the sample average approximation method for stochastic discrete optimization in this paper we study a monte carlo simulation based approach to stochastic discrete optimization problems the basic idea of such methods is that a random sample is generated and the expected value function is approximated by the corresponding sample average function the obtained sample average optimization problem is solved and the procedure is repeated several times until a stopping criterion is satisfied we discuss convergence rates stopping rules and computational complexity of this procedure and present a numerical example for the stochastic knapsack problem

on the uncertainty in the regularized solution regularization is a fundamental technique in the processing of measurement data these data are typically noisy and sparse and in some cases only a projection of the data on a low dimensional subspace is available very often direct analysis of measured data yields unphysical results due to sensitivity to noise and neglect of the underlying distribution of the measured physical phenomena therefore regularization is extensively applied in physics one of the most common regularization methods was developed by a few researchers most notably tikhonov and phillips during the 1940s 1960s in the early 1980s geman and geman reformulated regularization in a bayesian framework in which every candidate function f x is assigned a probability with the expectation of the random variable f x0 equal to the value of the regularized interpolating function at x0 taking this probabilistic approach to second order the variance of this random variable which reflects the amount of uncertainty inherent in the solution is computed when estimating physical phenomena not only the estimated value is important but also the uncertainty associated with it for example more measurements should yield a more certain solution this review provides a formal definition of this uncertainty explores some numerical issues with its computation and extends it to compute the uncertainties of the integral and derivative with the intriguing result that in some cases the uncertainty of the function and the derivative are highly uncorrelated for example there are cases in which the point at which the function is least reliable is the one at which the derivative is most reliable lastly a related definition of optimal sampling is discussed the relevance of optimal sampling to physics is that it offers a method to determine the location of the measuring devices which yields the most reliable restoration of the measured physical phenomenon

multilevel codes based on partitioning following v v ginzburg 1984 a hierarchy of codes is proposed to match the geometric partitioning of a signal set the authors show that coset codes including ungerboeck lattice and binary codes and indeed any codes which rely on partitioning of the signal set are all subclasses of the proposed coding scheme the combination of such codes in a multilevel scheme often leads to reduced complexity in comparison with previously published schemes a variety of decoder structures is discussed

provably near optimal sampling based policies for stochastic inventory control models in this paper we consider two fundamental inventory models the single period newsvendor problem and its multiperiod extension but under the assumption that the explicit demand distributions are not known and that the only information available is a set of independent samples drawn from the true distributions under the assumption that the demand distributions are given explicitly these models are well studied and relatively straightforward to solve however in most real life scenarios the true demand distributions are not available or they are too complex to work with thus a sampling driven algorithmic framework is very attractive both in practice and in theory we shall describe how to compute sampling based policies that is policies that are computed based only on observed samples of the demands without any access to or assumptions on the true demand distributions moreover we establish bounds on the number of samples required to guarantee that with high probability the expected cost of the sampling based policies is arbitrarily close i e with arbitrarily small relative error compared to the expected cost of the optimal policies which have full access to the demand distributions the bounds that we develop are general easy to compute and do not depend at all on the specific demand distributions

the data driven newsvendor problem new bounds and insights consider the newsvendor model but under the assumption that the underlying demand distribution is not known as part of the input instead the only information available is a random independent sample drawn from the demand distribution this paper analyzes the sample average approximation saa approach for the data driven newsvendor problem we obtain a new analytical bound on the probability that the relative regret of the saa solution exceeds a threshold this bound is significantly tighter than existing bounds and it matches the empirical accuracy of the saa solution observed in extensive computational experiments this bound reveals that the demand distribution s weighted mean spread affects the accuracy of the saa heuristic

soft decoding techniques for codes and lattices including the golay code and the leech lattice two kinds of algorithms are considered 1 if is a binary code of length n a soft decision decoding algorithm for changes an arbitrary point of r n into a nearest codeword nearest in euclidean distance 2 similarly a decoding algorithm for a lattice lambda in r n changes an arbitrary point of r n into a closest lattice point some general methods are given for constructing such algorithms ami are used to obtain new and faster decoding algorithms for the gosset lattice e 8 the golay code the leech lattice

a recursive approach to low complexity codes a method is described for constructing long error correcting codes from one or more shorter error correcting codes referred to as subcodes and a bipartite graph a graph is shown which specifies carefully chosen subsets of the digits of the new codes that must be codewords in one of the shorter subcodes lower bounds to the rate and the minimum distance of the new code are derived in terms of the parameters of the graph and the subeodes both the encoders and decoders proposed are shown to take advantage of the code s explicit decomposition into subcodes to decompose and simplify the associated computational processes bounds on the performance of two specific decoding algorithms are established and the asymptotic growth of the complexity of decoding for two types of codes and decoders is analyzed the proposed decoders are able to make effective use of probabilistic information supplied by the channel receiver e g reliability information without greatly increasing the number of computations required it is shown that choosing a transmission order for the digits that is appropriate for the graph and the subcodes can give the code excellent burst error correction abilities the construction principles

two component eigenfunction expansion for open systems described by the wave equation i completeness of expansion the concept of eigenfunction expansions for the wave equation is generalized to open systems in which waves escape to the outside these non conservative systems are non hermitian in the usual sense it is shown that the natural framework is an eigenfunction expansion within a two component formalism that treats the wavefunction and its conjugate momentum together provided the system approaches spatial infinity rapidly without tails and possesses spatial discontinuities the expansion in terms of the eigenfunctions which are now quasinormal modes is shown to be valid

approximation algorithms for stochastic inventory control models we consider two classical stochastic inventory control models the periodic review stochastic inventory control problem and the stochastic lot sizing problem the goal is to coordinate a sequence of orders of a single commodity aiming to supply stochastic demands over a discrete finite horizon with minimum expected overall ordering holding and backlogging costs in this paper we address the important problem of finding computationally efficient and provably good inventory control policies for these models in the presence of correlated nonstationary time dependent and evolving stochastic demands this problem arises in many domains and has many practical applications in supply chain management our approach is based on a new marginal cost accounting scheme for stochastic inventory control models combined with novel cost balancing techniques specifically in each period we balance the expected cost of overordering i e costs incurred by excess inventory against the expected cost of underordering i e costs incurred by not satisfying demand on time this leads to what we believe to be the first computationally efficient policies with constant worst case performance guarantees for a general class of important stochastic inventory control models that is there exists a constant c such that for any instance of the problem the expected cost of the policy is at most c times the expected cost of an optimal policy in particular we provide a worst case guarantee of two for the periodic review stochastic inventory control problem and a worst case guarantee of three for the stochastic lot sizing problem our results are valid for all of the currently known approaches in the literature to model correlation and nonstationarity of demands over time

an adaptive distribution free algorithm for the newsvendor problem with censored demands with applications to inventory and distribution we consider the problem of optimizing inventories for problems where the demand distribution is unknown and where it does not necessarily follow a standard form such as the normal we address problems where the process of deciding the inventory and then realizing the demand occurs repeatedly the only information we use is the amount of inventory left over rather than attempting to estimate the demand distribution we directly estimate the value function using a technique called the concave adaptive value estimation cave algorithm cave constructs a sequence of concave piecewise linear approximations using sample gradients of the recourse function at different points in the domain since it is a sampling based method cave does not require knowledge of the underlying sample distribution the result is a nonlinear approximation that is more responsive than traditional linear stochastic quasi gradient methods and more flexible than analytical techniques that require distribution information in addition we demonstrate near optimal behavior of the cave approximation in experiments involving two different types of stochastic programs the newsvendor stochastic inventory problem and two stage distribution problems

two component eigenfunction expansion for open systems described by the wave equation ii linear space structure for a broad class of open systems described by the wave equation the eigenfunctions which are quasinormal modes provide a complete basis for simultaneously expanding outgoing wavefunctions in this paper the linear space structure associated with this expansion is developed under a modified inner product the time evolution operator is self adjoint even though energy is not conserved for the system alone thus the eigenfunctions are mutually orthogonal consequently the usual tools of eigenfunction expansions can be transcribed to these open systems as an example the time independent perturbation theory is developed in straightforward analogy with quantum mechanics giving the shift in both the real part and the imaginary part of the eigenvalues

a class of optimum block codes in signal space by using convolutional coding coupled with expanded sets of channel signals ungerboeck showed that it is possible to obtain coding gains of several decibels at a given data rate without increasing the transmission bandwidth in this paper we construct a class of optimum signal space codes using short binary block as opposed to convolutional codes our results are compared to those obtained by ungerboeck a simplified decoding procedure that is asymptotically optimum is also presented

using stochastic approximation methods to compute optimal base stock levels in inventory control problems in this paper we consider numerous inventory control problems for which the base stock policies are known to be optimal and we propose stochastic approximation methods to compute the optimal base stock levels the existing stochastic approximation methods in the literature guarantee that their iterates converge but not necessarily to the optimal base stock levels in contrast we prove that the iterates of our methods converge to the optimal base stock levels moreover our methods continue to enjoy the well known advantages of the existing stochastic approximation methods in particular they only require the ability to obtain samples of the demand random variables rather than to compute expectations explicitly and they are applicable even when the demand information is censored by the amount of available inventory

determination of a finite range potential from discrete phase shift data by inverse scattering a practical method of inverse scattering is proposed to determine a finite range scattering potential from phase shift data evaluated at a set of discrete energies by using the asymptotic behaviour of the phase shifts and the connection between short and long wavelength scales a relatively few input data are found to be adequate the method works well for repulsive and moderately attractive potentials useful in many applications

channel coding with multilevel phase signals a coding technique is described which improves error performance of synchronous data links without sacrificing data rate or requiring more bandwidth this is achieved by channel coding with expanded sets of multilevel phase signals in a manner which increases free euclidean distance soft maximum likelihood ml decoding using the viterbi algorithm is assumed following a discussion of channel capacity simple hand designed trellis codes are presented for 8 phase shift keying psk and 16 quadrature amplitude shift keying qask modulation these simple codes achieve coding gains in the order of 3 4 db it is then shown that the codes can be interpreted as binary convolutional codes with a mapping of coded bits into channel signals which we call mapping by set partitioning based on a new distance measure between binary code sequences which efficiently lower bounds the euclidean distance between the corresponding channel signal sequences a search procedure for more powerful codes is developed codes with coding gains up to 6 db are obtained for a variety of multilevel phase modulation schemes simulation results are presented and an example of carrier phase tracking is discussed

coset codes i introduction and geometrical classification practically all known good constructive coding techniques for bandlimited channels including lattice codes and various trellis coded modulation schemes can be characterized as coset codes a coset code is defined by a lattice partition lambda lambda and by a binary encoder c that selects a sequence of cosets of the lattice lambda the fundamental coding gain of a coset code as well as other important parameters such as the error coefficient the decoding complexity and the constellation expansion factor are purely geometric parameters determined by c lambda lambda the known types of coset codes as well as a number of new classes that systematize and generalize known codes are classified and compared in terms of these parameters

multiple symbol differentially detected multilevel codes for the rayleigh fading channel conventional differential detection is known to perform poorly on the fast fading rayleigh fading channel due to the rapid variation of the channel state the technique of multiple symbol differential detection msdd improves the error performance by detecting over a sequence of symbols and exploiting the inherent correlation of the fading process to further reduce the error rate we need to introduce some form of coding successful coding for the rayleigh fading channel requires interleaving to decorrelate the channel a conflicting requirement to msdd we propose a solution consisting of interleaving blocks of l m ary phase shift keying mpsk symbols that are multiple symbol differentially detected the blocks are treated as elements of an m sup l symbol alphabet over which a multilevel code based on geometrically uniform partitions is defined

multilevel trellis mpsk modulation codes for the rayleigh fading channel the multilevel coding technique is used for constructing multilevel trellis m ary phase shift keying mpsk modulation codes for the rayleigh fading channel in the construction of a code all the factors which affect the code performance and its decoding complexity are considered the error performance of some of these codes based on both one stage optimum decoding and multistage suboptimum decoding has been simulated the simulation results show that these codes achieve good error performance with small decoding complexity

on the choice of the regularization parameter in nonlinear inverse problems this paper focuses on regularization techniques for nonlinear ill posed inverse problems tikhonov regularization and regularization due to the use of norm constraints are analyzed a model function technique is proposed to iteratively determine an optimal regularization parameter or the parameter characterizing the norm constraint and to estimate the error in the data if it is not known a priori

jordan blocks and generalized bi orthogonal bases realizations in open wave systems dissipation can sometimes be described by a non hermitian hamiltonian h whose left and right eigenvectors f j fj form a bi orthogonal basis bb for waves in a class of open systems this is known to lead to exact complete bb expansions if f j fj 0 for all j if not normalization seems impossible and many familiar formulae fail examples are given the problem is related to the merging of eigenmodes so that h can only be diagonalized to jordan blocks the resolution involves a generalized bb containing extra vectors whose dynamics are modified by polynomials in the time t the splitting of merged modes under a perturbation is also treated one thus obtains a non trivial extension of the bb formalism for open systems

coset codes ii binary lattices and related codes for pt i see ibid vol 34 no 5 p 1123 51 1988 the family of barnes wall lattices including d sub 4 and e sub 8 of lengths n 2 sup n and their principal sublattices which are useful in constructing coset codes are generated by iteration of a simple construction called the squaring construction the closely related reed muller codes are generated by the same construction the principal properties of these codes and lattices are consequences of the general properties of iterated squaring constructions which also exhibit the interrelationships between codes and lattices of different lengths an extension called the cubing construction generates good codes and lattices of lengths n 3 2 sup n including the golay code and leech lattice with the use of special bases for 8 space another related construction generates the nordstrom robinson code and an analogous 16 dimensional nonlattice packing these constructions are represented by trellis diagrams that display their structure and interrelationships and that lead to efficient maximum likelihood decoding algorithms

new multilevel codes over gf q set partitioning is applied to multidimensional signal spaces over gf q i e gf sup n1 q n1 or d are presented these codes use reed solomon codes as component codes longer multilevel block codes are also constructed using q ary block codes with block length longer than q 1 as component codes some quaternary multilevel block codes are presented with the same length and number of information symbols as but larger distance than the best previously known quaternary one level block codes it is proved that if all the component block codes are linear the multilevel block code is also linear low rate q ary convolutional codes word error correcting convolutional codes and binary to q ary convolutional codes can also be used to construct multilevel trellis codes over gf q or binary to q ary trellis codes

some ternary and quaternary codes and associated sphere packings tables of good ternary and quaternary codes are presented and they are used in the construction of dense sphere packings results include 1 tables of the best ternary and quaternary constacyclic codes including cyclic codes up to block length 50 2 a class of optimal n 2 codes over gf q 3 the u v w mod 2u v mod u construction a new ternary code construction technique that can be used to construct the ternary reed muller codes and others and 4 tables of linear ternary and quaternary codes obtained by modifying and combining various codes here and in the literature packings are generated in even dimensions up to 100 using these codes and a sphere packing construction in dimensions 36 and 60 new record densities appear to have been achieved

trellis coded modulation with multidimensional constellations trellis coded modulation schemes using four eight or 16 dimensional constellations have a number of potential advantages over the usual two dimensional schemes a smaller constituent two dimensional constellation easier tolerance to phase ambiguities and a better trade off between complexity and coding gain a number of such schemes are presented and evaluated starting with a variety of multidimensional lattices we show how to select multidimensional constellations how to partition them into subsets how to construct trellis codes using those subsets and how to map hits to constellation points simplifications of the viterbi decoding algorithm are presented we conclude that there are multidimensional trellis coded modulation schemes that perform better for the same complexity than do two dimensional schemes

the leech lattice the octacode and decoding algorithms new multilevel constructions of the golay code and the leech lattice are presented these are derived from the turyn construction and the holy construction with the octacode as the glue code further we show that the holy construction of the leech lattice with the octacode as the glue code is essentially different from the permuted turyn construction although both constructions rely on the octacode the turyn construction is based on an odd type of the octacode whereas any type of the octacode can be used in the holy construction moreover the multilevel representation of the holy construction leads to a novel lattice partition chain based on these structures we derive new bounded distance decoders for the golay code and the leech lattice whose effective error coefficient is smaller than that of any previously known bounded distance decoder we provide a general theorem for computing the effective error coefficient of coset decoding with bounded distance decoding for the subcode

waves in open systems via a biorthogonal basis dissipative quantum systems are sometimes phenomenologically described in terms of a non hermitian hamiltonian h with different left and right eigenvectors forming a biorthogonal basis it is shown that the dynamics of waves in open systems can be cast exactly into this form thus providing a well founded realization of the phenomenological description and at the same time placing these open systems into a well known framework the formalism leads to a generalization of norms and inner products for open systems which in contrast to earlier works is finite without the need for regularization the inner product allows transcription of much of the formalism for conservative systems including perturbation theory and second quantization

improved geometric goppa codes i basic theory in this paper we present a construction of improved geometric goppa codes which for the case of r 2g are often more efficient than the current geometric goppa codes derived from some varieties which include algebraic curves hyperplanes surfaces and other varieties for the special case of a plane in a three dimensional projective space the improved geometric goppa codes are reduced to linear multilevel codes for these improved geometric goppa codes a designed minimum distance can be easily determined and a decoding procedure which corrects up to half the designed minimum distance is also given

provably near optimal balancing policies for multi echelon stochastic inventory control models we develop the first algorithmic approach to compute provably good ordering policies for a multi echelon stochastic inventory system facing correlated nonstationary and evolving demands over a finite horizon specifically we study the serial system our approach is computationally efficient and provides worst case guarantees that is the expected cost of the algorithms is guaranteed to be within a constant factor of the optimal expected cost depending on the assumption the constant varies between two and three our algorithmic approach is based on an innovative scheme to account for costs in a multi echelon multi period environment as well as repeatedly balancing between opposing cost the cost accounting scheme called a cause effect cost accounting scheme is significantly different from traditional cost accounting schemes in that it reallocates costs with the goal of assigning every unit of cost to the decision that caused the cost to be incurred we believe it will have additional applications

on multilevel block modulation codes the multilevel technique for combining block coding and modulation is investigated a general formulation is presented for multilevel modulation codes in terms of component codes with appropriate distance measures a specific method for constructing multilevel block modulation codes with interdependency among component codes is proposed given a multilevel block modulation code c with no interdependency among the binary component codes the proposed method gives a multilevel block modulation code c that has the same rate as c a minimum squared euclidean distance not less than that of c a trellis diagram with the same number of states as that of c and a smaller number of nearest neighbor codewords than that of c finally a technique is presented for analyzing the error performance of block modulation codes for an additive white gaussian noise awgn channel based on soft decision maximum likelihood decoding error probabilities of some specific codes are evaluated by simulation and upper bounds based on their euclidean weight distributions

regret in the newsvendor model with partial information traditional stochastic inventory models assume full knowledge of the demand probability distribution however in practice it is often difficult to completely characterize the demand distribution especially in fast changing markets in this paper we study the newsvendor problem with partial information about the demand distribution e g mean variance symmetry unimodality in particular we derive the order quantities that minimize the newsvendor s maximum regret of not acting optimally most of our solutions are tractable which makes them attractive for practical application our analysis also generates insights into the choice of the demand distribution as an input to the newsvendor model in particular the distributions that maximize the entropy perform well under the regret criterion our approach can be extended to a variety of problems that require a robust but not conservative solution

on the performance evaluation of trellis codes generating function techniques for analyzing the error event and the bit error probabilities for trellis codes are considered the conventional state diagram approach for linear codes where the number of states is equal to the number of trellis states cannot be applied directly to arbitrary trellis codes and instead a state diagram where the number of states is equal to the square of the number of trellis states must be used it is shown that for an interesting class of trellis codes a modified generating function can be defined for which the number of states em is equal to the number of trellis states the class of codes considered includes trellis codes of rate r n 1 n based upon set partitioning whenever the first partition breaks the signal constellation into two subsets which have the same configuration matrix i e the same ordered set of mutual distances the complexity of calculating this modified generating function is the same as for the ordinary generating function of a convolutional code with the same number of trellis states bounds on the performance of some interesting codes are given based upon this method

coded phase frequency modulation channel coding combined with expanded signal sets has been shown 1 to improve error performance over uncoded modulation without expanding the bandwidth of the transmitted signals in this paper new coded modulation formats defined over an expanded set of signals varying both in phase and frequency are presented the new schemes combine fsk and psk modulation and make use of trellis coding and viterbi decoding to improve error performance over uncoded modulation the free euclidean distance is calculated for several classes of codes and upper bounds and simulation results are also presented for some simple codes the spectral characteristics of the proposed coded modulation formats are evaluated and compared to conventional two dimensional modulation formats differential encoding and various extensions of the basic scheme are also discussed

numerical algorithm for the determination of the potential of a conservative system from its normal mode spectra an efficient numerical algorithm is developed for determining the potential of a conservative system described by a klein gordon or schrodinger equation from its normal mode spectra the algorithm is based on first solving the linear problem of perturbative inversion by means of tikhonov regularization and then using a series of perturbative steps to approach the full potential by newton s method the algorithm is stable and convergence is rapid

two level coding based on trellis coded modulation and reed solomon codes a number of two level codes based on trellis coded modulation tcm schemes and their matched single or double error correcting reed solomon codes using qam constellations are presented and evaluated compared to the well known 4d 16 state tcm scheme with a comparable bandwidth efficiency these two level codes provide greater immunity from both gaussian and impulse noise at the cost of a reasonably higher complexity and a longer decoding delay on the order of a few hundred symbols the authors also show how to cope with phase ambiguities of the constellation in these two level codes both rotationally invariant coding and rotation detecting techniques are considered

finite state codes a class of codes called finite state fs codes is defined and investigated the codes which generalize both block and convolutional codes are defined by their encoders which are finite state machines with parallel inputs and outputs a family of upper bounds on the free distance of a given fs code is derived a general construction for fs codes is given and it is shown that in many cases the fs codes constructed in this way have a free distance that is the largest possible catastrophic error propagation cep for fs codes is also discussed it is found that to avoid cep one must solve the graph theoretic problem of finding a uniquely decodable edge labeling of the state diagram

profit estimation error in the newsvendor model under a parametric demand distribution we consider the newsvendor model in which uncertain demand is assumed to follow a probabilistic distribution with known functional form but unknown parameters these parameters are estimated unbia

new trellis codes based on lattices and cosets a new technique is proposed for constructing trellis codes which provides an alternative to ungerboeck s method of set partitioning the new codes use a signal constellation consisting of points from an n dimensional lattice lambda with an equal number of points from each coset of a sublattice lambda one part of the input stream drives a generalized convolutional code whose outputs are cosets of lambda while the other part selects points from these cosets several of the new codes are better than those previously known

two classes of convolutional codes over gf q for q ary orthogonal signaling two classes of nonbinary convolutional codes for error control in communication systems employing q ary orthogonal signaling are discussed these classes include as special cases the dual k codes and the codes treated by b d trumpis 1975 the types of code extensions of previous bounds on free distance the noncatastrophic condition for these codes and conditions for catastrophic error propagation are described codes obtained by a computer search optimal in that the truncated transfer function bound on the probability of symbol error is minimized are presented it is shown how good convolutional codes can be constructed from cyclic block codes the performance of some of these codes over additive white gaussian noise channels rayleigh channels and partial band interference channels is also discussed

multilevel codes and multistage decoding h imai and s hirakawa have proposed 1977 a multilevel coding method based on binary block codes that admits a staged decoding procedure the author extends the coding method to coset codes and shows how to calculate minimum squared distance and path multiplicity in terms of the norms and multiplicities of the different cosets the multilevel structure allows the redundancy in the coset selection procedure to be allocated efficiently among the different levels it also allows the use of suboptimal multistage decoding procedures that have performance complexity advantages over maximum likelihood decoding

a 2 approximation algorithm for stochastic inventory control models with lost sales in this paper we describe the first computationally efficient policies for stochastic inventory models with lost sales and replenishment lead times that admit worst case performance guarantees in particular we introduce dual balancing policies for lost sales models that are conceptually similar to dual balancing policies recently introduced for a broad class of inventory models in which demand is backlogged rather than lost that is in each period we balance two opposing costs the expected marginal holding costs against the expected marginal lost sales cost specifically we show that the dual balancing policies for the lost sales models provide a worst case performance guarantee of two under relatively general demand structures in particular the guarantee holds for independent not necessarily identically distributed demands and for models with correlated demands such as the ar 1 model and the multiplicative autoregressive demand model the policies and the worst case guarantee extend to models with capacity constraints on the size of the order and stochastic lead times our analysis has several novel elements beyond the balancing ideas for backorder models

a nonparametric asymptotic analysis of inventory planning with censored demand we study stochastic inventory planning with lost sales and instantaneous replenishment where contrary to the classical inventory theory knowledge of the demand distribution is not available furthermore we observe only the sales quantity in each period and lost sales are unobservable that is demand data are censored the manager must make an ordering decision in each period based only on historical sales data excess inventory is either perishable or carried over to the next period in this setting we propose nonparametric adaptive policies that generate ordering decisions over time we show that the t period average expected cost of our policy differs from the benchmark newsvendor cost the minimum expected cost that would have incurred if the manager had known the underlying demand distribution by at most o 1 t0 5