construction algorithms for polynomial lattice rules for multivariate integration we introduce a new construction algorithm for digital nets for integration in certain weighted tensor product hilbert spaces the first weighted hilbert space we consider is based on walsh functions dick and pillichshammer calculated the worst case error for integration using digital nets for this space here we extend this result to a special construction method for digital nets based on polynomials over finite fields this result allows us to find polynomials which yield a small worst case error by computer search we prove an upper bound on the worst case error for digital nets obtained by such a search algorithm which shows that the convergence rate is best possible and that strong tractability holds under some condition on the weights we extend the results for the weighted hilbert space based on walsh functions to weighted sobolev spaces in this case we use randomly digitally shifted digital nets the construction principle is the same as before only the worst case error is slightly different again digital nets obtained from our search algorithm yield a worst case error achieving the optimal rate of convergence and as before strong tractability holds under some condition on the weights these results show that such a construction of digital nets yields the until now best known results of this kind and that our construction methods are comparable to the construction methods known for lattice rules we conclude the article with numerical results comparing the expected worst case error for randomly digitally shifted digital nets with those for randomly shifted lattice rules

scrambled net variance for integrals of smooth functions hybrids of equidistribution and monte carlo methods of integration can achieve the superior accuracy of the former while allowing the simple error estimation methods of the latter one version randomized t m s nets has the property that the integral estimates are unbiased and that the variance is o 1 n for any square integrable integrand stronger assumptions on the integrand allow one to find rates of convergence this paper shows that for smooth integrands over s dimensions the variance is of order n 3 log n 3 1 compared to n 1 for ordinary monte carlo thus the integration errors are of order n 3 2 log n s 1 2 in probability this compares favorably with the rate n 1 log n s 1 for unrandomized t m s nets

the tent transformation can improve the convergence rate of quasi monte carlo algorithms using digital nets in this paper we investigate multivariate integration in reproducing kernel sobolev spaces for which the second partial derivatives are square integrable as quadrature points for our quasi monte carlo algorithm we use digital t m s nets over mathbb z 2 which are randomly digitally shifted and then folded using the tent transformation for this qmc algorithm we show that the root mean square worst case error converges with order 2 m 2 varepsilon for any 0 where 2m is the number of points a similar result for lattice rules has previously been shown by hickernell

approximating the single sink link installation problem in network design we initiate the algorithmic study of an important but np hard problem that arises commonly in network design the input consists of the following an undirected graph with one sink node and multiple source nodes a specified length for each edge and a specified demand demv for each source node v a small set of cable types where each cable type is specified by its capacity and its cost per unit length the cost per unit capacity per unit length of a high capacity cable may be significantly less than that of a low capacity cable reflecting an economy of scale i e the payoff for buying at bulk may be very high the goal is to design a minimum cost network that can simultaneously route all the demands at the sources to the sink by installing zero or more copies of each cable type on each edge of the graph an additional restriction is that the demand of each source must follow a single path the problem is to find a route from each source node to the sink and to assign capacity to each edge of the network such that the total costs of cables installed are minimized we call this problem the single sink link installation problem for the general problem we introduce a new moat type lower bound on the optimal value and we prove a useful structural property of near optimal solutions for every instance of our problem there is a near optimal solution whose graph is acyclic with a cost no more than twice the optimal cost we present efficient approximation algorithms for key special cases of the problem that arise in practice for points in the euclidean plane we give an approximation algorithm with performance guarantee o log d u1 where d is the total demand and u1 is the smallest cable capacity when the metric is arbitrary we consider the case where the network to be designed is restricted to be two level i e every source sink path has at most two edges for this problem we present an algorithm with performance guarantee o log n where n is the number of nodes in the input graph and also show that this performance guarantee is nearly best possible

the jackknife estimate of variance tukey s jackknife estimate of variance for a statistic s x 1 x 2 cdots x n which is a symmetric function of i i d random variables x i is investigated using an anova like decomposition of s it is shown that the jackknife variance estimate tends always to be biased upwards a theorem to this effect being proved for the natural jackknife estimate of operatorname var s x 1 x 2 cdots x n 1 based on x 1 x 2 cdots x n

quasi monte carlo methods for linear two stage stochastic programming problems quasi monte carlo qmc algorithms are studied for generating scenarios to solve two stage linear stochastic programming problems their integrands are piecewise linear quadratic but do not belong to the function spaces considered for qmc error analysis we show that under some weak geometric condition on the two stage model all terms of their anova decomposition except the one of highest order are continuously differentiable and second order mixed derivatives exist almost everywhere and belong to l 2 l2 this implies that randomly shifted lattice rules may achieve the optimal rate of convergence o n 1 delta o n 1 with delta in 0 frac 1 2 0 12 and a constant not depending on the dimension if the effective superposition dimension is less than or equal to two the geometric condition is shown to be satisfied for almost all covariance matrices if the underlying probability distribution is normal we discuss effective dimensions and techniques for dimension reduction numerical experiments for a production planning model with normal inputs show that indeed convergence rates close to the optimal rate are achieved when using randomly shifted lattice rules or scrambled sobol point sets accompanied with principal component analysis for dimension reduction

a method for approximating the solution set of a system of convex inequalities by polytopes abstract in this note a method for computing approximating by polytopes of the solution set q of a system of convex inequalities is presented it is shown that such approximations can be determined by an algorithm which converges in finitely many steps when the solution set of the given system of inequalities is bounded in this case the algorithm generates inner and outer approximations having the hausdorff distance to each other and to the set q no greater than an a priori fixed and having their extreme points in q and in the relative exterior of q respectively

block iterative projection methods for parallel computation of solutions to convex feasibility problems abstract an iterative method is proposed for solving convex feasibility problems each iteration is a convex combination of projections onto the given convex sets where the weights of the combination may vary from step to step it is shown that any sequence of iterations generated by the algorithm converges if the intersection of the given family of convex sets is nonempty and that the limit point of the sequence belongs to this intersection under mild conditions on the sequence of weight functions special cases are block iterative processes where in each iterative step a certain subfamily of the given family of convex sets is used in particular a block iterative version of the agmon motzkin schoenberg relaxation method for solving systems of linear inequalities is derived such processes lend themselves to parallel implementation and will be useful in various areas of applications including image reconstruction from projections image restoration and other fully discretized inversion problems

sampling and cost sharing approximation algorithms for stochastic optimization problems we consider two and multistage versions of stochastic combinatorial optimization problems with recourse in this framework the instance for the combinatorial optimization problem is drawn from a known probability distribution pi and is only revealed to the algorithm over two or multiple stages at each stage on receiving some more information about the instance the algorithm is allowed to build some partial solution since the costs of elements increase with each passing stage there is a natural tension between waiting for later stages to gain more information about the instance and purchasing elements in earlier stages to take advantages of lower costs we provide approximation algorithms for stochastic combinatorial optimization problems such as the steiner tree problem the steiner network problem and the vertex cover problem by means of a simple sampling based algorithm in every stage our algorithm samples the probability distribution of the requirements and constructs a partial solution to serve the resulting sample we show that if one can construct cost sharing functions associated with the algorithms used to construct these partial solutions then this strategy results in provable approximation guarantees for the overall stochastic optimization problem we also extend this approach to provide an approximation algorithm for the stochastic version of the uncapacitated facility location problem a problem that does not fit into the simpler framework of our main model

lattice rules in non periodic subspaces of sobolev spaces we investigate quasi monte carlo qmc integration over the s dimensional unit cube based on rank 1 lattice point sets in weighted non periodic sobolev spaces mathcal h k alpha varvec gamma s mathrm sob and their subspaces of high order smoothness alpha 1 where varvec gamma denotes a set of the weights a recent paper by dick nuyens and pillichshammer has studied qmc integration in half period cosine spaces with smoothness parameter alpha 1 2 consisting of non periodic smooth functions denoted by mathcal h k alpha varvec gamma s mathrm cos and also in the sum of half period cosine spaces and korobov spaces with common parameter alpha denoted by mathcal h k alpha varvec gamma s mathrm kor mathrm cos motivated by the results shown there we first study embeddings and norm equivalences on those function spaces in particular for an integer alpha we provide their corresponding norm equivalent subspaces of mathcal h k alpha varvec gamma s mathrm sob this implies that mathcal h k alpha varvec gamma s mathrm kor mathrm cos is strictly smaller than mathcal h k alpha varvec gamma s mathrm sob as sets for alpha ge 2 which solves an open problem by dick nuyens and pillichshammer then we study the worst case error of tent transformed lattice rules in mathcal h left k 2 varvec gamma s mathrm sob right and also the worst case error of symmetrized lattice rules in an intermediate space between mathcal h k alpha varvec gamma s mathrm kor mathrm cos and mathcal h k alpha varvec gamma s mathrm sob we show that the almost optimal rate of convergence can be achieved for both cases while a weak dependence of the worst case error bound on the dimension can be obtained for the former case

optimal quadrature for haar wavelet spaces this article considers the error of the scrambled equidistribution quadrature rules in the worst case random case and average case settings the underlying space of integrands is a hilbert space of multidimensional haar wavelet series hwav the asymptotic orders of the errors are derived for the case of the scrambled t m s nets and t s sequences these rules are shown to have the best asymptotic convergence rates for any random quadrature rule for the space of integrands hwav

construction of interlaced polynomial lattice rules for infinitely differentiable functions we study multivariate integration over the s dimensional unit cube in a weighted space of infinitely differentiable functions it is known from a recent result by suzuki that there exists a good quasi monte carlo qmc rule which achieves a super polynomial convergence of the worst case error in this function space and moreover that this convergence behavior is independent of the dimension under a certain condition on the weights in this paper we provide a constructive approach to finding a good qmc rule achieving such a dimension independent super polynomial convergence of the worst case error specifically we prove that interlaced polynomial lattice rules with an interlacing factor chosen properly depending on the number of points n and the weights can be constructed using a fast component by component algorithm in at most o sn log n 2 arithmetic operations to achieve a dimension independent super polynomial convergence the key idea for the proof of the worst case error bound is to use a variant of jensen s inequality with a purposely designed concave function

parallel application of block iterative methods in medical imaging and radiation therapy some row action algorithms which exploit special objective function and constraints structure have proven advantageous for solving huge and sparse feasibility or optimization problems recently developed block iterative versions of such special purpose methods enable parallel computation when the underlying problem is appropriately decomposed this opens the door for parallel computation in image reconstruction problems of computerized tomography and in the inverse problem of radiation therapy treatment planning all in their fully discretized modelling approach since there is more than one way of deriving block iterative versions of any row action method the choice has to be made with reference to the underlying real world problem

optimal quasi monte carlo rules on order 2 digital nets for the numerical integration of multivariate periodic functions we investigate quasi monte carlo rules for the numerical integration of multivariate periodic functions from besov spaces s r p q b mathbb t d sp qrb td with dominating mixed smoothness 1 p

a general approximation technique for constrained forest problems we present a general approximation technique for a large class of graph problems our technique mostly applies to problems of covering at minimum cost the vertices of a graph with trees cycles or paths satisfying certain requirements in particular many basic combinatorial optimization problems fit in this framework including the shortest path minimum cost spanning tree minimum weight perfect matching traveling salesman and steiner tree problems our technique produces approximation algorithms that run in o n 2 log n time and come within a factor of 2 of optimal for most of these problems for instance we obtain a 2 approximation algorithm for the minimum weight perfect matching problem under the triangle inequality our running time of o n 2 log n time compares favorably with the best strongly polynomial exact algorithms running in o n 3 time for dense graphs a similar result is obtained for the 2 matching problem and its variants we also derive the first approximation algorithms for many np complete problems including the non fixed point to point connection problem the exact path partitioning problem and complex location design problems moreover for the prize collecting traveling salesman or steiner tree problems we obtain 2 approximation algorithms therefore improving the previously best known performance guarantees of 2 5 and 3 respectively math programming 59 1993 pp 413 420

valid linear inequalities for fixed charge problems many problems in the operations research management science literature can be formulated with both zero one and continuous variables however the exact optimization of such mixed zero one models remains a computational challenge in this paper we propose to study mixed problems from a mathematical point of view that is similar in spirit to recent research on purely combinatorial problems that has investigated systems of defining linear inequalities or facets of the underlying polytope at least two numerical studies have validated this line of research computationally and the advances in the problem solving capabilities are considerable we expect that similar gains are possible for the mixed zero one problem more precisely we consider the mixed integer programs whose feasible region x is composed of i a simple additive constraint in the continuous variables xj for j 1 2 n and ii constraints 0 xj mjyj defined by binary variables yj for j 1 2 n this type of feasible region arises in a variety of mixed integer problems and particularly in network problems with fixed charges on the arcs we derive two classes of facet defining linear inequalities of the convex hull of x and show that the second of these classes gives a complete description of the convex hull when mj m for all j we also develop methods to detect violated inequalities from these classes so that these facets can be used as cutting planes to strengthen the formulations of certain mixed integer problems

an explicit construction of optimal order quasi monte carlo rules for smooth integrands in a recent paper by the authors it is shown that there exists a quasi monte carlo qmc rule which achieves the best possible rate of convergence for numerical integration in a reproducing kernel hilbert space consisting of smooth functions in this paper we provide an explicit construction of such an optimal order qmc rule our approach is to exploit both the decay and the sparsity of the walsh coefficients of the reproducing kernel simultaneously this can be done by applying digit interlacing composition due to dick to digital nets with large minimum hamming and niederreiter rosenbloom tsfasman metrics due to chen and skriganov to our best knowledge our construction gives the first qmc rule which achieves the best possible convergence in this function space

stochastic combinatorial optimization with controllable risk aversion level most of the recent work on 2 stage stochastic combinatorial optimization problems has focused on minimization of the expected cost or the worst case cost of the solution those two objectives can be viewed as two extreme ways of handling risk in this paper we propose to use a one parameter family of functionals to interpolate between them although such a family has been used in the mathematical finance and stochastic programming literature before its use in the context of approximation algorithms seems new we show that under standard assumptions a broad class of risk adjusted 2 stage stochastic programs can be efficiently treated by the sample average approximation saa method in particular our result shows that it is computationally feasible to incorporate some degree of robustness even when the underlying distribution can only be accessed in a black box fashion we also show that when combined with suitable rounding procedures our result yields new approximation algorithms for many risk adjusted 2 stage stochastic combinatorial optimization problems under the black box setting

duality theory and propagation rules for generalized digital nets digital nets are used in quasi monte carlo algorithms for approximating high dimensional integrals over the unit cube hence one wants to have explicit constructions of digital nets of high quality in this paper we consider the so called propagation rules for digital nets which state how one can obtain a new digital net of different size from existing digital nets this way one often can generate digital nets of higher quality than were previously known here we generalize existing propagation rules for classical digital nets to generalized digital nets as introduced by dick

cyclic digital nets hyperplane nets and multivariate integration in sobolev spaces cyclic nets are a special case of digital nets and were recently introduced by niederreiter here we present a construction algorithm for such nets where we use the root mean square worst case error of a randomly digitally shifted point set in a weighted sobolev space as a selection criterion this yields a feasible construction algorithm since for a cyclic net with qm points with fixed bijections and fixed ground field there are qm possible choices our results here match the convergence rate and strong tractability results for polynomial lattice rules hence providing us with an alternative construction algorithm further we improve upon previous results by including constructions over arbitrary finite fields and an arbitrary choice of bijections

higher order qmc petrov galerkin discretization for affine parametric operator equations with random field inputs we construct quasi monte carlo methods to approximate the expected values of linear functionals of petrov galerkin discretizations of parametric operator equations which depend on a possibly infinite sequence of parameters such problems arise in the numerical solution of differential and integral equations with random field inputs we analyze the regularity of the solutions with respect to the parameters in terms of the rate of decay of the fluctuations of the input field if p in 0 1 denotes the summability exponent corresponding to the fluctuations in affine parametric families of operators then we prove that deterministic interlaced polynomial lattice rules of order alpha lfloor 1 p rfloor 1 in s dimensions with n points can be constructed using a fast component by component algorithm in mathcal o alpha s n log n alpha 2 s 2 n operations to achieve a convergence rate of mathcal o n 1 p with the implied constant independent of s this dimension independen

convergence results for an accelerated nonlinear cimmino algorithm we present an accelerated version of cimmino s algorithm for solving the convex feasibility problem in finite dimension the algorithm is similar to that given by censor and elfving for linear inequalities we show that the nonlinear version converges locally to a weighted least squares solution in the general case and globally to a feasible solution in the consistent case applications to the linear problem are suggested

explicit constructions of quasi monte carlo rules for the numerical integration of high dimensional periodic functions in this paper we give explicit constructions of point sets in the s dimensional unit cube yielding quasi monte carlo algorithms which achieve the optimal rate of convergence of the worst case error for numerically integrating high dimensional periodic functions in the classical measure p alpha of the worst case error introduced by korobov the convergence for every even integer alpha ge 1 is of cal o n min alpha d log n s alpha 2 where d is a parameter of the construction which can be chosen arbitrarily large and n is the number of quadrature points this convergence rate is known to be the best possible up to some log n factors we prove the result for the deterministic and also a randomized setting the construction is based on a suitable extension of digital t m s nets over the finite field mathbb z b

walsh spaces containing smooth functions and quasi monte carlo rules of arbitrary high order we define a walsh space which contains all functions whose partial mixed derivatives up to order delta ge 1 exist and have finite variation in particular for a suitable choice of parameters this implies that certain sobolev spaces are contained in these walsh spaces for this walsh space we then show that quasi monte carlo rules based on digital t alpha s sequences achieve the optimal rate of convergence of the worst case error for numerical integration this rate of convergence is also optimal for the subspace of smooth functions explicit constructions of digital t alpha s sequences are given hence providing explicit quasi monte carlo rules which achieve the optimal rate of convergence of the integration error for arbitrarily smooth functions

when trees collide an approximation algorithm for thegeneralized steiner problem on networks we give the first approximation algorithm for the generalized network steiner problem a problem in network design an instance consists of a network with link costs and for each pair i j of nodes an edge connectivity requirement r ij the goal is to find a minimum cost network using the available links and satisfying the requirements our algorithm outputs a solution whose cost is within 2 lceil log 2 r 1 rceil of optimal where r is the highest requirement value in the course of proving the performance guarantee we prove a combinatorial min max approximate equality relating minimum cost networks to maximum packings of certain kinds of cuts as a consequence of the proof of this theorem we obtain an approximation algorithm for optimally packing these cuts we show that this algorithm has application to estimating the reliability of a probabilistic network

the discrete sell or hold problem with constraints on asset values the discrete sell or hold problem dshp which is introduced in he et al 6 is studied under the constraint that each asset can only take a constant number of different values we show that if each asset can only take two different values the problem becomes polynomial time solvable however even if each asset can only take three different values the dshp is still np hard an approximation algorithm which could potentially achieve a better approximation ratio than the one proposed in 6 is also given under this setting

tight error bounds for rank 1 lattice sampling in spaces of hybrid mixed smoothness we consider the approximate recovery of multivariate periodic functions from a discrete set of function values taken on a rank 1 lattice moreover the main result is the fact that any non linear reconstruction algorithm taking function values on any integration lattice of size m has a dimension independent lower bound of 2 alpha 1 2 m alpha 2 when considering the optimal worst case error with respect to function spaces of hybrid mixed smoothness alpha 0 on the d torus we complement this lower bound with upper bounds that coincide up to logarithmic terms these upper bounds are obtained by a detailed analysis of a rank 1 lattice sampling strategy where the rank 1 lattices are constructed by a component by component method the lattice group structure allows for an efficient approximation of the underlying function from its sampled values using a single one dimensional fast fourier transform this is one reason why these algorithms keep attracting significant interest we compare our results to recent almost optimal methods based upon samples on sparse grids

lattice rules for nonperiodic smooth integrands the aim of this paper is to show that one can achieve convergence rates of n alpha delta n for alpha 1 2 1 2 and for delta 0 0 arbitrarily small for nonperiodic alpha smooth cosine series using lattice rules without random shifting the smoothness of the functions can be measured by the decay rate of the cosine coefficients for a specific choice of the parameters the cosine series space coincides with the unanchored sobolev space of smoothness 1 we study the embeddings of various reproducing kernel hilbert spaces and numerical integration in the cosine series function space and show that by applying the so called tent transformation to a lattice rule one can achieve the almost optimal rate of convergence of the integration error the same holds true for symmetrized lattice rules for the tensor product of the direct sum of the korobov space and cosine series space but with a stronger dependence on the dimension in this case

volume approximation of convex bodies by inscribed polytopes a convex body c in d dimensional euclidean space e is a compact convex subset of le a with non empty interior we say that c is of differentiability class k if its boundary considered as a manifold is of class k the symmetric difference metric 6 s also called nikodym metric on the space of convex bodies is defined for any two convex bodies as the volume of their symmetric difference given a convex body c let c n d l d 2 denote the family of convex polytopes p having at most n vertices and being inscribed into c that is all vertices are on the boundary bdc of c it is well known that for each n there is a polytope p such that s c p s c inf s c p p

hedging uncertainty approximation algorithms for stochastic optimization problems we study two stage finite scenario stochastic versions of several combinatorial optimization problems and provide nearly tight approximation algorithms for them our problems range from the graph theoretic shortest path vertex cover facility location to set theoretic set cover bin packing and contain representatives with different approximation ratios the approximation ratio of the stochastic variant of a typical problem is found to be of the same order of magnitude as its deterministic counterpart furthermore we show that common techniques for designing approximation algorithms such as lp rounding the primal dual method and the greedy algorithm can be adapted to obtain these results

good lattice rules in weighted korobov spaces with general weights we study the problem of multivariate integration and the construction of good lattice rules in weighted korobov spaces with general weights these spaces are not necessarily tensor products of spaces of univariate functions sufficient conditions for tractability and strong tractability of multivariate integration in such weighted function spaces are found these conditions are also necessary if the weights are such that the reproducing kernel of the weighted korobov space is pointwise non negative the existence of a lattice rule which achieves the nearly optimal convergence order is proven a component by component cbc algorithm that constructs good lattice rules is presented the resulting lattice rules achieve tractability or strong tractability error bounds and achieve nearly optimal convergence order for suitably decaying weights we also study special weights such as finite order and order dependent weights for these special weights the cost of the cbc algorithm is polynomial numerical computations show that the lattice rules constructed by the cbc algorithm give much smaller worst case errors than the mean worst case errors over all quasi monte carlo rules or over all lattice rules and generally smaller worst case errors than the best korobov lattice rules in dimensions up to hundreds numerical results are provided to illustrate the efficiency of cbc lattice rules and korobov lattice rules with suitably chosen weights in particular for high dimensional finance problems

cyclic subgradient projections a cyclically controlled method of subgradient projections csp for the convex feasibility problem of solving convex inequalities is presented the features of this method make it an efficient tool in handling huge and sparse problems a particular application to an image reconstruction problem of emission computerized tomography is mentioned

stochastic survivable network design problems theory and practice abstract we study survivable network design problems with edge connectivity requirements under a two stage stochastic model with recourse and finitely many scenarios for the formulation in the natural space of edge variables we show that facet defining inequalities of the underlying polytope can be derived from the deterministic counterparts moreover by using graph orientation properties we introduce stronger cut based formulations for solving the proposed mixed integer programing models we suggest a two stage branch cut algorithm based on a decomposed model in order to accelerate the computations we suggest a new technique for strengthening the decomposed l shaped optimality cuts which is computationally fast and easy to implement a computational study shows the benefit of the decomposition and the cut strengthening which significantly reduces the number of master iterations and the computational running time moreover we evaluate the stability of the scenario generation method and analyze the value of the stochastic solution

lp rounding approximation algorithms for stochastic network design we study the steiner tree problem and the single cable single sink network design problem under a two stage stochastic model with recourse and finitely many scenarios in these models some edges are purchased in a first stage when only probabilistic information about the second stage is available in the second stage one of a finite number of specified scenarios is realized which results in the set of terminals becoming known and the opportunity to purchase additional edges under an inflated cost function to augment the first stage solution we provide constant factor approximation algorithms for these problems by rounding the linear relaxation of ip formulations of the problems our algorithms involve solving the linear relaxation first followed by a primal dual routine that is guided by the lp solution we also show that because our bounds are local the cost of each component is bounded by its cost in the lp solution we are able to obtain bounds that guard against a form of downside risk