an equilibrium existence theorem bewley 14 theorem l proved an infinite dimensional equilibrium existence theorem which is a significant extension of the classical finite dimensional theorem of arrow and debreu 11 the assumptions on technology and preferences are natural and applicable in a wide variety of cases the proof is based on a limit argument that makes direct use of the existence of equilibrium in the finite dimensional case this paper establishes the existence of equilibrium under assumptions which are essentially the same as those given by bewley with the additional assumption that the preference orderings of consumers are representable by real valued utility functions this approach is related to the welfare approach of negishi 9 and arrow and hahn 2 chap 51 in the finite dimensional case and simplifies the approach originally adopted by bewley 5 in addition to making clear the role played by each of the assumptions in establishing the existence of equilibrium this approach has the merit of constructing directly a certain real valued function that is maximised at an equilibrium a result that provides a powerful tool in the analysis of qualitative properties of an equilibrium a model of resource allocation in continuous time over an infinite horizon that may be viewed as an application of the model that follows is given in 181

brief constrained linear state estimation a moving horizon approach this article considers moving horizon strategies for constrained linear state estimation additional information for estimating state variables from output measurements is often available in the form of inequality constraints on states noise and other variables formulating a linear state estimation problem with inequality constraints however prevents recursive solutions such as kalman filtering and consequently the estimation problem grows with time as more measurements become available to bound the problem size we explore moving horizon strategies for constrained linear state estimation in this work we discuss some practical and theoretical properties of moving horizon estimation we derive sufficient conditions for the stability of moving horizon state estimation with linear models subject to constraints on the estimate we also discuss smoothing strategies for moving horizon estimation our framework is solely deterministic

duality and symmetry in constrained estimation and control problems the lagrangian dual of a constrained linear estimation problem is derived where the resulting dual system is a linear optimal control problem with projected cost variables the symmetry between constrained estimation and optimal control is revealed when the primal problem is expressed in an equivalent form involving projected variables

survey constrained model predictive control stability and optimality model predictive control is a form of control in which the current control action is obtained by solving at each sampling instant a finite horizon open loop optimal control problem using the current state of the plant as the initial state the optimization yields an optimal control sequence and the first control in this sequence is applied to the plant an important advantage of this type of control is its ability to cope with hard constraints on controls and states it has therefore been widely applied in petro chemical and related industries where satisfaction of constraints is particularly important because efficiency demands operating points on or close to the boundary of the set of admissible states and controls in this review we focus on model predictive control of constrained systems both linear and nonlinear and discuss only briefly model predictive control of unconstrained nonlinear and or time varying systems we concentrate our attention on research dealing with stability and optimality in these areas the subject has developed in our opinion to a stage where it has achieved sufficient maturity to warrant the active interest of researchers in nonlinear control we distill from an extensive literature essential principles that ensure stability and use these to present a concise characterization of most of the model predictive controllers that have been proposed in the literature in some cases the finite horizon optimal control problem solved on line is exactly equivalent to the same problem with an infinite horizon in other cases it is equivalent to a modified infinite horizon optimal control problem in both situations known advantages of infinite horizon optimal control accrue

lagrangian duality between constrained estimation and control we show that the lagrangian dual of a constrained linear estimation problem is a particular nonlinear optimal control problem the result has an elegant symmetry which is revealed when the constrained estimation problem is expressed as an equivalent nonlinear optimisation problem the results extend and enhance known connections between the linear quadratic regulator and linear quadratic state estimation problems

riccati equations in optimal filtering of nonstabilizable systems having singular state transition matrices until recently it was believed that a necessary and sufficient condition for convergence of the riccati difference equation of optimal filtering was that the system be both delectable and stabilizable recently it has been shown that the stabilizability condition can be removed but convergence has only established under restrictive assumptions including the requirement that the state transition matrix be nonsingular the present paper generalizes these results in several directions first properties of the algebraic riccati equation are established for the case of singular state transition matrix second several assumptions previously imposed in establishing convergence of the riccati difference equation for systems with unreachable modes on the unit circle are relaxed including replacing observability by detectability weakening the conditions on the initial covariance and allowing the state transition matrix to be singular third results on the convergence and properties of the riccati equations are expressed as both necessary and sufficient conditions whereas previous results were only sufficient these extensions mean that the results have wider applicability including fixed lag smoothing problems and filtering for systems with time delays the implications of the results in the dual problem of optimal control are also studied

suboptimal receding horizon estimation via noise blocking abstract for discrete time linear systems we propose a suboptimal approach to constrained estimation so that the associated computation burden is reduced this is achieved by enforcing a move blocking mb structure in the estimated process noise sequence pns we show that full information estimation fie and receding horizon estimation rhe with mb are both stable in the sense of an observer the techniques in proving stability are inspired by those that have been proposed for standard rhe to be specific stability results are mainly achieved by i carefully embellishing the general assumptions for standard rhe to accommodate the mb requirement ii exploiting the principle of optimality as well as convexity of the quadratic programs qps associated with fie and rhe iii relying on the fact that the kalman filter is the best linear estimator in the least squares sense a crucial requirement in achieving stability for mb rhe is that the segment structure ss of the pns of mb fie for the optimization steps within the receding horizon i e steps between t n and t 1 has to be enforced in the mb rhe optimization as a result the mb rhe strategy becomes a dynamic estimator with a periodically varying computational complexity the theoretical results have been illustrated with examples

tight bounds for online class constrained packing we consider class constrained packing problems in which we are given a set of bins each having a capacity v and c compartments and n items of m different classes and the same unit size we need to fill the bins with items subject to capacity constraints such that items of different classes are placed in separate compartments thus each bin can contain items of at most c distinct classes we consider two optimization goals in the class constrained bin packing problem ccbp our goal is to pack all the items in a minimal number of bins in the class constrained multiple knapsack problem ccmk we wish to maximize the total number of items packed in m bins for m 1 the ccbp and ccmk problems model fundamental resource allocation problems in computer and manufacturing systems both are known to be strongly np hard in this paper we derive tight bounds for the online variants of these problems we first present a lower bound of 1 on the competitive ratio of any deterministic algorithm for the online ccbp where 0 1 depends on v c m and n we show that this ratio is achieved by the algorithm first fit we then consider the temporary ccbp in which items may be packed for a bounded time interval that is unknown in advance we obtain a lower bound of v c on the competitive ratio of any deterministic or randomized algorithm we show that this ratio is achieved by all any fit algorithms finally tight bounds are derived for the online ccmk and the temporary ccmk problems

brief paper symmetry between constrained reference tracking and constrained state estimation this paper exposes a symmetry relationship between constrained output reference tracking and constrained state estimation problems the symmetry which is different from the traditional duality relationship between control and estimation is provided by means of two tables that give a complete translation of all variables of one problem into the variables of the other an example is provided to illustrate the behaviour of the optimal solutions to the reference tracking and state estimation problems the symmetry relationship leads to interesting interpretations of several of the resulting trajectories

on a resource allocation model with infinite horizon in this paper we treat a resource allocation model defined on an infinite interval we show that the solution of the corresponding problem with finite horizon cannot be extended to a solution of the infinite horizon problem since the resource allocation problem in the unmodified setting does not have a solution on an unbounded interval to change this situation we bring an additional state constraint into the model which contains a weight function the new problem called now the adapted resource allocation problem has an optimal solution which will be identified by means of the duality concept of klotzler

abstract dynamic programming models under commutativity conditions the unifying purpose of the abstract dynamic programming models is to find sufficient conditions on the recursive definition of the objective function that guarantee the validity of the dynamic programming iteration this paper presents backward forward and backward forward models that weaken previous sufficient conditions and that include but are not restricted to optimization problems the backward forward model is devoted to the simultaneous solution of a collection of interrelated sequential problems based on the independent computation of a cost to arrive function and a cost to go function several extremization and nonextremization problems illustrate the applicability of the proposed models

pontryagin type maximum principle for budget constrained infinite horizon optimal control problems with linear dynamics abstract in this paper a class of infinite horizon optimal control problems with an isoperimetrical constraint also interpreted as a budget constraint is considered herein a linear both in the state and in the control dynamic is allowed the problem setting includes a weighted sobolev space as the state space for this class of problems we establish the necessary optimality conditions in form of a pontryagin type maximum principle including a transversality condition the proved theoretical result is applied to a linear quadratic regulator problem

on the use of constraints in least squares estimation and control in this article we examine the effect of constraints on estimation and control methods based on quadratic penalty functions we begin with estimation theory and analyze how constraints alter the statistical properties of the least squares estimates it is shown that constraints can be used to formulate maximum likelihood mle and maximum a posteriori map estimators for a variety of unimodal distributions this provides greater flexibility over the assumption of normality inherent in the mle and map interpretation of traditional least squares we discuss how these ideas apply to state space models of dynamic systems possible applications for controllers that handle constraints are also discussed a parameter estimation example is given to demonstrate the potential for improved performance over unconstrained least squares

weighted functional spaces approach in infinite horizon optimal control problems a systematic analysis of hidden opportunities and advantages abstract this paper constitutes the advantages of using the weighted sobolev and weighted lebesgue spaces in optimal control problems defined on an infinite time interval based on numerous examples it demonstrates how the introduction of certain weight or density functions into the problem statement may influence the existence of an optimal solution and the optimality of a chosen candidate the positive effects of the proper relation between the state and the co state functional spaces for the necessary optimality conditions as well as for the development of numerical schemes are discussed this paper is based on over one decade of research in this field and summarizes the main findings concerning the use of weight functions in optimal control problems

bounds for online bin packing with cardinality constraints abstract we study a bin packing problem in which a bin can contain at most k items of total size at most 1 where k 2 is a given parameter items are presented one by one in an online fashion we analyze the best absolute competitive ratio of the problem and prove tight bounds of 2 for any k 4 additionally we present bounds for relatively small values of k with respect to the asymptotic competitive ratio and the absolute competitive ratio in particular we provide tight bounds on the absolute competitive ratio of first fit for k 2 3 4 and improve the known lower bounds on asymptotic competitive ratios for multiple values of k our method for obtaining a lower bound on the asymptotic competitive ratio using a certain type of an input is general and we also use it to obtain an alternative proof of the known lower bound on the asymptotic competitive ratio of standard online bin packing

on the online bin packing problem a new framework for analyzing online bin packing algorithms is presented this framework presents a unified way of explaining the performance of algorithms based on the harmonic approach within this framework it is shown that a new algorithm harmonic has asymptotic performance ratio at most 1 58889 it is also shown that the analysis of harmonic 1 presented in richey 1991 is incorrect this is a fundamental logical flaw not an error in calculation or an omitted case the asymptotic performance ratio of harmonic 1 is at least 1 59217 thus harmonic provides the best upper bound for the online bin packing problem to date

class constrained bin packing revisited we study the following variant of the bin packing problem we are given a set of items where each item has a non negative size and a color we are also given an integer parameter k and the goal is to partition the items into a minimum number of subsets such that for each subset s in the solution the total size of the items in s is at most 1 as in the classical bin packing problem and the total number of colors of the items in s is at most k which distinguishes our problem from the classical version we follow earlier work on this problem and study the problem in both offline and online scenarios

constrained state estimation for nonlinear discrete time systems stability and moving horizon approximations state estimator design for a nonlinear discrete time system is a challenging problem further complicated when additional physical insight is available in the form of inequality constraints on the state variables and disturbances one strategy for constrained state estimation is to employ online optimization using a moving horizon approximation we propose a general theory for constrained moving horizon estimation sufficient conditions for asymptotic and bounded stability are established we apply these results to develop a practical algorithm for constrained linear and nonlinear state estimation examples are used to illustrate the benefits of constrained state estimation our framework is deterministic

new lower bounds for certain classes of bin packing algorithms on line algorithms have been extensively studied for the one dimensional bin packing problem in this paper we investigate two classes of one dimensional bin packing algorithms and we give better lower bounds for their asymptotic worst case behavior for on line algorithms so far the best lower bound was given by van vliet in 1992 12 he proved that there is no on line bin packing algorithm with better asymptotic performance ratio than 1 54014 in this paper we give an improvement on this bound to 248161 1 54037 and we investigate the parametric case as well for those lists where the elements are preprocessed according to their sizes in non increasing order csirik et al 1983 1 proved that no on line algorithm can have an asymptotic performance ratio smaller than 87 we improve this result to 5447

the class constrained bin packing problem with applications to video on demand in this paper we present approximation results for the class constrained bin packing problem that has applications to video on demand systems in this problem we are given bins of size b with c compartments and n items of q different classes each item i 1 n with class c i and size s i the problem is to pack the items into bins where each bin contains at most c different classes and has total items size at most b we present several approximation algorithms for offline and online versions of the problem

moving horizon state estimation for nonlinear discrete time systems new stability results and approximation schemes a moving horizon state estimation problem is addressed for a class of nonlinear discrete time systems with bounded noises acting on the system and measurement equations as the statistics of such disturbances and of the initial state are assumed to be unknown we use a generalized least squares approach that consists in minimizing a quadratic estimation cost function defined on a recent batch of inputs and outputs according to a sliding window strategy for the resulting estimator the existence of bounding sequences on the estimation error is proved in the absence of noises exponential convergence to zero is obtained moreover suboptimal solutions are sought for which a certain error is admitted with respect to the optimal cost value the approximate solution can be determined either on line by directly minimizing the cost function or off line by using a nonlinear parameterized function simulation results are presented to show the effectiveness of the proposed approach in comparison with the extended kalman filter

afptas results for common variants of bin packing a new method for handling the small items we consider two well known natural variants of bin packing and show that these packing problems admit asymptotic fully polynomial time approximation schemes afptass in bin packing problems a set of one dimensional items of size at most 1 is to be assigned packed to subsets of sum at most 1 bins it has been known for a while that the most basic problem admits an afptas in this paper we develop new methods that allow us to extend this result to other variants of bin packing consisting of a family of two parameter bin packing problems we demonstrate the use of our methods by designing afptass for the following problems the first problem is bin packing with cardinality constraints where a parameter k is given such that a bin may contain up to k items the goal is to minimize the number of bins used the second problem is bin packing with rejection where every item has a rejection penalty associated with it an item needs to be either packed to a bin or rejected and the goal is to minimize the number of bins used and the total rejection penalty of unpacked items this resolves the complexity of two important variants of the bin packing problem our approximation schemes use a novel method for packing the small items this new method is the core of the improved running times of our schemes over the running times of the previous results which are only asymptotic polynomial time approximationschemes

different interpretations of the improper integral objective in an infinite horizon control problem abstract we provide an example of a convex infinite horizon problem with a linear objective functional where the different interpretations of the improper integral 0 f t x t u t d t in either lebesgue or riemann sense lead to different but finite optimal values

a simple on line bin packing algorithm the one dimensional on line bin packing problem is considered a simple o 1 space and o n time algorithm called harmonic m is presented it is shown that this algorithm can achieve a worst case performance ratio of less than 1 692 which is better than that of the o n space and o n log n time algorithm first fit also shown is that 1 691 is a lower bound for all 0 1 space on line bin packing algorithms finally a revised version of harmonic m an o n space and o n time algorithm is presented and is shown to have a worst case performance ratio of less than 1 636 d

monotonicity and stabilizability properties of solutions of the riccati difference equation propositions lemmas theorems fallacious conjectures and counterexamples abstract the problem considered is that of selecting an initial covariance matrix for the kalman filter to ensure that the closed loop filter at every subsequent time instant is exponentially asymptotically stable as a time invariant filter sufficient conditions are derived based on monotonicity properties of the solution of the riccati difference equation the results have application in observer design and the cases of filtering for nonstabilizable systems and systems with singular system matrices are included