on a multistage discrete stochastic optimization problem with stochastic constraints and nested sampling we consider a multistage stochastic discrete program in which constraints on any stage might involve expectations that cannot be computed easily and are approximated by simulation we study a sample average approximation saa approach that uses nested sampling in which at each stage a number of scenarios are examined and a number of simulation replications are performed for each scenario to estimate the next stage constraints this approach provides an approximate solution to the multistage problem to establish the consistency of the saa approach we first consider a two stage problem and show that in the second stage problem given a scenario the optimal values and solutions of the saa converge to those of the true problem with probability one when the sample sizes go to infinity these convergence results do not hold uniformly over all possible scenarios for the second stage problem we are nevertheless able to prove that the optimal values and solutions of the saa converge to the true ones with probability one when the sample sizes at both stages increase to infinity we also prove exponential convergence of the probability of a large deviation for the optimal value of the saa the true value of an optimal solution of the saa and the probability that any optimal solution to the saa is an optimal solution of the true problem all of these results can be extended to a multistage setting and we explain how to do it our framework and saa results cover a large variety of resource allocation problems for which at each stage after the first one new information becomes available and the allocation can be readjusted under constraints that involve expectations estimated by monte carlo as an illustration we apply this saa method to a staffing problem in a call center in which the goal is to optimize the numbers of agents of each type under some constraints on the quality of service qos the staffing allocation has to be decided under an uncertain arrival rate with a prior distribution in the first stage and can be adjusted at some additional cost when better information on the arrival rate becomes available in later stages

fuzzy model based control stability robustness and performance issues a nonlinear controller based on a fuzzy model of mimo dynamical systems is described and analyzed the fuzzy model is based on a set of arx models that are combined using a fuzzy inference mechanism the controller is a discrete time nonlinear decoupler which is analyzed both for the adaptive and the fixed parameter cases a detailed stability analysis is carried out and the main result is that the closed loop is globally stable and robust with respect to unstructured uncertainty which may include modeling error and disturbances in addition bounds on the asymptotic and transient performance are given the main assumptions on the system and model are that they must not have strong nonminimum phase effects except time delay and the unstructured uncertainty must not be too large a simulation example illustrates some of the properties of the modeling method and model based control structure

global stochastic optimization with low dispersion point sets this study concerns a generic model free stochastic optimization problem requiring the minimization of a risk function defined on a given bounded domain in a euclidean space smoothness assumptions regarding the risk function are hypothesized and members of the underlying space of probabilities are presumed subject to a large deviation principle however the risk function may well be nonconvex and multimodal a general approach to finding the risk minimizer on the basis of decision observation pairs is proposed it consists of repeatedly observing pairs over a collection of design points principles are derived for choosing the number of these design points on the basis of an observation budget and for allocating the observations between these points in both prescheduled and adaptive settings on the basis of these principles large deviation type bounds of the minimizer in terms of sample size are established

lines in space times we construct a complete timelike maximal geodesic line in a timelike geodesically complete spacetimem containing a compact acausal spacelike hypersurfaces which lies in the past of somes ray ans ray is a future complete geodesic starting ons which maximizes lorentzian distance froms to any of its points if the timelike convergence condition strong energy condition holds a line exists only ifm is static i e it splits geometrically as space time so timelike completeness must fail for a nonstatic spacetime with strong energy condition which contains a closed universe s with the above properties

a stochastic approach to stability in stochastic programming abstract a random optimization problem p 0 min x 0 0 x is approximated by a sequence of random surrogate problems p n n n with p n min x n n x p a given probability space we investigate the convergence almost surely and in probability of the optimal values and the solution sets the results can be regarded as random versions of well known stability statements of parametric programming semicontinuous convergence almost surely in probability of sequences of random functions is a crucial assumption in this framework and will be investigated in more detail

approximation properties of piece wise parabolic functions fuzzy logic systems spline functions are widely used in the field of approximation theory and its applications in this article we propose the use of p functions which are spline functions of degree two as membership functions of the input variables at first we show by a constructive theory that multi input single output miso sugeno fuzzy logic systems with p input variables membership functions are universal approximators to every continuously differentiable function defined on a compact subset of r n moreover they are of first order approximation accuracy then we show that with appropriately chosen parameters of the p functions fuzzy logic systems with p input variables membership functions are universal approximators not only for the function itself but also for its first order derivative

approximation accuracy analysis of fuzzy systems as function approximators this paper establishes the approximation error bounds for various classes of fuzzy systems i e fuzzy systems generated by different inferential and defuzzification methods based on these bounds the approximation accuracy of various classes of fuzzy systems is analyzed and compared it is seen that the class of fuzzy systems generated by the product inference and the center average defuzzifier has better approximation accuracy and properties than the class of fuzzy systems generated by the min inference and the center average defuzzifier and the class of fuzzy systems defuzzified by the mom defuzzifier in addition it is proved that fuzzy systems can represent any linear and multilinear function and explicit expressions of fuzzy systems generated by the mom defuzzified method are given

approximation accuracy of some neuro fuzzy approaches many methods have been proposed in the literature for designing fuzzy systems from input output data the so called neuro fuzzy methods but very little was done to analyze the performance of the methods from a rigorous mathematical point of view in this paper we establish approximation bounds for two of these methods the table lookup scheme proposed by wang et al 1992 and the clustering method studied by wang 1993 1997 we derive detailed formulas of the error bounds between the nonlinear function to be approximated and the fuzzy systems designed using the methods based on input output data these error bounds show explicitly how the parameters in the two methods influence their approximation capability we also propose modified versions for the two methods such that the designed fuzzy systems are well defined over the whole input domain

generation of a probabilistic fuzzy rule base by learning from examples this study considers probabilistic fuzzy systems constructed using mamdani probabilistic fuzzy rules as a generalisation of deterministic fuzzy systems mamdani probabilistic fuzzy systems better model practical complex systems involving uncertainty because they combine the interpretability of fuzzy systems with the statistical properties of probabilistic systems using probabilistic fuzzy rules both probabilistic uncertainty and linguistic ambiguity are handled simultaneously with a single framework considering that the information available often consists of a training set of input output data pairs a general method for generating mamdani probabilistic fuzzy rule bases from numerical data pairs is presented a fuzzy reasoning method is used on the generated probabilistic fuzzy rule base to derive a map leading from the input space to the output space and a probabilistic fuzzy system is constructed we use this probabilistic fuzzy modelling method for nonlinear regression analysis the effectiveness of the proposed method is demonstrated by a comparison with similar regression techniques

rate based daily arrival process models with application to call centers we propose develop and compare new stochastic models for the daily arrival rate in a call center following standard practice the day is divided into time periods of equal length e g 15 or 30 minutes the arrival rate is assumed random but constant in time in each period and the arrivals are from a poisson process conditional on the rate the random rate for each period is taken as a deterministic base rate or expected rate multiplied by a random busyness factor having mean 1 models in which the busyness factors are independent across periods or in which a common busyness factor applies to all periods have been studied previously but they are not sufficiently realistic we examine alternative models for which the busyness factors have some form of dependence across periods maximum likelihood parameter estimation for these models is not easy mainly because the arrival rates themselves are never observed we develop specialized techniques to perform this estimation we compare the goodness

variance bounds and existence results for randomly shifted lattice rules we study the convergence of the variance for randomly shifted lattice rules for numerical multiple integration over the unit hypercube in an arbitrary number of dimensions we consider integrands that are square integrable but whose fourier series are not necessarily absolutely convergent for such integrands a bound on the variance is expressed through a certain type of weighted discrepancy we prove existence and construction results for randomly shifted lattice rules such that the variance bounds are almost o n a where n is the number of function evaluations and a 1 depends on our assumptions on the convergence speed of the fourier coefficients these results hold for general weights arbitrary n and any dimension with additional conditions on the weights we obtain a convergence that holds uniformly in the dimension and this provides sufficient conditions for strong tractability of the integration problem we also show that lattice rules that satisfy these bounds are not difficult to construct explicitly and we provide numerical illustrations of the behaviour of construction algorithms

constrained optimization in expensive simulation novel approach this article presents a novel heuristic for constrained optimization of computationally expensive random simulation models one output is selected as objective to be minimized while other outputs must satisfy given threshold values moreover the simulation inputs must be integer and satisfy linear or nonlinear constraints the heuristic combines i sequentialized experimental designs to specify the simulation input combinations ii kriging or gaussian process or spatial correlation modeling to analyze the global simulation input output data resulting from these designs and iii integer nonlinear programming to estimate the optimal solution from the kriging metamodels the heuristic is applied to an s s inventory system and a call center simulation and compared with the popular commercial heuristic optquest embedded in the arena versions 11 and 12 in these two applications the novel heuristic outperforms optquest in terms of number of simulated input combinations and quality of the estimated optimum

sufficient conditions on general fuzzy systems as function approximators abstract in a constructive way we have found sufficient conditions under which general fuzzy systems can uniformly approximate any real continuous function on a compact domain to any degree of accuracy more importantly we have revealed the underlying mechanism of such function approximation the fuzzy systems can uniformly approximate any polynomials which according to the weierstrass approximation theorem can uniformly approximate any continuous function on a compact domain these findings lead to the following practical results 1 explicit fuzzy rules of fuzzy systems can now be easily obtained according to functions to be approximated and 2 formulas are derived which can calculate the number of input fuzzy sets output fuzzy sets and fuzzy rules needed in order to satisfy any given approximation accuracy the number increases as required approximation error decreases and as approximation error approaches zero the number approaches these formulas also reveal that the number is minimal when the maximum number of intersection between adjacent input fuzzy sets is one practical implications of these results will be discussed especially in the context of fuzzy control and fuzzy modeling two examples are provided to demonstrate how to design fuzzy systems to approximate given functions with required approximation accuracy

on the asymptotics for the vacuum einstein constraint equations in this paper we prove density of asymptotically flat solutions with special asymptotics in general classes of solutions of the vacuum constraint equations the first type of special asymptotic form we consider is called harmonic asymptotics this generalizes in a natural way the conformally flat asymptotics for the k 0 constraint equations we show that solutions with harmonic asymptotics form a dense subset in a suitable weighted sobolev topology of the full set of solutions an important feature of this construction is that the approximation allows large changes in the angular momentum the second density theorem we prove allows us to approximate asymptotically flat initial data on a three manifold m for the vacuum einstein field equation by solutions which agree with the original data inside a given domain and are identical to that of a suitable kerr slice or identical to a member of some other admissible family of solutions outside a large ball in a given end the construction generalizes work in c where the time symmetric k 0 case was studied

evolutionary fuzzy modeling this study is concerned with a general methodology of identification of fuzzy models unlike numeric models fuzzy models operate at a level of information granules fuzzy sets and this aspect brings up an important design requirement of transparency of the model we propose a three phase development framework by distinguishing between structural and parametric optimization processes the underlying topology of the model dwells on fuzzy neural networks architectures governed by fuzzy logic and equipped with parametric flexibility two general optimization mechanisms are explored the structural optimization is realized via genetic programming whereas for the ensuing detailed parametric optimization we proceed with gradient based learning the main advantages of this approach are discussed in detail the study is illustrated with the aid of a numeric example that provides a detailed insight into the performance of the fuzzy models and quantifies crucial design issues

scalar curvature deformation and a gluing construction for the einstein constraint equations on a compact manifold the scalar curvature map at generic metrics is a local surjection f m we show that this result may be localized to compact subdomains in an arbitrary riemannian manifold the method is extended to establish the existence of asymptotically flat scalar flat metrics on n n 3 which are spherically symmetric hence schwarzschild at infinity i e outside a compact set such metrics provide cauchy data for the einstein vacuum equations which evolve into nontrivial vacuum spacetimes which are identically schwarzschild near spatial infinity

implications of energy conditions on standard static space times abstract in the framework of standard static space times we state a family of sufficient or necessary conditions for a set of physically reasonable energy and convergence conditions in relativity and related theories we concentrate our study on questions about the sub harmonicity of the warping function the scalar curvature map conformal hyperbolicity conjugate points and the time like diameter of this class of space times

fuzzy basis functions universal approximation and orthogonal least squares learning fuzzy systems are represented as series expansions of fuzzy basis functions which are algebraic superpositions of fuzzy membership functions using the stone weierstrass theorem it is proved that linear combinations of the fuzzy basis functions are capable of uniformly approximating any real continuous function on a compact set to arbitrary accuracy based on the fuzzy basis function representations an orthogonal least squares ols learning algorithm is developed for designing fuzzy systems based on given input output pairs then the ols algorithm is used to select significant fuzzy basis functions which are used to construct the final fuzzy system the fuzzy basis function expansion is used to approximate a controller for the nonlinear ball and beam system and the simulation results show that the control performance is improved by incorporating some common sense fuzzy control rules

staffing multiskill call centers via linear programming and simulation we study an iterative cutting plane algorithm on an integer program for minimizing the staffing costs of a multiskill call center subject to service level requirements that are estimated by simulation we solve a sample average version of the problem where the service levels are expressed as functions of the staffing for a fixed sequence of random numbers driving the simulation an optimal solution of this sample problem is also an optimal solution to the original problem when the sample size is large enough several difficulties are encountered when solving the sample problem especially for large problem instances and we propose practical heuristics to deal with these difficulties we report numerical experiments with examples of different sizes the largest example corresponds to a real life call center with 65 types of calls and 89 types of agents skill groups

approximation theory of fuzzy systems spl minus siso case in this paper the approximation properties of mimo fuzzy systems generated by the product inference are discussed we first give an analysis of fuzzy basic functions fbf s and present several properties of fbf s based on these properties of fbf s we obtain several basic approximation properties of fuzzy systems 1 basic approximation property which reveals the basic approximation mechanism of fuzzy systems 2 uniform approximation bounds which give the uniform approximation bounds between the desired control or decision functions and fuzzy systems 3 uniform convergent property which shows that fuzzy systems with defined approximation accuracy can always be obtained by dividing the input space into finer fuzzy regions and 4 universal approximation property which shows that fuzzy systems are universal approximators and extends some previous results on this aspect the similarity between fuzzy systems and mathematical approximation is discussed and an idea to improve approximation accuracy is suggested based on uniform approximation bounds

on a data driven method for staffing large call centers we consider a call center model with multiple customer classes and multiple server pools calls arrive randomly over time and the instantaneous arrival rates are allowed to vary both temporally and stochastically in an arbitrary manner the objective is to minimize the sum of personnel costs and expected abandonment penalties by selecting an appropriate staffing level for each server pool we propose a simple and computationally tractable method for solving this problem that requires as input only a few system parameters and historical call arrival data for each customer class in this sense the method is said to be data driven the efficacy of the proposed method is illustrated via numerical examples an asymptotic analysis establishes that the prescribed staffing levels achieve near optimal performance and characterizes the magnitude of the optimality gap

uniqueness of l 1 solutions for the laplace equation and the heat equation on riemannian manifolds on deduit une hypothese geometrique optimale sur une variete de riemann complete qui assure des proprietes d unicite pour des solutions l 1 de l equation de laplace et de la chaleur

the wm method completed a flexible fuzzy system approach to data mining in this paper the so called wang mendel wm method for generating fuzzy rules from data is enhanced to make it a comprehensive and flexible fuzzy system approach to data description and prediction in the description part the core ideas of the wm method are used to develop three methods to extract fuzzy if then rules from data the first method shows how to extract rules for the user specified cases the second method generates all the rules that can be generated directly from the data and the third method extrapolates the rules generated by the second method over the entire domain of interest in the prediction part two fuzzy predictive models are constructed based on the fuzzy if then rules extracted by the methods of the description part the first model gives a continuous output and is suitable for predicting continuous variables and the second model gives a piecewise constant output and is suitable for predicting categorical variables we show that by comparing the prediction accuracy of the fuzzy predictive models with different numbers of fuzzy sets covering the input variables we can rank the importance of the input variables we also propose an algorithm to optimize the fuzzy predictive models and show how to use the models to solve pattern recognition problems throughout this paper we use a set of real data from a steel rolling plant to demonstrate the ideas and test the models