min max min robust combinatorial optimization the idea of k adaptability in two stage robust optimization is to calculate a fixed number k of second stage policies here and now after the actual scenario is revealed the best of these policies is selected this idea leads to a min max min problem in this paper we consider the case where no first stage variables exist and propose to use this approach to solve combinatorial optimization problems with uncertainty in the objective function we investigate the complexity of this special case for convex uncertainty sets we first show that the min max min problem is as easy as the underlying certain problem if k is greater than the number of variables and if we can optimize a linear function over the uncertainty set in polynomial time we also provide an exact and practical oracle based algorithm to solve the latter problem for any underlying combinatorial problem on the other hand we prove that the min max min problem is np hard for every fixed number k even when the uncertainty set is a polyhedron given by an inner description for the case that k is smaller or equal to the number of variables we finally propose a fast heuristic algorithm and evaluate its performance

adjustable robust solutions of uncertain linear programs we consider linear programs with uncertain parameters lying in some prescribed uncertainty set where part of the variables must be determined before the realization of the uncertain parameters non adjustable variables while the other part are variables that can be chosen after the realization adjustable variables we extend the robust optimization methodology 1 3 6 9 13 14 to this situation by introducing the adjustable robust counterpart arc associated with an lp of the above structure often the arc is significantly less conservative than the usual robust counterpart rc however in most cases the arc is computationally intractable np hard this difficulty is addressed by restricting the adjustable variables to be affine functions of the uncertain data the ensuing affinely adjustable robust counterpart aarc problem is then shown to be in certain important cases equivalent to a tractable optimization problem typically an lp or a semidefinite problem and in other cases having a tight approximation which is tractable the aarc approach is illustrated by applying it to a multi stage inventory management problem

distributionally robust optimization and its tractable approximations in this paper we focus on a linear optimization problem with uncertainties having expectations in the objective and in the set of constraints we present a modular framework to obtain an approximate solution to the problem that is distributionally robust and more flexible than the standard technique of using linear rules our framework begins by first affinely extending the set of primitive uncertainties to generate new linear decision rules of larger dimensions and is therefore more flexible next we develop new piecewise linear decision rules that allow a more flexible reformulation of the original problem the reformulated problem will generally contain terms with expectations on the positive parts of the recourse variables finally we convert the uncertain linear program into a deterministic convex program by constructing distributionally robust bounds on these expectations these bounds are constructed by first using different pieces of information on the distribution of the underlying uncertainties to develop separate bounds and next integrating them into a combined bound that is better than each of the individual bounds

an empirical analysis of scenario generation methods for stochastic optimization this work presents an empirical analysis of popular scenario generation methods for stochastic optimization including quasi monte carlo moment matching and methods based on probability metrics as well as a new method referred to as voronoi cell sampling solution quality is assessed by measuring the error that arises from using scenarios to solve a multi dimensional newsvendor problem for which analytical solutions are available in addition to the expected value the work also studies scenario quality when minimizing the expected shortfall using the conditional value at risk to quickly solve problems with millions of random parameters a reformulation of the risk averse newsvendor problem is proposed which can be solved via benders decomposition the empirical analysis identifies voronoi cell sampling as the method that provides the lowest errors with particularly good results for heavy tailed distributions a controversial finding concerns evidence for the ineffectiveness of widely used methods based on minimizing probability metrics under high dimensional randomness

the c 3 theorem and a d 2 algorithm for large scale stochastic mixed integer programming set convexification this paper considers the two stage stochastic integer programming problem with an emphasis on instances in which integer variables appear in the second stage drawing heavily on the theory of disjunctive programming we characterize convexifications of the second stage problem and develop a decomposition based algorithm for the solution of such problems in particular we verify that problems with fixed recourse are characterized by scenario dependent second stage convexifications that have a great deal in common we refer to this characterization as the c3 common cut coefficients theorem based on the c3 theorem we develop a decomposition algorithm which we refer to as disjunctive decomposition d2 in this new class of algorithms we work with master and subproblems that result from convexifications of two coupled disjunctive programs we show that when the second stage consists of 0 1 milp problems we can obtain accurate second stage objective function estimates after finitely many steps this result implies the convergence of the d2 algorithm

distributionally robust convex optimization distributionally robust optimization is a paradigm for decision making under uncertainty where the uncertain problem data are governed by a probability distribution that is itself subject to uncertainty the distribution is then assumed to belong to an ambiguity set comprising all distributions that are compatible with the decision maker s prior information in this paper we propose a unifying framework for modeling and solving distributionally robust optimization problems we introduce standardized ambiguity sets that contain all distributions with prescribed conic representable confidence sets and with mean values residing on an affine manifold these ambiguity sets are highly expressive and encompass many ambiguity sets from the recent literature as special cases they also allow us to characterize distributional families in terms of several classical and or robust statistical indicators that have not yet been studied in the context of robust optimization we determine conditions under which distributionally robust optimization problems based on our standardized ambiguity sets are computationally tractable we also provide tractable conservative approximations for problems that violate these conditions

data driven distributionally robust optimization using the wasserstein metric performance guarantees and tractable reformulations we consider stochastic programs where the distribution of the uncertain parameters is only observable through a finite training dataset using the wasserstein metric we construct a ball in the space of multivariate and non discrete probability distributions centered at the uniform distribution on the training samples and we seek decisions that perform best in view of the worst case distribution within this wasserstein ball the state of the art methods for solving the resulting distributionally robust optimization problems rely on global optimization techniques which quickly become computationally excruciating in this paper we demonstrate that under mild assumptions the distributionally robust optimization problems over wasserstein balls can in fact be reformulated as finite convex programs in many interesting cases even as tractable linear programs leveraging recent measure concentration results we also show that their solutions enjoy powerful finite sample performance guarantees our theoretical results are exemplified in mean risk portfolio optimization as well as uncertainty quantification

the ordered k median problem surrogate models and approximation algorithms in the last two decades a steady stream of research has been devoted to studying various computational aspects of the ordered k median problem which subsumes traditional facility location problems such as median center p centrum etc through a unified modeling approach given a finite metric space the objective is to locate k facilities in order to minimize the ordered median cost function in its general form this function penalizes the coverage distance of each vertex by a multiplicative weight depending on its ranking or percentile in the ordered list of all coverage distances while antecedent literature has focused on mathematical properties of ordered median functions integer programming methods various heuristics and special cases this problem was not studied thus far through the lens of approximation algorithms in particular even on simple network topologies such as trees or line graphs obtaining non trivial approximation guarantees is an open question the main contribution of this paper is to devise the first provably good approximation algorithms for the ordered k median problem we develop a novel approach that relies primarily on a surrogate model where the ordered median function is replaced by a simplified ranking invariant functional form via efficient enumeration surprisingly while this surrogate model is varomega n varomega 1 hard to approximate on general metrics we obtain an o log n approximation for our original problem by employing local search methods on a smooth variant of the surrogate function in addition an improved guarantee of 2 epsilon is obtained on tree metrics by optimally solving the surrogate model through dynamic programming finally we show that the latter optimality gap is tight up to an o epsilon term

k adaptability in two stage robust binary programming over the last two decades robust optimization has emerged as a computationally attractive approach to formulate and solve single stage decision problems affected by uncertainty more recently robust optimization has been successfully applied to multistage problems with continuous recourse this paper takes a step toward extending the robust optimization methodology to problems with integer recourse which have largely resisted solution so far to this end we approximate two stage robust binary programs by their corresponding k adaptability problems in which the decision maker precommits to k second stage policies here and now and implements the best of these policies once the uncertain parameters are observed we study the approximation quality and the computational complexity of the k adaptability problem and we propose two mixed integer linear programming reformulations that can be solved with off the shelf software we demonstrate the effectiveness of our reformulations for stylized instances o

application of wall functions to generalized nonorthogonal curvilinear coordinate systems a method has been developed for the application of wall functions to generalized curvilinear coordinate systems with nonorthogonal grids two test cases have been computed using this method with the k e turbulence model flow over a flat plate at 0 deg angle of attack using a nonorthogonal grid at the wall and flow over a prolate hemispheroid with a hemispherical nose cap at 0 deg angle of attack all results are compared with experimental data in addition the hemispheroid results are compared with computations using the baldwin lomax algebraic turbulence model and the chien low reynolds number k e turbulence model

improved combinatorial algorithms for facility location problems we present improved combinatorial approximation algorithms for the uncapacitated facility location problem two central ideas in most of our results are cost scaling and greedy improvement we present a simple greedy local search algorithm which achieves an approximation ratio of 2 414 epsilon in tilde o n 2 epsilon time this also yields a bicriteria approximation tradeoff of 1 gamma 1 2 gamma for facility cost versus service cost which is better than previously known tradeoffs and close to the best possible combining greedy improvement and cost scaling with a recent primal dual algorithm for facility location due to jain and vazirani we get an approximation ratio of 1 853 in tilde o n 3 time this is very close to the approximation guarantee of the best known algorithm which is linear programming lp based further combined with the best known lp based algorithm for facility location we get a very slight improvement in the approximation factor for facility location achieving 1 728 we also consider a variant of the capacitated facility location problem and present improved approximation algorithms for this

multistage robust mixed integer optimization with adaptive partitions we present a new partition and bound method for multistage adaptive mixed integer optimization amio problems that extends previous work on finite adaptability the approach analyzes the optimal solution to a static nonadaptive version of an amio problem to gain insight into which regions of the uncertainty set are restricting the objective function value we use this information to construct partitions in the uncertainty set leading to a finitely adaptable formulation of the problem we use the same information to determine a lower bound on the fully adaptive solution the method repeats this process iteratively to further improve the objective until a desired gap is reached we provide theoretical motivation for this method and characterize its convergence properties and the growth in the number of partitions using these insights we propose and evaluate enhancements to the method such as warm starts and smarter partition creation we describe in detail how to apply finite adaptability to multista

sampling based approximation algorithms for multistage stochastic optimization stochastic optimization problems provide a means to model uncertainty in the input data where the uncertainty is modeled by a probability distribution over the possible realizations of the actual data we consider a broad class of these problems in which the realized input is revealed through a series of stages and hence are called multistage stochastic programming problems multistage stochastic programming and in particular multistage stochastic linear programs with full recourse is a domain that has received a great deal of attention within the operations research community mostly from the perspective of computational results in application settings our main result is to give the first fully polynomial approximation scheme for a broad class of multistage stochastic linear programming problems with any constant number of stages the algorithm analyzed known as the sample average approximation method is quite simple and is the one most commonly used in practice the algorithm accesses the input by

approximations of semicontinuous functions with applications to stochastic optimization and statistical estimation upper semicontinuous usc functions arise in the analysis of maximization problems distributionally robust optimization and function identification which includes many problems of nonparametric statistics we establish that every usc function is the limit of a hypo converging sequence of piecewise affine functions of the difference of max type and illustrate resulting algorithmic possibilities in the context of approximate solution of infinite dimensional optimization problems in an effort to quantify the ease with which classes of usc functions can be approximated by finite collections we provide upper and lower bounds on covering numbers for bounded sets of usc functions under the attouch wets distance the result is applied in the context of stochastic optimization problems defined over spaces of usc functions we establish confidence regions for optimal solutions based on sample average approximations and examine the accompanying rates of convergence examples from nonparametric statistics illustrate the results

complexity of local search for the p median problem abstract we study the complexity of finding local minima for the p median problem the relationship between swap local optima 0 1 local saddle points and classical karush kuhn tucker conditions is presented it is shown that the local search problems with some neighborhoods are tight pls complete moreover the standard local descent algorithm takes exponential number of iterations in the worst case regardless of the tie breaking and pivoting rules used to illustrate this property we present a family of instances where some local minima may be hard to find computational results with different pivoting rules for random and euclidean test instances are discussed these empirical results show that the standard local descent algorithm is polynomial in average for some pivoting rules

least squares quantization in pcm it has long been realized that in pulse code modulation pcm with a given ensemble of signals to handle the quantum values should be spaced more closely in the voltage regions where the signal amplitude is more likely to fall it has been shown by panter and dite that in the limit as the number of quanta becomes infinite the asymptotic fractional density of quanta per unit voltage should vary as the one third power of the probability density per unit voltage of signal amplitudes in this paper the corresponding result for any finite number of quanta is derived that is necessary conditions are found that the quanta and associated quantization intervals of an optimum finite quantization scheme must satisfy the optimization criterion used is that the average quantization noise power be a minimum it is shown that the result obtained here goes over into the panter and dite result as the number of quanta become large the optimum quautization schemes for 2 b quanta b 1 2 cdots 7 are given numerically for gaussian and for laplacian distribution of signal amplitudes

on min max r gatherings we consider a min max version of the previously studied r gathering problem with unit demands the problem we consider is a metric facility location problem in which each open facility must serve at least r customers and the maximum of all the facility and connection costs should be minimized rather than their sum this problem is motivated by scenarios in which r customers are required for a facility to be worth opening and the costs represent the time until the facility connection will be available i e we want to have the complete solution ready as soon as possible we present a 3 approximation algorithm for this problem and prove that it cannot be approximated better assuming p np next we consider this problem with the additional natural requirement that each customer will be assigned to a nearest open facility and present a 9 approximation algorithm we further consider previously introduced special cases and variants and obtain improved algorithmic and hardness results

approximating k median via pseudo approximation we present a novel approximation algorithm for k median that achieves an approximation guarantee of 1 sqrt 3 epsilon improving upon the decade old ratio of 3 epsilon our improved approximation ratio is achieved by exploiting the power of pseudo approximation more specifically our approach is based on two components each of which we believe is of independent interest first we show that in order to give an alpha approximation algorithm for k median it is sufficient to give a pseudo approximation algorithm that finds an alpha approximate solution by opening k o 1 facilities this is a rather surprising result as there exist instances for which opening k 1 facilities may lead to a significantly smaller cost than that of opening only k facilities second we give such a pseudo approximation algorithm with alpha 1 sqrt 3 epsilon prior to our work it was not even known whether opening k o k facilities would help improve the approximation ratio

supermodularity and affine policies in dynamic robust optimization this paper considers a particular class of dynamic robust optimization problems where a large number of decisions must be made in the first stage which consequently fix the constraints and cost structure underlying a one dimensional linear dynamical system we seek to bridge two classical paradigms for solving such problems namely 1 dynamic programming dp and 2 policies parameterized in model uncertainties also known as decision rules obtained by solving tractable convex optimization problems we show that if the uncertainty sets are integer sublattices of the unit hypercube the dp value functions are convex and supermodular in the uncertain parameters and a certain technical condition is satisfied then decision rules that are affine in the uncertain parameters are optimal we also derive conditions under which such rules can be obtained by optimizing simple i e linear objective functions over the uncertainty sets our results suggest new modeling paradigms for dynamic robust optimiz

a 3 approximation algorithm for the facility location problem with uniform capacities we consider the facility location problem where each facility can serve at most u clients we analyze a local search algorithm for this problem which uses only the operations of add delete and swap and prove that any locally optimum solution is no more than 3 times the global optimum this improves on a result of chudak and williamson who proved an approximation ratio of 3 2 sqrt 2 for the same algorithm we also provide an example which shows that any local search algorithm which uses only these three operations cannot achieve an approximation guarantee better than 3

scenario tree generation for multiperiod financial optimization by optimal discretization multiperiod financial optimization is usually based on a stochastic model for the possible market situations there is a rich literature about modeling and estimation of continuous state financial processes but little attention has been paid how to approximate such a process by a discrete state scenario process and how to measure the pertaining approximation error in this paper we show how a scenario tree may be constructed in an optimal manner on the basis of a simulation model of the underlying financial process by using a stochastic approximation technique consistency relations for the tree may also be taken into account

tractable stochastic analysis in high dimensions via robust optimization modern probability theory whose foundation is based on the axioms set forth by kolmogorov is currently the major tool for performance analysis in stochastic systems while it offers insights in understanding such systems probability theory in contrast to optimization has not been developed with computational tractability as an objective when the dimension increases correspondingly some of its major areas of application remain unsolved when the underlying systems become multidimensional queueing networks auction design in multi item multi bidder auctions network information theory pricing multi dimensional options among others we propose a new approach to analyze stochastic systems based on robust optimization the key idea is to replace the kolmogorov axioms and the concept of random variables as primitives of probability theory with uncertainty sets that are derived from some of the asymptotic implications of probability theory like the central limit theorem in addition we observe that several desired system properties such as incentive compatibility and individual rationality in auction design are naturally expressed in the language of robust optimization in this way the performance analysis questions become highly structured optimization problems linear semidefinite mixed integer for which there exist efficient practical algorithms that are capable of solving problems in high dimensions we demonstrate that the proposed approach achieves computationally tractable methods for a analyzing queueing networks b designing multi item multi bidder auctions with budget constraints and c pricing multi dimensional options

stochastic programming with integer variables including integer variables into traditional stochastic linear programs has considerable implications for structural analysis and algorithm design starting from mean risk approaches with different risk measures we identify corresponding two and multi stage stochastic integer programs that are large scale block structured mixed integer linear programs if the underlying probability distributions are discrete we highlight the role of mixed integer value functions for structure and stability of stochastic integer programs when applied to the block structures in stochastic integer programming well known algorithmic principles such as branch and bound lagrangian relaxation or cutting plane methods open up new directions of research we review existing results in the field and indicate departure points for their extension

one dimensional k center on uncertain data problems on uncertain data have attracted significant attention due to the imprecise nature of many measurement data in this paper we consider the k center problem on one dimensional uncertain data the input is a set p of weighted uncertain points on a real line and each uncertain point is specified by its probability density function pdf which is a piecewise uniform function i e a histogram the goal is to find a set q of k points on the line to minimize the maximum expected distance from the uncertain points of p to their expected closest points in q we present efficient algorithms for this uncertain k center problem and their running times almost match those for the deterministic k center problem

piecewise static policies for two stage adjustable robust linear optimization in this paper we consider two stage adjustable robust linear optimization problems under uncertain constraints and study the performance of piecewise static policies these are a generalization of static policies where we divide the uncertainty set into several pieces and specify a static solution for each piece we show that in the worst case there is no piecewise static policy with a polynomial number of pieces that has a significantly better performance than an optimal static policy this is quite surprising as piecewise static policies are significantly more general than static policies the proof is based on a combinatorial argument and the structure of piecewise static policies

recovery guarantees for exemplar based clustering for a certain class of distributions we prove that the linear programming relaxation of k medoids clustering a variant of k means clustering where means are replaced by exemplars from within the dataset distinguishes points drawn from nonoverlapping balls with high probability once the number of points drawn and the separation distance between any two balls are sufficiently large our results hold in the nontrivial regime where the separation distance is small enough that points drawn from different balls may be closer to each other than points drawn from the same ball in this case clustering by thresholding pairwise distances between points can fail we also exhibit numerical evidence of high probability recovery in a substantially more permissive regime

design of near optimal decision rules in multistage adaptive mixed integer optimization in recent years decision rules have been established as the preferred solution method for addressing computationally demanding multistage adaptive optimization problems despite their success existing decision rules a are typically constrained by their a priori design and b do not incorporate in their modeling adaptive binary decisions to address these problems we first derive the structure for optimal decision rules involving continuous and binary variables as piecewise linear and piecewise constant functions respectively we then propose a methodology for the optimal design of such decision rules that have a finite number of pieces and solve the problem robustly using mixed integer optimization we demonstrate the effectiveness of the proposed methods in the context of two multistage inventory control problems we provide global lower bounds and show that our approach is i practically tractable and ii provides high quality solutions that outperform alternative methods

data stability in clustering a closer look we consider the model introduced by bilu and linial 2010 13 who study problems for which the optimal clustering does not change when distances are perturbed they show that even when a problem is np hard it is sometimes possible to obtain efficient algorithms for instances resilient to certain multiplicative perturbations e g on the order of o n for max cut clustering awasthi et al 2012 6 consider center based objectives and balcan and liang 2012 9 analyze the k median and min sum objectives giving efficient algorithms for instances resilient to certain constant multiplicative perturbations here we are motivated by the question of to what extent these assumptions can be relaxed while allowing for efficient algorithms we show there is little room to improve these results by giving np hardness lower bounds for both the k median and min sum objectives on the other hand we show that constant multiplicative resilience parameters can be so strong as to make the clustering problem trivial leaving only a narrow range of resilience parameters for which clustering is interesting we also consider a model of additive perturbations and give a correspondence between additive and multiplicative notions of stability our results provide a close examination of the consequences of assuming stability in data

implicit multiple grid solution of the compressible navier stokes equations using k epsilon turbulence closure in the present work an algorithm for the numerical solution of the two dimensional mass weighted ensemble averaged compressible navier stokes equations is presented turbulence closure for the reynolds stresses is obtained using a low reynolds n umber two equation k s model which includes near wall effects the same algorithm is used for the integration of both the navier stokes and the turbulence transport equations the algorithm consists of a basic explicit finite volume scheme whose convergence is accelerated using local time step ping and multiple grid techniques convergence is further enhanced by the use of an implicit residual smoothing technique applied both in the fine and the coarser grids the application of the numerical scheme to the solution of the k s equations is examined in detail comparisons of computations with experimental data are presented for three shock wave turbulent boundary layer interaction flows algorithm convergence rate and cpu time require ments are discussed he numerical solution of the navier stokes equations for the simulation of turbulent flowfields is a subject of intense research and code development work 1 interest is focused both on code applicability to variegated config urations and on computational rapidity although direct simulation of turbulent flows2 or large eddy numerical sim ulations using subgrid turbulence modeling 3 4 have appeared in the literature current status of algorithm development and computer power indicate that for practical applications the ensemble averaged navier stokes equations should be used in conjunction with transport equations for turbulence closure 5 7 algebraic or mixing length models may provide satisfactory results for relatively simple flow geometries8 but require the determination of the shear layer thickness and of the distance of each grid point from the solid boundary thus they lack both geometric invariance and locality which are highly desirable properties from a computational viewpoint when addressing complex flowfields with many interacting solid boundaries although research on reynolds stre ss transport closure appears quite promising 9 10 its extension to three dimensions involves considerable increase of the number of transport equations to be solved also further research con cerning wall effects is required the more so since the use of wall functions is considered inadequate 11 on the other hand the use of two equation closures such as a 12 14 is quite widespread notwithstanding the fact that the use of the boussinesq hypothesis15 introduces a question able simplification of the flow physics 16 the major drawback of the k s model is probably the near wall formulation which fails to reproduce correctly the effects of the solid boundary on turbulence several variants of the original jones launder model12 have been developed in an attempt to improve upon near wall modeling some ten such models have been thor oughly examined in a critical review article by patel et al 17 who concluded that the models of launder and sharma 14

better bounds for incremental medians in the incremental version of the well known k medianproblem the objective is to compute an incremental sequence of facility sets f email protected f email protected email protected f n where each f k contains at most k facilities we say that this incremental medians sequence is r competitive if the cost of each f k is at most r times the optimum cost of k facilities the smallest such r is called the competitive ratio of the sequence f k mettu and plaxton ramgopal r mettu c greg plaxton the online median problem in proc 41st symposium on foundations of computer science focs ieee 2000 pp 339 348 ramgopal r mettu c greg plaxton the online median problem siam journal on computing 32 3 2003 816 832 presented a polynomial time algorithm that computes an incremental sequence with competitive ratio 30 they also showed a lower bound of 2 the upper bound on the ratio was improved to 8 in guolong lin chandrashekha nagarajan rajmohan rajamaran david p williamson a general approach for incremental approximation and hierarchical clustering in proc 17th symposium on discrete algorithms soda 2006 pp 1147 1156 and marek chrobak claire kenyon john noga neal young online medians via online bidding in proc 7th latin american theoretical informatics symposium latin in lecture notes in computer science vol 3887 2006 pp 311 322 we improve both bounds in this paper we first show that no incremental sequence can have competitive ratio better than 2 01 and we give a probabilistic construction of a sequence whose competitive ratio is at most 2 42 7 656 we also propose a new approach to the problem that for instances that we refer to as equable achieves an optimal ratio of 2

quantitative stability analysis for minimax distributionally robust risk optimization this paper considers distributionally robust formulations of a two stage stochastic programming problem with the objective of minimizing a distortion risk of the minimal cost incurred at the second stage we carry out a stability analysis by looking into variations of the ambiguity set under the wasserstein metric decision spaces at both stages and the support set of the random variables in the case when the risk measure is risk neutral the stability result is presented with the variation of the ambiguity set being measured by generic metrics of zeta structure which provides a unified framework for quantitative stability analysis under various metrics including total variation metric and kantorovich metric when the ambiguity set is structured by a zeta ball we find that the hausdorff distance between two zeta balls is bounded by the distance of their centers and difference of their radii the findings allow us to strengthen some recent convergence results on distributionally robust optimization where the center of the wasserstein ball is constructed by the empirical probability distribution

scenario reduction in stochastic programming an approach using probability metrics given a convex stochastic programming problem with a discrete initial probability distribution the problem of optimal scenario reduction is stated as follows determine a scenario subset of prescribed cardinality and a probability measure based on this set that is the closest to the initial distribution in terms of a natural or canonical probability metric arguments from stability analysis indicate that fortet mourier type probability metrics may serve as such canonical metrics efficient algorithms are developed that determine optimal reduced measures approximately numerical experience is reported for reductions of electrical load scenario trees for power management under uncertainty for instance it turns out that after 50 reduction of the scenario tree the optimal reduced tree still has about 90 relative accuracy

decomposition with branch and cut approaches for two stage stochastic mixed integer programming decomposition has proved to be one of the more effective tools for the solution of large scale problems especially those arising in stochastic programming a decomposition method with wide applicability is benders decomposition which has been applied to both stochastic programming as well as integer programming problems however this method of decomposition relies on convexity of the value function of linear programming subproblems this paper is devoted to a class of problems in which the second stage subproblem s may impose integer restrictions on some variables the value function of such integer subproblem s is not convex and new approaches must be designed in this paper we discuss alternative decomposition methods in which the second stage integer subproblems are solved using branch and cut methods one of the main advantages of our decomposition scheme is that stochastic mixed integer programming smip problems can be solved by dividing a large problem into smaller mip subproblems that can be solved in parallel this paper lays the foundation for such decomposition methods for two stage stochastic mixed integer programs

a nearly linear time approximation scheme for the euclidean k median problem this paper provides a randomized approximation scheme for the k median problem when the input points lie in the d dimensional euclidean space the worst case running time is o 2 o log 1 epsilon varepsilon d 1 n log d 6 n which is nearly linear for any fixed varepsilon and d moreover our method provides the first polynomial time approximation scheme for and uncapacitated facility location instances in d dimensional euclidean space for any fixed d 2 our work extends techniques introduced originally by arora for the euclidean traveling salesman problem tsp to obtain the improvement we develop a structure theorem to describe hierarchical decomposition of solutions the theorem is based on an adaptive decomposition scheme which guesses at every level of the hierarchy the structure of the optimal solution and accordingly modifies the parameters of the decomposition we believe that our methodology is of independent interest and may find applications to further geometric problems

incremental algorithms for facility location and k median in the incremental versions of facility location and k median the demand points arrive one at a time and the algorithm maintains a good solution by either adding each new demand to an existing cluster or placing it in a new singleton cluster the algorithm can also merge some of the existing clusters at any point in time for facility location we consider the case of uniform facility costs where the cost of opening a facility is the same for all points and present the first incremental algorithm which achieves a constant performance ratio using this algorithm as a building block we obtain the first incremental algorithm for k median which achieves a constant performance ratio using o k medians the algorithm is based on a novel merge rule which ensures that the algorithm s configuration monotonically converges to the optimal facility locations according to a certain notion of distance using this property we reduce the general case to the special case when the optimal solution consists of a single facility

l shaped decomposition of two stage stochastic programs with integer recourse we consider two stage stochastic programming problems with integer recourse the l shaped method of stochastic linear programming is generalized to these problems by using generalized benders decomposition nonlinear feasibility and optimality cuts are determined via general duality theory and can be generated when the second stage problem is solved by standard techniques finite convergence of the method is established when gomory s fractional cutting plane algorithm or a branch and bound algorithm is applied

prediction of curved channel flow with an extended k epsilon model of turbulence a l aide d une approximation algebrique pour les equations de contraintes de reynolds on calcule une expression plus generale pour le coefficient c de l equation de viscosite turbulente qui rend compte simultanement des effets de courbure de ligne de courant et de la deformation due a la pression dans l ecoulement le dernier incluant les effets d amortissement a la paroi

stability of varepsilon approximate solutions to convex stochastic programs an analysis of convex stochastic programs is provided when the underlying probability distribution is subjected to small perturbations it is shown in particular that varepsilon approximate solution sets of convex stochastic programs behave lipschitz continuously with respect to certain distances of probability distributions that are generated by the relevant integrands it is shown that these results apply to linear two stage stochastic programs with random recourse we discuss the consequences on associating fortet mourier metrics to two stage models and on the asymptotic behavior of empirical estimates of such models respectively

a probability metrics approach for reducing the bias of optimality gap estimators in two stage stochastic linear programming monte carlo sampling based estimators of optimality gaps for stochastic programs are known to be biased when bias is a prominent factor estimates of optimality gaps tend to be large on average even for high quality solutions this diminishes our ability to recognize high quality solutions in this paper we present a method for reducing the bias of the optimality gap estimators for two stage stochastic linear programs with recourse via a probability metrics approach motivated by stability results in stochastic programming we apply this method to the averaged two replication procedure a2rp by partitioning the observations in an effort to reduce bias which can be done in polynomial time in sample size we call the resulting procedure the averaged two replication procedure with bias reduction a2rp b we provide conditions under which a2rp b produces strongly consistent point estimators and an asymptotically valid confidence interval we illustrate the effectiveness of our approach analytically on a newsvendor problem and test the small sample behavior of a2rp b on a number of two stage stochastic linear programs from the literature our computational results indicate that the procedure effectively reduces bias we also observe variance reduction in certain circumstances

limitations of the near wall k epsilon turbulence model comparisons are given at two different reynolds numbers between the measured turbulent kinetic energy in channel flow and the predictions of several near wall variants of the k e closure included among these are new calculations based on a form of the e equation that is consistent with the physically correct boundary condition it is observed that all of the approaches fail to account for the large peak k value in the wall region that is evident in the experimental data by contrasting the computed energy budget with its measured values it is shown that this defect may be attributed to a fundamental inconsistency in the commonly used model for the pressure diffu sion term in the k equation near the boundary he k e closure is widely utilized today in the prediction of the mean properties of turbulent flowfields in recent years extensions of its basic form have been developed to permit treatment of complex fluid motions such as those found in curved pipes 1 compressible boundary layers 2 com bustion chambers 3 etc a strong incentive for the continued development and application of the k e model has been the remarkable success it has achieved in the prediction of mean velocity fields in wall bounded shear flows such as occur in channels pipes and boundary layers 4 6 the success of the k e closure in these relatively simple flows is not complete however since the high accuracy obtained for mean velocity is not duplicated in the case of the turbulent kinetic energy k a notable defect4 7 9 is the 25 or more underprediction of the large peak k value shown by clark 10 and kreplin and eckelmann11 to occur at y 15 in a channel flow for ap plications where predictions of turbulence levels are impor tant such as in internal combustion engine simulations 12 this failure can be of great practical significance it may be expected that major errors in the prediction of k stem from deficiencies in the low turbulent reynolds number forms of the k e equations which have been developed for the region adjacent to boundaries jones and launder4 for mulated the first such model that permitted k and e to be calculated down to a solid wall without the use of wall func tions several more recent studies including those of lam and bremhorst 7 hassid and poreh 8 and chien 9 have at tempted to improve upon the jones and launder approach but these authors report only modest gains in the accuracy of k the object of the present study is to elucidate the cause of this discrepancy by making direct comparisons between the terms in the exact and modeled energy equations to ac complish this a set of new calculations was performed in which e was required to satisfy the physically correct bound ary condition at the wall it was revealed that the failure of the k e model to predict the correct peak k value is most likely due to reliance on a flawed model for the pressure velocity correlation in the k equation this relation will be shown to be inconsistent with the experimentally measured values of the exact term as found by kreplin and eckelmann 11 and may be directly implicated in suppressing the peak k value removal of this limitation should allow for better predictions of k near solid boundaries

exponential map algorithm for stress updates in anisotropic multiplicative elastoplasticity for single crystals this paper presents a new stress update algorithm for large strain rate independent single crystal plasticity the theoretical frame is the well established continuum slip theory based on the multiplicative decomposition of the deformation gradient into elastic and plastic parts a distinct feature of the present formulation is the introduction and computational exploitation of a particularly simple hyperelastic stress response function based on a further multiplicative decomposition of the elastic deformation gradient into spherical and unimodular parts resulting in a very convenient representation of the schmid resolved shear stresses on the crystallographic slip systems in terms of a simple inner product of eulerian vectors the key contribution of this paper is an algorithmic formulation of the exponential map exp sl 3 sl 3 for updating the special linear group sl 3 of unimodular plastic deformation maps this update preserves exactly the plastic incompressibility condition of the anisotropic plasticity model under consideration the resulting fully implicit stress update algorithm treats the possibly redundant constraints of single crystal plasticity by means of an active set search it exploits intrinsically the simple representation of the schmid stresses by formulating the return algorithm and the associated consistent elastoplastic moduli in terms of eulerian vectors updates the performance of the proposed algorithm is demonstrated by means of a representative numerical example

clustering under approximation stability a common approach to clustering data is to view data objects as points in a metric space and then to optimize a natural distance based objective such as the k median k means or min sum score for applications such as clustering proteins by function or clustering images by subject the implicit hope in taking this approach is that the optimal solution for the chosen objective will closely match the desired target clustering e g a correct clustering of proteins by function or of images by who is in them however most distance based objectives including those mentioned here are np hard to optimize so this assumption by itself is not sufficient assuming p np to achieve clusterings of low error via polynomial time algorithms in this article we show that we can bypass this barrier if we slightly extend this assumption to ask that for some small constant c not only the optimal solution but also all c approximations to the optimal solution differ from the target on at most some e fraction of points we call this c e approximation stability we show that under this condition it is possible to efficiently obtain low error clusterings even if the property holds only for values c for which the objective is known to be np hard to approximate specifically for any constant c 1 c e approximation stability of k median or k means objectives can be used to efficiently produce a clustering of error o e with respect to the target clustering as can stability of the min sum objective if the target clusters are sufficiently large thus we can perform nearly as well in terms of agreement with the target clustering as if we could approximate these objectives to this np hard value

second order cone programming second order cone programming socp problems are convex optimization problems in which a linear function is minimized over the intersection of an affine linear manifold with the cartesian product of second order lorentz cones linear programs convex quadratic programs and quadratically constrained convex quadratic programs can all be formulated as socp problems as can many other problems that do not fall into these three categories these latter problems model applications from a broad range of fields from engineering control and finance to robust optimization and combinatorial optimization on the other hand semidefinite programming sdp that is the optimization problem over the intersection of an affine set and the cone of positive semidefinite matrices includes socp as a special case therefore socp falls between linear lp and quadratic qp programming and sdp like lp qp and sdp problems socp problems can be solved in polynomial time by interior point methods the computational effort per iteration required by these methods to solve socp problems is greater than that required to solve lp and qp problems but less than that required to solve sdp s of similar size and structure because the set of feasible solutions for an socp problem is not polyhedral as it is for lp and qp problems it is not readily apparent how to develop a simplex or simplex like method for socp while socp problems can be solved as sdp problems doing so is not advisable both on numerical grounds and computational complexity concerns for instance many of the problems presented in the survey paper of vandenberghe and boyd vb96 as examples of sdps can in fact be formulated as socps and should be solved as such in 2 3 below we give socp formulations for four of these examples the convex quadratically constrained quadratic programming qcqp problem problems involving fractional quadratic functions rutcor rutgers university e mail alizadeh rutcor rutgers edu research supported in part by the u s national science foundation grant ccr 9901991 ieor columbia university e mail gold ieor columbia edu research supported in part by the department of energy grant de fg02 92er25126 national science foundation grants dms 94 14438 cda 97 26385 and dms 01 04282

uncertain linear programs extended affinely adjustable robust counterparts in this paper we introduce the extended affinely adjustable robust counterpart to modeling and solving multistage uncertain linear programs with fixed recourse our approach first reparameterizes the primitive uncertainties and then applies the affinely adjustable robust counterpart proposed in the literature in which recourse decisions are restricted to be linear in terms of the primitive uncertainties we propose a special case of the extended affinely adjustable robust counterpart the splitting based extended affinely adjustable robust counterpart and illustrate both theoretically and computationally that the potential of the affinely adjustable robust counterpart method is well beyond the one presented in the literature similar to the affinely adjustable robust counterpart our approach ends up with deterministic optimization formulations that are tractable and scalable to multistage problems

a new partitioned staggered scheme for flexible multibody interactions with strong inertial effects abstract in generic flexible multibody interaction problems the differences in the mass damping and stiffness of the bodies can be very large of particular interest is a small change in the higher mass structure which triggers relatively large inertial effects on the lower mass structure in this work a new partitioned staggered time integration scheme is developed and its applicability is extended to the problems involving low mass ratios in coupled multibody problems the equation of motion of body with larger inertia is integrated by implicit newmark method whereas the smaller inertia is solved using a robust self starting explicit integration procedure the coupled formulation includes explicit correction terms that adjusts the amount of interfacial velocity which plays a key role in the stability and accuracy of the simulations a wide range of mass 10 6 10 1 damping and stiffness 10 3 10 3 ratios are chosen to assess the stability and accuracy characteristics of the proposed scheme the closed form expressions for the coupling parameter have been constructed for both matching and non matching time stepping through the godunov ryabenkii normal mode analysis the stability of the proposed method is observed as a function of the relative properties and temporal discretisation of the coupled system the time step of the heavier body significantly influences the stability more than the lighter body to maintain the staggering error minimal an optimal range of the coupling parameter is identified the error computed with uniform variation in the time step revealed that the present method is more accurate than the existing methods the partitioned method is an energy preserving integration method for the dynamic analysis of multibody systems as a potential application of this scheme flexible multibody problems in the field of offshore engineering viz wave energy converter and floater mooring systems are presented and validated with experimental measurements

on generalizations of network design problems with degree bounds iterative rounding and relaxation have arguably become the method of choice in dealing with unconstrained and constrained network design problems in this paper we extend the scope of the iterative relaxation method in two directions 1 by handling more complex degree constraints in the minimum spanning tree problem namely laminar crossing spanning tree and 2 by incorporating degree bounds in other combinatorial optimization problems such as matroid intersection and lattice polyhedra we give new or improved approximation algorithms hardness results and integrality gaps for these problems our main result is a 1 b o log n approximation algorithm for the minimum crossing spanning tree mcst problem with laminar degree constraints the laminar mcst problem is a natural generalization of the well studied bounded degree mst and is a special case of general crossing spanning tree we give an additive log c m hardness of approximation for general mcst even in the absence of costs c 0 is a fixed constant and m is the number of degree constraints this also leads to a multiplicative log c m hardness of approximation for the robust k median problem anthony et al in math oper res 35 79 101 2010 improving over the previously known factor 2 hardness we then consider the crossing contra polymatroid intersection problem and obtain a 2 2b 1 approximation algorithm where is the maximum element frequency this models for example the degree bounded spanning set intersection in two matroids finally we introduce the crossing latticep olyhedron problem and obtain a 1 b 2 1 approximation algorithm under certain condition this result provides a unified framework and common generalization of various problems studied previously such as degree bounded matroids

solving stochastic programs with integer recourse by enumeration a framework using gro bner basis reductions in this paper we present a framework for solving stochastic programs with complete integer recourse and discretely distributed right hand side vector using grobner basis methods from computational algebra to solve the numerous second stage integer programs using structural properties of the expected integer recourse function we prove that under mild conditions an optimal solution is contained in a finite set furthermore we present a basic scheme to enumerate this set and suggest improvements to reduce the number of function evaluations needed

scenario reduction in stochastic programming given a convex stochastic programming problem with a discrete initial probability distribution the problem of optimal scenario reduction is stated as follows determine a scenario subset of prescribed cardinality and a probability measure based on this set that is the closest to the initial distribution in terms of a natural or canonical probability metric arguments from stability analysis indicate that fortet mourier type probability metrics may serve as such canonical metrics efficient algorithms are developed that determine optimal reduced measures approximately numerical experience is reported for reductions of electrical load scenario trees for power management under uncertainty for instance it turns out that after 50 reduction of the scenario tree the optimal reduced tree still has about 90 relative accuracy

an improved 1 1 evolutionary algorithm for k median clustering problem with performance guarantee abstract evolutionary algorithms eas have been widely studied in numerical experiments and successfully applied in practice however most existing eas are designed without a theoretical analysis for their performance guarantee in this paper we define a fitness function to guide the well known 1 1 evolutionary algorithm called 1 1 ea to optimize the np hard k median problem we prove that the 1 1 ea can obtain a performance guarantee of 5 for k median problem in polynomial expected runtime o m n 2 d m a x if all distances between data points and cluster centers are integers where m and n are respectively the cardinalities of the data set and cluster center set and d m a x denotes the largest distance between data points and cluster centers to tackle the general case that the distances between data points and cluster centers are real number we propose an improved 1 1 ea which employs a distance scaling scheme during the evolutionary process our rigorous theoretical analysis shows that the improved 1 1 ea obtains a performance guarantee of 5 e in expected runtime o e 1 m n 2 log m d m a x where e 0 is a constant this study reveals that an appropriate scaling scheme can help eas to obtain a constant performance guarantee in polynomial expected runtime and provides insights into design eas for some np hard problems in addition we conduct experiment to compare the efficiency of the improved 1 1 ea with three clustering algorithms on optimizing three uci datasets

a robust and efficient substepping scheme for the explicit numerical integration of a rate dependent crystal plasticity model the hood doors and other panels of a car can be made by aluminium sheets through complex forming operations design of forming processes requires trustable models and simulations forming properties are different in different directions there are two principle ways to describe this plastic anisotropy of metals the first one is the continuum theories for materials the flow properties in various directions are described by the yield function it is a parametric function calibrated by experimental tests unfortunately only a few mechanical tests can be directly used the second approach takes into account that metals consist of many small crystals the properties of each crystal are determined by crystal planes but crystals are rotated in many directions the crystal models either resolve all the details of thousands of crystals in a small volume element requiring huge computer capacity or simplified statistical models can be applied in both cases the models are much more demanding than continuum models in practice these models can be used to perform virtual experiments for calibration of continuum models in his phd work kai zhang kai zhang ntnu no implemented and tested both type of models more efficient numerical algorithms for computer simulations were developed and programmed for the detailed crystal plasticity approach three different cases were investigated experimentally in one of the cases it was found that even for the detailed computer implementation the predictions could not replace experiments for design purposes a combination of real and virtual experiments was recommended for calibration for the other two cases the virtual experiments could be trusted it was found that the best statistical models that are more than thousand times faster can replace detailed simulations of the crystal structure with high accuracy

scenario reduction revisited fundamental limits and guarantees the goal of scenario reduction is to approximate a given discrete distribution with another discrete distribution that has fewer atoms we distinguish continuous scenario reduction where the new atoms may be chosen freely and discrete scenario reduction where the new atoms must be chosen from among the existing ones using the wasserstein distance as measure of proximity between distributions we identify those n point distributions on the unit ball that are least susceptible to scenario reduction i e that have maximum wasserstein distance to their closest m point distributions for some prescribed m n we also provide sharp bounds on the added benefit of continuous over discrete scenario reduction finally to our best knowledge we propose the first polynomial time constant factor approximations for both discrete and continuous scenario reduction as well as the first exact exponential time algorithms for continuous scenario reduction

finite element implementation of non linear elastoplastic constitutive laws using local and global explicit algorithms with automatic error control implicit stress integration algorithms have been demonstrated to provide a robust formulation for finite element analyses in computational mechanics but are difficult and impractical to apply to increasingly complex non linear constitutive laws this paper discusses the performance of fully explicit local and global algorithms with automatic error control used to integrate general non linear constitutive laws into a non linear finite element computer code the local explicit stress integration procedure falls under the category of return mapping algorithm with standard operator split and does not require the determination of initial yield or the use of any form of stress adjustment to prevent drift from the yield surface the global equations are solved using an explicit load stepping with automatic error control algorithm in which the convergence criterion is used to compute automatically the coarse load increment size the proposed numerical procedure is illustrated here through the implementation of a set of elastoplastic constitutive relations including isotropic and kinematic hardening as well as small strain hysteretic non linearity a series of numerical simulations confirm the robustness accuracy and efficiency of the algorithms at the local and global level published in 2001 by john wiley sons ltd

on the numerical implementation of 3d rate dependent single crystal plasticity formulations in this paper several important numerical issues are addressed for three dimensional 3d rate dependent single crystal plasticity after a thorough comparison of different constitutive algorithms we classify the integration methods into three approaches namely the implicit elastic plastic deformation gradient approach the implicit slip rate approach and the explicit slip rate approach as part of this algorithmic study we focus on five different schemes to enforce the plastic incompressibility four ways to update the texture and one convergence criteria the numerical performance of these different methods is illustrated the contribution of this study is three fold a stable scheme for the incompressibility enforcement an improved implicit algorithm and a fully explicit algorithm

bfc a branch and fix coordination algorithmic framework for solving some types of stochastic pure and mixed 0 1 programs we present a framework for solving some types of 0 1 multi stage scheduling planning problems under uncertainty in the objective function coe cients and the right hand side a scenario analysis scheme with full recourse is used the solution o ered for each scenario group at each stage takes into account all scenarios but without subordinating to any of them the constraints are modelled by a splitting variables representation via scenarios so a 0 1 model for each scenario is consid ered plus the non anticipativity constraints that equate the 0 1 variables from the same group of scenarios in each stage the mathematical representation of the model is very amenable for the proposed framework to deal with the 0 1 character of the variables a branch and fix coordi nation approach is introduced for coordinating the selection of the branching nodes and branching variables in the scenario subproblems to be jointly optimized some computational experience is reported for di erent types of problems

solving stochastic programs with integer recourse by enumeration a framework using grobner basis in this paper we present a framework for solving stochastic programs with complete integer recourse and discretely distributed right hand side vector using grobner basis methods from computational algebra to solve the numerous second stage integer programs using structural properties of the expected integer recourse function we prove that under mild conditions an optimal solution is contained in a finite set furthermore we present a basic scheme to enumerate this set and suggest improvements to reduce the number of function evaluations needed

a geometric characterization of the power of finite adaptability in multistage stochastic and adaptive optimization in this paper we show a significant role that geometric properties of uncertainty sets such as symmetry play in determining the power of robust and finitely adaptable solutions in multistage stochastic and adaptive optimization problems we consider a fairly general class of multistage mixed integer stochastic and adaptive optimization problems and propose a good approximate solution policy with performance guarantees that depend on the geometric properties of the uncertainty sets in particular we show that a class of finitely adaptable solutions is a good approximation for both the multistage stochastic and the adaptive optimization problem a finitely adaptable solution generalizes the notion of a static robust solution and specifies a small set of solutions for each stage the solution policy implements the best solution from the given set depending on the realization of the uncertain parameters in past stages therefore it is a tractable approximation to a fully adaptable solution for the multistage problems to the best of our knowledge these are the first approximation results for the multistage problem in such generality moreover the results and the proof techniques are quite general and also extend to include important constraints such as integrality and linear conic constraints

an approximation algorithm for the k median problem with uniform penalties via pseudo solution abstract we present a 1 3 approximation algorithm for the k median problem with uniform penalties extending the recent result by li and svensson for the classical k median problem without penalties one important difference of this work from that of li and svensson is a new definition of sparse instance to exploit the combinatorial structure of our problem

on optimal allocation of indivisibles under uncertainty the optimal allocation of indivisible resources is formalized as a stochastic optimization problem involving discrete decision variables a general stochastic search procedure is proposed which develops the concept of the branch and bound method the main idea is to process large collections of possible solutions and to devote more attention to the most promising groups by gathering more information to reduce the uncertainty and by narrowing the search area the optimal solution can be found with probability one special techniques for calculating stochastic lower and upper bounds are discussed the results are illustrated by a computational experiment

polyhedral properties of the k median problem on a tree the polyhedral structure of the k median problem on a tree is examined even for very small connected graphs we show that additional constraints are needed to describe the integer polytope a complete description is given of those trees for which an optimal integer lp solution is guaranteed to exist we present a new and simpler demonstration that an lp characterization of the 2 median problem is complete also we provide a simpler proof of the value of a tight worst case bound for the lp relaxation a new class of valid inequalities is identified these inequalities describe a subclass of facets for the k median problem on a general graph also we provide polyhedral descriptions for several types of trees as part of this work we summarize most known results for the k median problem on a tree

facility location with matroid or knapsack constraints in the classical k median problem we are given a metric space and want to open k centers so as to minimize the sum over all the vertices of the distance of each vertex to its nearest open center in this paper we present the first constant factor approximation algorithms for two natural generalizations of this problem that handle matroid or knapsack constraints in the matroid median problem there is an underlying matroid on the vertices and the set of open centers is constrained to be independent in this matroid when the matroid is uniform we recover the k median problem another previously studied special case is the red blue median problem where we have a partition matroid with two parts our algorithm for matroid median is based on rounding a natural linear programming relaxation in two stages and it relies on a connection to matroid intersection in the knapsack median problem centers have weights and the total weight of open centers is constrained to be at most a given capacity when all wei

improved algorithms for joint optimization of facility locations and network connections abstract this paper studies a k median steiner forest problem that jointly optimizes the opening of at most k facility locations and their connections to the client locations so that each client is connected by a path to an open facility with the total connection cost minimized the problem has wide applications in the telecommunication and transportation industries but is strongly np hard in the literature only a 2 approximation algorithm is known it being based on a lagrangian relaxation of the problem and using a sophisticated primal dual schema in this study we have developed an improved approximation algorithm using a simple transformation from an optimal solution of a minimum spanning tree problem compared with the existing 2 approximation algorithm our new algorithm not only achieves a better approximation ratio that is easier to be proved but also guarantees to produce solutions of equal or better quality up to 50 percent improvement in some cases in addition for two non trivial special cases where either every location contains a client or all the locations are in a tree shaped network we have developed for the first time in the literature new algorithms that can solve the problem to optimality in polynomial time

improved approximations for two stage min cut and shortest path problems under uncertainty in this paper we study the robust and stochastic versions of the two stage min cut and shortest path problems introduced in dhamdhere et al in how to pay come what may approximation algorithms for demand robust covering problems in focs pp 367 378 2005 and give approximation algorithms with improved approximation factors specifically we give a 2 approximation for the robust min cut problem and a 4 approximation for the stochastic version for the two stage shortest path problem we give a 3 39 3 39 approximation for the robust version and 6 78 6 78 approximation for the stochastic version our results significantly improve the previous best approximation factors for the problems in particular we provide the first constant factor approximation for the stochastic min cut problem our algorithms are based on a guess and prune strategy that crucially exploits the nature of the robust and stochastic objective in particular we guess the worst case second stage cost and based on the guess select a subset of costly scenarios for the first stage solution to address the second stage solution for any scenario is simply the min cut or shortest path problem in the residual graph the key contribution is to show that there is a near optimal first stage solution that completely satisfies the subset of costly scenarios that are selected by our procedure while the guess and prune strategy is not directly applicable for the stochastic versions we show that using a novel lp formulation we can adapt a guess and prune algorithm for the stochastic versions our algorithms based on the guess and prune strategy provide insights about the applicability of this approach for more general robust and stochastic versions of combinatorial problems

on the power of robust solutions in two stage stochastic and adaptive optimization problems we consider a two stage mixed integer stochastic optimization problem and show that a static robust solution is a good approximation to the fully adaptable two stage solution for the stochastic problem under fairly general assumptions on the uncertainty set and the probability distribution in particular we show that if the right hand side of the constraints is uncertain and belongs to a symmetric uncertainty set such as hypercube ellipsoid or norm ball and the probability measure is also symmetric then the cost of the optimal fixed solution to the corresponding robust problem is at most twice the optimal expected cost of the two stage stochastic problem furthermore we show that the bound is tight for symmetric uncertainty sets and can be arbitrarily large if the uncertainty set is not symmetric we refer to the ratio of the optimal cost of the robust problem and the optimal cost of the two stage stochastic problem as the stochasticity gap we also extend the bound on the stochasticity gap for another class of uncertainty sets referred to as positive if both the objective coefficients and right hand side are uncertain we show that the stochasticity gap can be arbitrarily large even if the uncertainty set and the probability measure are both symmetric however we prove that the adaptability gap ratio of optimal cost of the robust problem and the optimal cost of a two stage fully adaptable problem is at most four even if both the objective coefficients and the right hand side of the constraints are uncertain and belong to a symmetric uncertainty set the bound holds for the class of positive uncertainty sets as well moreover if the uncertainty set is a hypercube special case of a symmetric set the adaptability gap is one under an even more general model of uncertainty where the constraint coefficients are also uncertain

an approximation scheme for stochastic linear programming and its application to stochastic integer programs stochastic optimization problems attempt to model uncertainty in the data by assuming that the input is specified by a probability distribution we consider the well studied paradigm of 2 stage models with recourse first given only distributional information about some of the data one commits on initial actions and then once the actual data is realized according to the distribution further recourse actions can be taken we show that for a broad class of 2 stage linear models with recourse one can for any e 0 in time polynomial in 1 e and the size of the input compute a solution of value within a factor 1pe of the optimum in spite of the fact that exponentially many second stage scenarios may occur in conjunction with a suitable rounding scheme this yields the first approximation algorithms for 2 stage stochastic integer optimization problems where the underlying random data is given by a black box and no restrictions are placed on the costs in the two stages our rounding approach for stochastic integer programs shows that an approximation algorithm for a deterministic analogue yields with a small constant factor loss provably near optimal solutions for the stochastic generalization among the range of applications we consider are stochastic versions of the multicommodity flow set cover vertex cover and facility location problems

a plant location guide for the unsure approximation algorithms for min max location problems this paper studies an extension of the k median problem under uncertain demand we are given an n vertex metric space v d and m client sets documentclass aastex usepackage amsbsy usepackage amsfonts usepackage amssymb usepackage bm usepackage mathrsfs usepackage pifont usepackage stmaryrd usepackage textcomp usepackage portland xspace usepackage amsmath amsxtra pagestyle empty declaremathsizes 10 9 7 6 begin document s i subseteq v i 1 m end document the goal is to open a set of k facilities f such that the worst case connection cost over all the client sets is minimized i e documentclass aastex usepackage amsbsy usepackage amsfonts usepackage amssymb usepackage bm usepackage mathrsfs usepackage pifont usepackage stmaryrd usepackage textcomp usepackage portland xspace usepackage amsmath amsxtra pagestyle empty declaremathsizes 10 9 7 6 begin document min f subseteq v f k max i in m left sum j in s i d

primal and dual linear decision rules in stochastic and robust optimization linear stochastic programming provides a flexible toolbox for analyzing real life decision situations but it can become computationally cumbersome when recourse decisions are involved the latter are usually modeled as decision rules i e functions of the uncertain problem data it has recently been argued that stochastic programs can quite generally be made tractable by restricting the space of decision rules to those that exhibit a linear data dependence in this paper we propose an efficient method to estimate the approximation error introduced by this rather drastic means of complexity reduction we apply the linear decision rule restriction not only to the primal but also to a dual version of the stochastic program by employing techniques that are commonly used in modern robust optimization we show that both arising approximate problems are equivalent to tractable linear or semidefinite programs of moderate sizes the gap between their optimal values estimates the loss of optimality incurred by the linear decision rule approximation our method remains applicable if the stochastic program has random recourse and multiple decision stages it also extends to cases involving ambiguous probability distributions

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

local search heuristics for k median and facility location problems we analyze local search heuristics for the metric k median and facility location problems we define the locality gap of a local search procedure for a minimization problem as the maximum ratio of a locally optimum solution obtained using this procedure to the global optimum for k median we show that local search with swaps has a locality gap of 5 furthermore if we permit up to p facilities to be swapped simultaneously then the locality gap is 3 2 p this is the first analysis of a local search for k median that provides a bounded performance guarantee with only k medians this also improves the previous known 4 approximation for this problem for uncapacitated facility location we show that local search which permits adding dropping and swapping a facility has a locality gap of 3 this improves the bound of 5 given by m korupolu c plaxton and r rajaraman analysis of a local search heuristic for facility location problems technical report 98 30 dimacs 1998 we also consider a capacitated facility location problem where each facility has a capacity and we are allowed to open multiple copies of a facility for this problem we introduce a new local search operation which opens one or more copies of a facility and drops zero or more facilities we prove that this local search has a locality gap between 3 and 4

an approximation framework for two stage ambiguous stochastic integer programs under mean mad information abstract we consider two stage recourse models in which only limited information is available on the probability distributions of the random parameters in the model if all decision variables are continuous then we are able to derive the worst case and best case probability distributions under the assumption that only the means and mean absolute deviations of the random parameters are known contrary to most existing results in the literature these probability distributions are the same for every first stage decision the ambiguity set that we use in this paper also turns out to be particularly suitable for ambiguous recourse models involving integer decisions variables for such problems we develop a general approximation framework and derive error bounds for using these approximatons we apply this approximation framework to mixed ambiguous mixed integer recourse models in which some of the probability distributions of the random parameters are known and others are ambiguous to illustrate these results we carry out numerical experiments on a surgery block allocation problem

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