a hierarchy of bounds for stochastic mixed integer programs we consider general two stage smips with recourse in which integer variables are allowed in both stages of the problem and randomness is allowed in the objective function the constraint matrices i e the technology matrix and the recourse matrix and the right hand side we develop a hierarchy of lower and upper bounds for the optimal objective value of an smip by generalizing the wait and see solution and the expected result of using the expected value solution these bounds become progressively stronger but generally more difficult to compute our numerical study indicates the bounds we develop in this paper can be strong relative to those provided by linear relaxations hence this new bounding approach is a complementary tool to the current bounding techniques used in solving smips particularly for large scale and poorly formulated problems

a primal dual approach to inexact subgradient methods for optimization problems with computationally demanding objective functions and subgradients inexact subgradient methods ixs have been introduced by using successive approximation schemes within subgradient optimization methods au et al 1994 in this paper we develop alternative solution procedures when the primal dual information of ixs is utilized this approach is especially useful when the projection operation onto the feasible set is difficult we also demonstrate its applicability to stochastic linear programs

optimal lossless data compression non asymptotics and asymptotics this paper provides an extensive study of the behavior of the best achievable rate and other related fundamental limits in variable length strictly lossless compression in the non asymptotic regime the fundamental limits of fixed to variable lossless compression with and without prefix constraints are shown to be tightly coupled several precise quantitative bounds are derived connecting the distribution of the optimal code lengths to the source information spectrum and an exact analysis of the best achievable rate for arbitrary sources is given fine asymptotic results are proved for arbitrary not necessarily prefix compressors on general mixing sources nonasymptotic explicit gaussian approximation bounds are established for the best achievable rate on markov sources the source dispersion and the source varentropy rate are defined and characterized together with the entropy rate the varentropy rate serves to tightly approximate the fundamental nonasymptotic limits of fixed to variable compression for all but very small block lengths

stability of solutions for stochastic programs with complete recourse quantitative continuity of optimal solution sets to convex stochastic programs with linear complete recourse and random right hand sides is investigated when the underlying probability measure varies in a metric space the central result asserts that under a strong convexity condition for the expected recourse in the unperturbed problem optimal tenders behave holder continuous with respect to a wasserstein metric for linear stochastic programs this carries over to the hausdorff distance of optimal solution sets a general sufficient condition for the crucial strong convexity assumption is given and verified for recourse problems with separable and nonseparable objectives

inertial extragradient algorithms for strongly pseudomonotone variational inequalities abstract the purpose of this paper is to study and analyze two different kinds of inertial type iterative methods for solving variational inequality problems involving strongly pseudomonotone and lipschitz continuous operators in hilbert spaces the projection method is used to design the algorithms which can be computed more easily the construction of solution approximations and the proof of convergence of the algorithms are performed without prior knowledge of the modulus of strong pseudomonotonicity and the lipschitz constant of cost operator instead of that the algorithms use variable stepsize sequences which are diminishing and non summable the numerical behaviors of the proposed algorithms on a test problem are illustrated and compared with several previously known algorithms

on the convergence of algorithms with implications for stochastic and nondifferentiable optimization studies of the convergence of algorithms often revolve around the existence of a function with respect to which monotonic descent is required in this paper we show that under relatively lenient conditions stage dependent descent not necessarily monotonic is sufficient to guarantee convergence this development also provides the impetus to examine optimization algorithms we show that one of the important avenues in the study of convergence namely the theory of epi convergence imposes stronger conditions than are necessary to establish the convergence of an optimization algorithm working from a relaxation of epi convergence we introduce the notion of compatibility and prove several results that permit relaxations of conditions imposed by previous approaches to algorithmic convergence finally to illustrate the usefulness of the concepts we combine stage dependent descent with results derivable from compatibility to provide a basis for the convergence of a general algorithmic statement that might be used for stochastic and nondifferentiable optimization

a modified forward backward splitting method for maximal monotone mappings we consider the forward backward splitting method for finding a zero of the sum of two maximal monotone mappings this method is known to converge when the inverse of the forward mapping is strongly monotone we propose a modification to this method in the spirit of the extragradient method for monotone variational inequalities under which the method converges assuming only the forward mapping is lipschitz continuous on some closed convex subset of its domain the modification entails an additional forward step and a projection step at each iteration applications of the modified method to decomposition in convex programming and monotone variational inequalities are discussed

stochastic decomposition an algorithm for two stage linear programs with recourse we present a cutting plane algorithm for two stage stochastic linear programs with recourse motivated by benders decomposition our method uses randomly generated observations of random variables to construct statistical estimates of supports of the objective function in general the resulting piecewise linear approximations do not agree with the objective function in finite time however certain subsequences of the estimated supports are shown to accumulate at supports of the objective function with probability one from this we establish the convergence of the algorithm under relatively mild assumptions

ten ways to generate the il in and related schemes abstract for one dimensional singularly perturbed boundary value problems there exist a remarkable variety of possibilities to generate uniformly convergent schemes we describe the basic ideas of ten different approaches and the advantages and disadvantages of several methods

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

high order compact exponential finite difference methods for convection diffusion type problems a class of high order compact hoc exponential finite difference fd methods is proposed for solving one and two dimensional steady state convection diffusion problems the newly proposed hoc exponential fd schemes have nonoscillation property and yield high accuracy approximation solution as well as are suitable for convection dominated problems the o h 4 compact exponential fd schemes developed for the one dimensional 1d problems produce diagonally dominant tri diagonal system of equations which can be solved by applying the tridiagonal thomas algorithm for the two dimensional 2d problems o h 4 k 4 compact exponential fd schemes are formulated on the nine point 2d stencil and the line iterative approach with alternating direction implicit adi procedure enables us to deal with diagonally dominant tridiagonal matrix equations which can be solved by application of the one dimensional tridiagonal thomas algorithm with a considerable saving in computing time to validate the present hoc exponential fd methods three linear and nonlinear problems mostly with boundary or internal layers where sharp gradients may appear due to high peclet or reynolds numbers are numerically solved comparisons are made between analytical solutions and numerical results for the currently proposed hoc exponential fd methods and some previously published hoc methods the present hoc exponential fd methods produce excellent results for all test problems it is shown that besides including the excellent performances in computational accuracy efficiency and stability the present method has the advantage of better scale resolution the method developed in this article is easy to implement and has been applied to obtain the numerical solutions of the lid driven cavity flow problem governed by the 2d incompressible navier stokes equations using the stream function vorticity formulation

projected reflected gradient methods for monotone variational inequalities this paper is concerned with some new projection methods for solving variational inequality problems with monotone and lipschitz continuous mapping in hilbert space first we propose the projected reflected gradient algorithm with a constant stepsize it is similar to the projected gradient method namely the method requires only one projection onto the feasible set and only one value of the mapping per iteration this distinguishes our method from most other projection type methods for variational inequalities with monotone mapping also we prove that it has r linear rate of convergence under the strong monotonicity assumption the usual drawback of algorithms with constant stepsize is the requirement to know the lipschitz constant of the mapping to avoid this we modify our first algorithm so that the algorithm needs at most two projections per iteration in fact our computational experience shows that such cases with two projections are very rare this scheme at least theoretically seems to be ve

spectral resolutioned exponential compact higher order scheme srechos for convection diffusion equations a spectral resolutioned exponential compact higher order scheme srechos is developed to solve stationary convection diffusion type of differential equations with constant convection and diffusion coefficients the scheme is o h6 for one dimensional problems and produces a tri diagonal system of equations that can be solved efficiently using thomas algorithm for two dimensional problems the scheme produces an o h4 k4 accuracy over a compact nine point stencil which can be solved using any line iterative approach with alternate direction implicit adi procedure the efficiency of the developed scheme is measured using wave number analysis the analysis shows that the srechos has a much better spectral resolution than any of the existing higher order schemes numerical examples are solved and better performance of the srechos in terms of accuracy over the existing schemes in the literature is demonstrated

skew hermitian based iterations for nine point approximations of convection diffusion problems we study skew hermitian based splitting methods for the iterative solution of nonsymmetric linear systems arising from high order compact hoc approximations of two dimensional convection diffusion problems such discretisations lead to nine point system matrices with block tridiagonal structures the nonsymmetric discretisation matrix for a constant coefficient grid aligned flow problem is shown to be positive definite and thus hermitian and skew hermitian splitting hss methods can be considered for the solution of the corresponding linear system for the solution of this linear system using the hss iteration we derive an analytical expression for the optimal value for the upper bound of the contraction factor comparison of the hss iteration with triangular and skew hermitian splitting tss methods and their corresponding block variants btss are carried out for various test problems cpu timings for the different splitting methods using experimentally determined optimal acceleration parameters are also given our results show that the combination of high order compact discretisations and skew hermitian based iterative methods for solving the corresponding linear systems provide efficient procedures for approximating the solutions of two dimensional convection diffusion equations

two sided bounds for nonuniform rates of convergence in the central limit theorem two sided bounds are derived for nonuniform rates of convergence in the central limit theorem and are applied to solve several problems on rates of convergence in nonuniform metrics in particular an expression for the fastest rate of convergence is obtained and rates of convergence using different norming constants are compared a useful estimate of the rate of convergence in the 1 x 2 metric is derived which does not require the assumption of 2 th order moments

a piecewise linear upper bound on the network recourse function we consider the optimal value of a pure minimum cost network flow problem as a function of supply demand and arc capacities we present a new piecewise linear upper bound on this function which is called the network recourse function the bound is compared to the standard madansky bound and is shown computationally to be a little weaker but much faster to find the amount of work is linear in the number of stochastic variables not exponential as is the case for the madansky bound therefore the reduction in work increases as the number of stochastic variables increases computational results are presented

total variation bounds on the expectation of periodic functions with applications to recourse approximations we derive a lower and upper bound for the expectation of periodic functions depending on the total variation of the probability density function of the underlying random variable using worst case analysis we derive tighter bounds for functions that are periodically monotone these bounds can be used to evaluate the performance of approximations for both continuous and integer recourse models in this paper we introduce a new convex approximation for totally unimodular recourse models and we show that this convex approximation has the best worst case error bound possible improving previous bounds with a factor 2 moreover we use similar analysis to derive error bounds for two types of discrete approximations of continuous recourse models with continuous random variables furthermore we derive a tractable lipschitz constant for pure integer recourse models

edgeworth expansion for u statistics under minimal conditions berry esseen bounds for u statistics under the optimal moment conditions were derived by koroljuk and borovskich and friedrich under the same optimal moment assumptions with an additional nonlattice condition we establish a one term edgeworth expansion with remainder o n 1 2 for u statistics

strong convergence theorem for uniformly l lipschitzian asymptotically pseudocontractive mapping in real banach space abstract let e be a real banach space let k be a nonempty closed and convex subset of e t k k a uniformly l lipschitzian asymptotically pseudocontractive mapping with sequence k n n 0 1 lim n k n 1 such that f t let n n 0 0 1 be such that n 0 n n 0 n 2 and n 0 n k n 1 suppose x n n 0 is iteratively defined by x n 1 1 n x n n t n x n n 0 and suppose there exists a strictly increasing continuous function 0 0 0 0 such that t n x x j x x k n x x 2 x x x k it is proved that x n n 0 converges strongly to x f t it is also proved that the sequence of iteration x n defined by x n 1 a n x n b n t n x n c n u n n 0 where u n n 0 is a bounded sequence in k and a n n 0 b n n 0 c n n 0 are sequences in 0 1 satisfying appropriate conditions converges strongly to a fixed point of t

regularized and inertial algorithms for common fixed points of nonlinear operators this paper deals with a general fixed point iteration for computing a point in some nonempty closed and convex solution set included in the common fixed point set of a sequence of mappings on a real hilbert space the proposed method combines two strategies viscosity approximations regularization and inertial type extrapolation the first strategy is known to ensure the strong convergence of some successive approximation methods while the second one is intended to speed up the convergence process under classical conditions on the operators and the parameters we prove that the sequence of iterates generated by our scheme converges strongly to the element of minimal norm in the solution set this algorithm works for instance for approximating common fixed points of infinite families of demicontractive mappings including the classes of quasi nonexpansive operators and strictly pseudocontractive ones

asymptotically optimal multistage tests of simple hypotheses a family of variable stage size multistage tests of simple hypotheses is described based on efficient multistage sampling procedures using a loss function that is a linear combination of sampling costs and error probabilities these tests are shown to minimize the integrated risk to second order as the costs per stage and per observation approach zero a numerical study shows significant improvement over group sequential tests in a binomial testing problem

tight bounds for stochastic convex programs variable and row aggregation as a technique of simplifying a mathematical program is utilized to develop bounds for two stage stochastic convex programs with random right hand sides if one is able to utilize the problem structure along with only first moment information a tighter bound than the usual mean model bound based on jensen s inequality may be obtained moreover it is possible to construct examples for which the mean model bound will be arbitrarily poor consequently one can tighten jensen s bound for stochastic programs when the distribution has a compact support this bound may be improved further by partitioning the support using conditional first moments with regard to first moment upper bounds the gassmann ziemba inequality is used for the stochastic convex program to seek a model which can be solved using standard convex programming techniques moreover it allows one to easily construct upper bounds using the solution of the lower bounding problem finally the results are extended to multistage stochastic convex programming problems

second order lower bounds on the expectation of a convex function we develop a class of lower bounds on the expectation of a convex function the bounds utilize the first two moments of the underlying random variable whose support is contained in a bounded interval or hyperrectangle our bounds have applications to stochastic programs whose random parameters are known only through limited moment information computational results are presented for two stage stochastic linear programs

new nonasymptotic channel coding theorems for structured codes new nonasymptotic random coding theorems with error probability epsilon and finite block length n based on gallager parity check ensemble and general parity check ensembles are derived in this paper the resulting nonasymptotic achievability bounds when combined with nonasymptotic equipartition properties developed in this paper can be easily computed analytically these nonasymptotic achievability bounds are shown to be asymptotically tight up to the second order of the coding rate as n goes to infinity with either constant or subexponentially decreasing epsilon in the case of gallager parity check ensemble and to imply that low density parity check ldpc codes be capacity achieving in the case of ldpc ensembles numerically they are also compared favorably for finite n and epsilon of practical interest with existing nonasymptotic achievability bounds in the literature

new second order bounds on the expectation of saddle functions with applications to stochastic linear programming this paper develops new bounds on the expectation of a convex concave saddle function of a random vector with compact domains the bounds are determined by replacing the underlying distribution by unique discrete distributions constructed using second order moment information the results extend directly to new second moment lower bounds in closed form for the expectation of a convex function these lower bounds are better than jensen s bound the only previously known lower bound for the convex case under limited moment information application of the second moment bounds to two stage stochastic linear programming is reported computational experiments using randomly generated stochastic programs indicate that the new bounds may easily outperform the usual first order bounds

bounds for two stage stochastic programs with fixed recourse this paper develops upper and lower bounds on two stage stochastic linear programs using limited moment information the case considered is when both the right hand side as well as the objective coefficients of the second stage problem are random random variables are allowed to have arbitrary multivariate probability distributions with bounded support first upper and lower bounds are obtained using first and cross moments from which we develop bounds using only first moments the bounds are shown to solve the respective general moment problems

uniform high order difference schemes for a singularly perturbed two point boundary value problem on construit une famille de schemas aux differences finies uniformement precis pour le probleme du type eu a x u b x u f x on donne une analyse complete de l erreur de discretisation basee sur les resultats de stabilite on presente des essais numeriques

un critere sur les petites boules dans le theoreme limite central we show that a banach space valued random variablex such that mathop lim limits x to infty t 2 mathbb p left left x right t right 0 satisfies the central limit theorem if and only if the following criterion on small balls is fulfilled mathop lim limits x to infty t 2 mathbb p left left x right t right 0

exponential higher order compact scheme for 3d steady convection diffusion problem abstract in this paper the exponential higher order compact ehoc finite difference schemes proposed by tian and dai 2007 10 for solving the one and two dimensional steady convection diffusion equations with constant or variable convection coefficients are extended to the three dimensional case the proposed ehoc scheme has the feature that it provides very accurate solution exact in case of constant convection coefficient in 1d of the homogeneous equation that is responsible for the fundamental singularity of the homogeneous solution while approximates the particular part of the solution by fourth order accuracy over the nineteen point compact stencil the key properties of this scheme are its stability accuracy and efficiency so that high gradients near the boundary layers can be effectively resolved even on coarse uniform meshes to validate the present ehoc method three test problems mostly with boundary or internal layers are solved comparisons are made between numerical results for the present ehoc scheme and other available methods in the literature

barycentric scenario trees in convex multistage stochastic programming this work deals with the approximation of convex stochastic multistage programs allowing prices and demand to be stochastic with compact support based on earlier results sequences of barycentric scenario trees with associated probability trees are derived for minorizing and majorizing the given problem error bounds for the optimal policies of the approximate problem and duality analysis with respect to the stochastic data determine the scenarios which improve the approximation convergence of the approximate solutions is proven under the stated assumptions preliminary computational results are outlined

short note a high order pade adi method for unsteady convection diffusion equations a high order alternating direction implicit adi method for computations of unsteady convection diffusion equations is proposed by using fourth order pade schemes for spatial derivatives the present scheme is fourth order accurate in space and second order accurate in time the solution procedure consists of a number of tridiagonal matrix operations which make the computation cost effective the method is unconditionally stable and shows higher accuracy and better phase and amplitude error characteristics than the standard second order adi method d w peaceman h h rachford jr the numerical solution of parabolic and elliptic differential equations journal of the society of industrial and applied mathematics 3 1959 28 41 and the fourth order adi scheme of karaa and zhang high order adi method for solving unsteady convection diffusion problem journal of computational physics 198 2004 1 9

exact convergence rate and leading term in central limit theorem for student s t statistic the leading term in the normal approximation to the distribution of student s t statistic is derived in a general setting with the sole assumption being that the sampled distribution is in the domain of attraction of a normal law the form of the leading term is shown to have its origin in the way in which extreme data influence properties of the studentized sum the leading term approximation is used to give the exact rate of convergence in the central limit theorem up to order n 1 2 where n denotes sample size it is proved that the exact rate uniformly on the whole real line is identical to the exact rate on sets of just three points moreover the exact rate is identical to that for the non studentized sum when the latter is normalized for scale using a truncated form of variance but when the corresponding truncated centering constant is omitted examples of characterizations of convergence rates are also given it is shown that in some instances their validity uniformly on the whole real line is equivalent to their validity on just two symmetric points

on the asymptotic equivalence of l p metrics for convergence to normality this paper is concerned with the rate of convergence to zero of the l p metrics np l p constructed out of differences between distribution functions for departure from normality for normed sums of independent and identically distributed random variables with zero mean and unit variance it is shown that the np are under broad conditions asymptotically equivalent in the strong sense that for l p p np np is universally bounded away from zero and infinity as n

an upper bound on the expectation of simplicial functions of multivariate random variables we introduce an upper bound on the expectation of a special class of sublinear functions of multivariate random variables defined over the entire euclidean space without an independence assumption the bound can be evaluated easily requiring only the solution of systems of linear equations thus permitting implementations in high dimensional space only knowledge on the underlying distribution means and second moments is necessary we discuss pertinent techniques on dominating general sublinear functions by using simpler sublinear and polyhedral functions and second order quadratic functions

on the redundancy of lossy source coding with abstract alphabets the redundancy problem of lossy source coding with abstract source and reproduction alphabets is considered for coding at a fixed rate level it is shown that for any fixed rate r 0 and any memoryless abstract alphabet source p satisfying some mild conditions there exists a sequence c sub n sub n 1 sup spl infin of block codes at the rate r such that the distortion redundancy of c sub n defined as the difference between the performance of c sub n and the distortion rate function d p r of p is upper bounded by spl part d p r spl part r ln n 2n o ln n n for coding at a fixed distortion level it is demonstrated that for any d 0 and any memoryless abstract alphabet source p satisfying some mild conditions there exists a sequence c sub n sub n 1 sup spl infin of block codes at the fixed distortion d such that the rate redundancy of c sub n defined as the difference between the performance of c sub n and the rate distortion function r p d of p is upper bounded by 7ln n 6n o ln n n these results strengthen the traditional berger s 1968 1971 abstract alphabet source coding theorem and extend the positive redundancy results of zhang yang and wei see ibid vol 43 no 1 p 71 91 1997 and ibid vol 42 p 803 21 1996 on lossy source coding with finite alphabets and the redundancy result of wyner see ibid vol 43 p 1452 64 1997 on block coding of memoryless gaussian sources

solving slp recourse problems with arbitrary multivariate distributions the dependent case bounding the expectation of a convex function of a multivariate random variable is of great importance for solving stochastic linear programming problems slp with recourse the classic bounds are those of jensen and edmundson madansky whereby jensen s inequality is available in the independent as well as in the dependent case in contrast with the edmundson madansky e m inequality madansky a 1959 bounds on the expectation of a convex function of multivariate random variable ann math statist 30 743 746 which is valid only in the independent case this paper presents a straightforward extension of the e m inequality for the dependent case yielding a discrete distribution that is extremal with respect to a partial ordering of a set of distribution functions further we prove that the bounds for the expectation behave monotonically by applying the obtained inequality to smaller and smaller subintervals as this requires the evaluation of conditional probabilities and expectations we replace the given distribution by a discrete one resulting from sampling and determine the sampling size that keeps the statistical error negligible finally we conclude with the application to slp recourse problems and state some computational results illustrating the effort for solving recourse problems

the ugc hardness threshold of the lp grothendieck problem for p 2 we consider the problem of given an n n matrix a aij whose diagonal entries vanish approximating in polynomial time the number rm opt p a max left sum i j 1 na ij x ix j x 1 ldots x n in mathbb r n wedge left sum i 1 n x i p right 1 p le 1 right when p 2 this is simply the problem of computing the maximum eigenvalue of a whereas for p actually it suffices to take p log n it is the grothendieck problem on the complete graph which was shown to have a o log n approximation algorithm in nemirovski et al nemirovski a c roos t terlaky 1999 on maximization of quadratic form over intersection of ellipsoids with common center math program ser a86 3 463 473 megretski megretski a 2001 relaxations of quadratic programs in operator theory and system analysis systems approximation singular integral operators and related topics bordeaux 2000 vol 129 operator theory advances and applications birkhauser basel 365 392 charikar and wirth charikar m a wirth 2004 maximizing quadratic programs extending grothendieck s inequality proc 45th annual sympos foundations comput sci ieee computer society 54 60 and was used in the work of charikar and wirth noted above to design the best known algorithm for the problem of computing the maximum correlation in correlation clustering thus the problem of approximating optp a interpolates between the spectral p 2 case and the correlation clustering p case from a physics point of view this problem corresponds to computing the ground states of spin glasses in a hard wall potential well we design a polynomial time algorithm which given p 2 and an n n matrix a aij with zeros on the diagonal computes optp a up to a factor p e 30 log p on the other hand assuming the unique games conjecture ugc we show that it is np hard to approximate optp a up to a factor smaller than p e hence as p the ugc hardness threshold for computing optp a is exactly p e 1 o 1

on variable step relaxed projection algorithm for variational inequalities projection algorithms are practically useful for solving variational inequalities vi however some among them require the knowledge related to vi in advance such as lipschitz constant usually it is impossible in practice this paper studies the variable step basic projection algorithm and its relaxed version under weakly co coercive condition the algorithms discussed need not know constant function associated with the co coercivity or weak co coercivity and the step size is varied from one iteration to the next under certain conditions the convergence of the variable step basic projection algorithm is established for the practical consideration we also give the relaxed version of this algorithm in which the projection onto a closed convex set is replaced by another projection at each iteration and latter is easy to calculate the convergence of relaxed scheme is also obtained under certain assumptions finally we apply these two algorithms to the split feasibility problem sfp

inexact subgradient methods with applications in stochastic programming in many instances the exact evaluation of an objective function and its subgradients can be computationally demanding by way of example we cite problems that arise within the context of stochastic optimization where the objective function is typically defined via multi dimensional integration in this paper we address the solution of such optimization problems by exploring the use of successive approximation schemes within subgradient optimization methods we refer to this new class of methods as inexact subgradient algorithms with relatively mild conditions imposed on the approximations we show that the inexact subgradient algorithms inherit properties associated with their traditional i e exact counterparts within the context of stochastic optimization the conditions that we impose allow a relaxation of requirements traditionally imposed on steplengths in stochastic quasi gradient methods additionally we study methods in which steplengths may be defined adaptively in a manner that reflects the improvement in the objective function approximations as the iterations proceed we illustrate the applicability of our approach by proposing an inexact subgradient optimization method for the solution of stochastic linear programs

numerical approach to monotone variational inequalities by a one step projected reflected gradient method with line search procedure abstract in this paper we revisit a recent approach to classical monotone variational inequalities by means of a projected reflected gradient type method in r n a line search procedure is incorporated for possible varying step sizes and no lipschitz continuity condition on the operator is required a main feature of the proposed method is that it requires only one projection step onto the feasible set at each iteration

restricted recourse bounds for stochastic linear programming we consider the problem of bounding the expected value of a linear program lp containing random coefficients with applications to solving two stage stochastic programs an upper bound for minimizations is derived from a restriction of an equivalent penalty based formulation of the primal stochastic lp and a lower bound is obtained from a restriction of a reformulation of the dual our restricted recourse bounds are more general and more easily computed than most other bounds because random coefficients may appear anywhere in the lp neither independence nor boundedness of the coefficients is needed and the bound is computed by solving a single lp or nonlinear program analytical examples demonstrate that the new bounds can be stronger than complementary jensen bounds an upper bound is complementary to a lower bound and vice versa in computational work we apply the bounds to a two stage stochastic program for semiconductor manufacturing with uncertain demand and production rates

stopping rules for a class of sampling based stochastic programming algorithms monte carlo sampling based algorithms hold much promise for solving stochastic programs with many scenarios a critical component of such algorithms is a stopping criterion to ensure the quality of the solution in this paper we develop a stopping rule theory for a class of algorithms that estimate bounds on the optimal objective function value by sampling we provide rules for selecting sample sizes and terminating the algorithm under which asymptotic validity of confidence intervals for the quality of the proposed solution can be verified empirical coverage results are given for a simple example

edgeworth expansion of a function of sample means many important statistics can be written as functions of sample means of vector variables a fundamental contribution to the edgeworth expansion for functions of sample means was made by bhattacharya and ghosh in their work the crucial cramer c condition is assumed on the joint distribution of all the components of the vector variable however in many practical situations only one or a few of the components satisfy conditionally this condition while the rest do not such a case is referred to as satisfying the partial cramer c condition the purpose of this paper is to establish edgeworth expansions for functions of sample means when only the partial cramer c condition is satisfied

sublinear upper bounds for stochastic programs with recourse separable sublinear functions are used to provide upper bounds on the recourse function of a stochastic program the resulting problem s objective involves the inf convolution of convex functions a dual of this problem is formulated to obtain an implementable procedure to calculate the bound function evaluations for the resulting convex program only require a small number of single integrations in contrast with previous upper bounds that require a number of function evaluations that grows exponentially in the number of random variables the sublinear bound can often be used when other suggested upper bounds are intractable computational results indicate that the sublinear approximation provides good efficient bounds on the stochastic program objective value

on the effect of random norming on the rate of convergence in the central limit theorem it is shown that studentizing i e normalizing by the sample standard deviation rather than the population standard deviation can improve the rate of convergence in the central limit theorem this provides concise confirmation of one feature of the folklore that a studentized sum is in some sense more robust than a normed sum the case of infinite population standard deviation is also examined 1 introduction our purpose in this paper is to show that studentizing the operation of replacing a population standard deviation in a norming constant by its sample estimate can improve the rate of convergence in the central limit theorem thus we provide concise confirmation of one feature of the folklore that a studentized sum is in some sense more robust than a normed sum the reason is that large deviations of summands which are responsible for slow rates of convergence when ordinary norming constants are used cancel from the sample mean and sample standard deviation remark 3 7 in section 3 will give an intuitive explanation of the mechanism behind this cancellation note that the large deviations have a particularly subtle influence since under the condition that the distribution is attracted to the normal law the largest summand is negligible in comparison with the sum itself section 2 will briefly survey results on rates of convergence in the ordinary i e nonstudentized central limit theorem and will discuss their connection with our work section 3 will state our main results and summarize their consequences and section 4 will give proofs

upper bounds on the expected value of a convex function using gradient and conjugate function information new upper bounds are given for the expected value of a convex function the bounds employ subgradient information and the conjugate function in contrast to most other bounds explicit moment information is not needed we derive the bounds and compare them with previous bounds with different information requirements

convergence determining sets in the central limit theorem a complete characterisation is given of the class of all doublets which determine the rate of convergence in the central limit theorem this enables a number of important properties of convergence determining sets to be deduced in particular it is shown that no singleton can be convergence determining and any set consisting of four or more distinct points is convergence determining numerical and analytic methods are used to derive the geometry of the class of all convergence determining doublets

problem based optimal scenario generation and reduction in stochastic programming scenarios are indispensable ingredients for the numerical solution of stochastic programs earlier approaches to optimal scenario generation and reduction are based on stability arguments involving distances of probability measures in this paper we review those ideas and suggest to make use of stability estimates based only on problem specific data for linear two stage stochastic programs we show that the problem based approach to optimal scenario generation can be reformulated as best approximation problem for the expected recourse function which in turn can be rewritten as a generalized semi infinite program we show that the latter is convex if either right hand sides or costs are random and can be transformed into a semi infinite program in a number of cases we also consider problem based optimal scenario reduction for two stage models and optimal scenario generation for chance constrained programs finally we discuss problem based scenario generation for the classical newsvendor problem