizon optimization we consider sequential decision problems over an infinite horizon the forecast or solution horizon approach to solving such problems requires that the optimal initial decision be unique we show that multiple optimal initial decisions can exist in general and refer to their existence as degeneracy we then present a conceptual cost perturbation algorithm for resolving degeneracy and identifying a forecast horizon we also present a general near optimal forecast horizon

dual based methods for solving infinite horizon nonstationary deterministic dynamic programs we develop novel dual ascent and primal dual methods to solve infinite horizon nonstationary deterministic dynamic programs these methods are finitely implementable and converge in value to optimality moreover the dual ascent method produces a sequence of improving dual solutions that pointwise converge to an optimal dual solution while the primal dual algorithm provides a sequence of primal basic feasible solutions with value error bounds from optimality that converge to zero our dual based methods work on a more general class of infinite network flow problems that include the shortest path formulation of dynamic programs as a special case to our knowledge these are the first dual based methods proposed in the literature to solve infinite horizon nonstationary deterministic dynamic programs

a new optimality criterion for nonhomogeneous markov decision processes we propose a new definition of optimality for nonhomogeneous markov decision processes called periodic forecast horizon pfh optimality using measures of discounting and ergodicity we establish conditions under which pfh optimal strategies exist and pfh optimality implies the more conventional notions of optimality and average optimality finally we use this definition of optimality as the basis for a direct development of forecast horizon results for discounted and undiscounted nonhomogeneous markov decision processes

families of hadamard bbz_ 2 bbz_ 4 q_ 8 codes a z2z4q8 code is the binary image after a gray map of a subgroup of z2k1 z4k2 q8k3 where q8 is the quaternion group on eight elements such z2z4q8 codes are translation invariant propelinear codes as are the well known z4 linear or z2z4 linear codes in this paper we show that there exist pure z2z4q8 codes that is codes that do not admit any abelian translation invariant propelinear structure we study the dimension of the kernel and rank of the z2z4q8 codes and we give upper and lower bounds for these parameters we give tools to construct a new class of hadamard codes formed by several families of z2z4q8 codes we classify such codes from an algebraic point of view and we improve the upper and lower bounds for the rank and the dimension of the kernel when the codes are hadamard

conditions for the existence of planning horizons we consider the general class of discounted problems involving sequential decision making over an unbounded horizon under the assumptions of a finite set of policy alternatives at each decision time and costs that are eventually uniformly bounded by some exponential existence of an optimal infinite horizon strategy is established moreover it is shown that there is a finite horizon version whose first policy decision agrees with an infinite horizon optimal policy under the additional assumption that the infinite horizon optimal strategy is eventually cyclic the stronger result follows that the infinite horizon optimal solution is unique for almost all interest rates and that therefore a planning horizon exists algorithmic implications are explored for a restricted subclass of these problems

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

a linear programming approach to nonstationary infinite horizon markov decision processes nonstationary infinite horizon markov decision processes mdps generalize the most well studied class of sequential decision models in operations research namely that of stationary mdps by relaxing the restrictive assumption that problem data do not change over time linear programming lp has been very successful in obtaining structural insights and devising solution methods for stationary mdps however an lp approach for nonstationary mdps is currently missing this is because the lp formulation of a nonstationary infinite horizon mdp includes countably infinite variables and constraints and research on such infinite dimensional lps has traditionally faced several hurdles for instance duality results may not hold an extreme point may not be a basic feasible solution and in the context of a simplex algorithm a pivot operation may require infinite data and computations and a sequence of improving extreme points need not converge in value to optimal in this paper we tackle these challenges a

a weighted markov decision process the two most commonly considered reward criteria for markov decision processes are the discounted reward and the long term average reward the first tends to neglect the future concentrating on the short term rewards while the second one tends to do the opposite we consider a new reward criterion consisting of the weighted combination of these two criteria thereby allowing the decision maker to place more or less emphasis on the short term versus the long term rewards by varying their weights the mathematical implications of the new criterion include the deterministic stationary policies can be outperformed by the randomized stationary policies which in turn can be outperformed by the nonstationary policies an optimal policy might not exist we present an iterative algorithm for computing an e optimal nonstationary policy with a very simple structure

a tie breaking rule for discrete infinite horizon optimization we study discrete infinite horizon optimization problems without the common assumption of a unique optimum a method based on solution set convergence is employed for finding optimal initial decisions by solving finite horizon problems this method is applicable to general discrete decision models that satisfy a weak reachability condition the algorithm together with a stopping rule is applied to production planning and capacity expansion and computational results are reported

denumerable state nonhomogeneous markov decision processes we consider denumerable state nonhomogeneous markov decision processes and extend results from both denumerable state homogeneous and finite state nonhomogeneous problems we show that under weak ergodicity accumulation points of finite horizon optima termed algorithmic optima are average cost optimal we also establish the existence of solution horizons finally an algorithm is presented to solve problems of this class for the case where there is a unique algorithmic optimum

ranks and kernels of codes from generalized hadamard matrices the ranks and kernels of generalized hadamard matrices are studied it is proved that any generalized hadamard matrix h q lambda over mathbb f q q 3 or q 3 and gcd 3 lambda not 1 generates a self orthogonal code this result puts a natural upper bound on the rank of the generalized hadamard matrices lower and upper bounds are given for the dimension of the kernel of the corresponding generalized hadamard codes for specific ranks and dimensions of the kernel within these bounds generalized hadamard codes are constructed

convergence of selections with applications in optimization we consider the problem of finding an easily implemented tie breaking rule for a convergent set valued algorithm i e a sequence of compact non empty subsets of a metric space converging in the hausdorff metric our tie breaking rule is determined by nearest point selections detined by uniqueness points in the space i e points having a unique best approximation in the limit set of the convergent algorithm convergence of the algorithm is shown to be equivalent to convergence of all such nearest point selections under reasonable additional hypotheses all points in the metric space have the uniqueness property consequently all points yield convergent nearest point selections i e tie breaking rules for a convergent algorithm we then show how to apply these results to approximate solutions for the following types of problems infinite systems of inequalities semi infinite mathematical programming non convex optimization and infinite horizon optimization

the rank and kernel of extended 1 perfect spl zopf sub 4 linear and additive non spl zopf sub 4 linear codes a binary extended 1 perfect code of length n 1 2 sup t is additive if it is a subgroup of spl zopf sub 2 sup spl alpha spl times spl zopf sub 4 sup spl beta the punctured code by deleting a spl zopf sub 2 coordinate if there is one gives a perfect additive code 1 perfect additive codes were completely characterized and by using that characterization we compute the possible parameters spl alpha spl beta rank and dimension of the kernel for extended 1 perfect additive codes a very special case is that of extended 1 perfect spl zopf sub 4 linear codes

rolling horizon procedures in nonhomogeneous markov decision processes by far the most common planning procedure found in practice is to approximate the solution to an infinite horizon problem by a series of rolling finite horizon solutions although many empirical studies have been done this so called rolling horizon procedure has been the subject of few analytic studies we provide a cost error bound for a general rolling horizon algorithm when applied to infinite horizon nonhomogeneous markov decision processes both in the discounted and average cost cases we show that a doeblin coefficient of ergodicity acts much like a discount factor to reduce this error in particular we show that the error goes to zero for any fixed rolling horizon as this doeblin measure of control over the future decreases the theory is illustrated through an application to vehicle deployment

concepts of forecast and decision horizons applications to dynamic stochastic optimization problems in this paper we develop a rigorous new framework for the concepts of forecast and decision horizons these concepts are conditional in nature and in turn enable us to unify the existing concepts of strong and weak horizons moreover we are able to precisely state the questions of existence of forecast decision horizons finally we are able to further develop the theory of existence of forecast decision horizons in the discounted case and procedures to compute these horizons

rank and kernel of binary hadamard codes in this paper the rank and the dimension of the kernel for binary hadamard codes of length a power of two are studied in general it is well known that the rank of a hadamard code of length n 2 sup t is a value in t 1 n 2 in the present paper the range of possible values for the dimension of the kernel is computed and a construction of hadamard codes of length n 2 sup t for each one of these values is given lower and upper bounds for the rank and dimension of the kernel of a hadamard code of length n 2 sup t are also established finally we construct hadamard codes for all possible ranks and dimension of kernels between these bounds

a regularized stabilized mixed finite element formulation for viscoplasticity of bingham type abstract in this article we present a new mixed stabilized regularized finite element formulation in primitive variables with continuous velocity and discontinuous pressure interpolations for the steady flow of an incompressible fluid of bingham type this formulation is based on an augmented lagrangian regularization technique and a least squares stabilization method mathematical analyses are performed for the new formulation in terms of stability existence and uniqueness of the solution optimal orders of convergence are obtained mathematically improving on classical methods which also present limitations as regards the values of the yield stress numerical results are presented confirming the theory developed here and they show the robustness of the new method with stability obtained for the velocity and especially for the pressure when the yield stress is very high