bounding the expectation of a saddle function with application to stochastic programming we previously obtained tight upper and lower bounds to the expectation of a saddle function of multivariate random variables using first and cross moments of the random variables without independence assumptions these bounds are applicable when domains of the random vectors are compact sets in the euclidean space in this paper we extend the results to the case of unbounded domains similar in spirit to the extensions by birge and wets in the pure convex case the relationship of these bounds to a certain generalized moment problem is also investigated finally for solving stochastic linear programs utilizing the above bounding procedures a computationally more appealing order cone decomposition scheme is proposed which behaves quadratically in the number of random variables moreover the resulting upper and lower approximations are amenable to efficient solution techniques

tight formulations for some simple mixed integer programs and convex objective integer programs we study the polyhedral structure of simple mixed integer sets that generalize the two variable set s z r 1 z 1 s z b these sets form basic building blocks that can be used to derive tight formulations for more complicated mixed integer programs for four such sets we give a complete description by valid inequalities and or an integral extended formulation and we also indicate what constraints can be added without destroying integrality we apply these results to provide tight formulations for certain piecewise linear convex objective integer programs and in a companion paper we exploit them to provide polyhedral descriptions and computationally effective mixed integer programming formulations for discrete lot sizing problems

laplacian growth on a random lattice abstract several properties of dla and dbm clusters grown with infinite noise reduction on a recently proposed vectorizable random lattice are described the purpose of this work is twofold to see how the cluster shapes are affected when these random lattices are continuously changed into regular lattices and on the other side to study the properties of these lattices themselves for dbm growth the shapes of the clusters are continuously changed into the ordered shapes that one finds on a regular lattice dla rules on the other hand present a sharp transition at zero disorder even on infinitesimally disordered lattices will dla models give rise to random looking clusters and only for zero disorder lattices one recovers the usual needle type structures the origin of this behavior is discussed for full disorder in the lattices we analyze the statistical properties of the mass distribution of four possible growth models finding that a mild anisotropy is still present in these lattices dla with counters on sites sdla is the model which most strongly reflects this anisotropy

laplacian growth with infinite noise reduction abstract four models of laplacian growth with infinite noise reduction on a square lattice are investigated these are the four possible combinations of dla and dbm with counters on bonds or sites which we call sdla bdla sdbm and bdbm all four give rise to compact patterns with nontrivial scaling exponents which we roughly estimate while bdla and bdbm are nearly identical and sdla is very similar to both sdbm produces a different pattern resembling the effect of a finite surface tension also the scaling exponent for this case could be different from the rest

multiresolution wavelet coarsening and analysis of transport in heterogeneous media abstract modeling transport in a disordered medium with broadly distributed and correlated heterogeneities has been hampered by the fact that a very large set of equations must be solved if the heterogeneities are distributed over several distinct length scales we recently proposed a novel method based on the application of wavelet transformations phys rev lett 79 1997 4385 that can be used for simulation of transport processes in such disordered media in this paper we extend the method so as to be able to study transport in two classes of highly heterogeneous media that are otherwise very difficult to simulate namely disordered anisotropic media and those that contain more than one distinct family of transport paths for example natural rock with both pores and fractures we report the results of extensive computer simulations of transport in such disordered media in which the heterogeneities are broadly distributed and contain long range correlations we show that even for the type of disordered media that we consider in this paper the wavelet based method drastically reduces by at least two orders of magnitude the number of equations to be solved without neglecting any important information about the media s morphology hence providing highly accurate estimates of their transport properties thus we conclude that the wavelet based method is applicable to any disordered medium with any type of heterogeneities

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

semi infinite programming theory methods and applications starting from a number of motivating and abundant applications in 2 including control of robots eigenvalue computations mechanical stress of materials and statistical design the authors describe a class of optimization problems which are referred to as semi infinite because their constraints bound functions of a finite number of variables on a whole region in 3 5 first and second order optimality conditions are derived for general nonlinear problems as well as a procedure for reducing the problem locally to one with only finitely many constraints another main effort for achieving simplification is through duality in 6 there algebraic properties of finite linear programming are brought to bear on duality theory in semi infinite programming section 7 treats numerical methods based on either discretization or local reduction with the emphasis on the design of superlinearly convergent sqp type methods taking this differentiable point of view this paper can be considered to be complementar

effective renormalization group algorithm for transport in oil reservoirs abstract a real space renormalization group algorithm is used to evaluate macroscopic permeabilities the algorithm was tested on many different distributions with large spatial correlations and anisotropies and the results are very close to the expected exact values

a functional approach to the stationary waiting time and idle period distributions of the gi g 1 queue the gi g 1 queueing model is regarded as a functional which maps the service and interarrival time distributions onto output quantities of interest such as the stationary waiting time distribution for the case where the input distributions have densities techniques from infinite dimensional analysis are used to obtain derivatives and taylor series expansions for the functionals these yield approximations to the output distributions which can be viewed as nonparametric alternatives to parametric approximations such as those provided by infinitesimal perturbation analysis or the phase method

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

on generalized semi infinite optimization and bilevel optimization the paper studies the connections and differences between bilevel problems bl and generalized semi infinite problems gsip under natural assumptions gsip can be seen as a special case of a bl we consider the so called reduction approach for bl and gsip leading to optimality conditions and newton type methods for solving the problems we show by a structural analysis that for gsip problems the regularity assumptions for the reduction approach can be expected to hold generically at a solution but for general bl problems not the genericity behavior of bl and gsip is in particular studied for linear problems

second order asymptotic behaviour of subordinated sequences with longtailed subordinator abstract suppose that a n is a discrete probability distribution on the set n 0 0 1 2 and p n is some non negative sequence defined on the same set the equation b n k 0 p k a k n defines a new sequence b n here a k n denotes the k fold convolution of the distribution a n in the paper the asymptotic behaviour of the sequence b n is investigated it is known that for the large classes of the sequences a n and p n b n p n where 1 is the mean of the distribution a n the main object of the present work is to estimate the difference b n p n for some classes of the sequences a n and p n

optimization of a continuous distillation process under random inflow rate abstract the paper deals with a continuous distillation process under stochastic rate of inflows collected in a feed tank the aim of analysis is to find a robust control of extracting feed from the tank over a certain time horizon such that without knowledge of future realizations of the inflow rate some level constraints in the feed tank will be met with high probability this approach relies on formulating and numerically treating probabilistic constraints the inflow rate is considered as a stochastic process for which two basically different model assumptions are made the first model assumes a gaussian process and thus reflects the superposition of many independent elementary inflows the second model treats maybe the simplest case of a single elementary inflow profile namely rectangular inflows with fixed rate and duration but stochastic starting time numerical results illustrating both assumptions are presented and advantages over the simple anticipation of nominal inflow profiles are highlighted

robust algorithm for random resistor networks using hierarchical domain structure in this study we discuss methods for solving random resistor networks and similar problems we discuss the node elimination method and we demonstrate its equivalence to the gaussian elimination scheme finding a good elimination order which makes the method highly efficient the transfer matrix method is shown to be a special case of the node elimination method with an ordering that is far from optimal we compare the performance of these exact methods with a state of the art conjugate gradient solver in general the node elimination method is the faster method

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

the price of robustness a robust approach to solving linear optimization problems with uncertain data was proposed in the early 1970s and has recently been extensively studied and extended under this approach we are willing to accept a suboptimal solution for the nominal values of the data in order to ensure that the solution remains feasible and near optimal when the data changes a concern with such an approach is that it might be too conservative in this paper we propose an approach that attempts to make this trade off more attractive that is we investigate ways to decrease what we call the price of robustness in particular we flexibly adjust the level of conservatism of the robust solutions in terms of probabilistic bounds of constraint violations an attractive aspect of our method is that the new robust formulation is also a linear optimization problem thus we naturally extend our methods to discrete optimization problems in a tractable way we report numerical results for a portfolio optimization problem a knapsack problem and a problem from the net lib library

an efficient algorithm for simulating fracture using large fuse networks the high computational cost involved in modelling of the progressive fracture simulations using large discrete lattice networks stems from the requirement to solve a new large set of linear equations every time a new lattice bond is broken to address this problem we propose an algorithm that combines the multiple rank sparse cholesky downdating algorithm with the rank p inverse updating algorithm based on the sherman morrison woodbury formula for the simulation of progressive fracture in disordered quasi brittle materials using discrete lattice networks using the present algorithm the computational complexity of solving the new set of linear equations after breaking a bond reduces to the same order as that of a simple backsolve forward elimination and backward substitution using the already lu factored matrix that is the computational cost is o nnz l where nnz l denotes the number of non zeros of the cholesky factorization l of the stiffness matrix a this algorithm using the direct sparse solver is faster than the fourier accelerated preconditioned conjugate gradient pcg iterative solvers and eliminates the critical slowing down associated with the iterative solvers that is especially severe close to the critical points numerical results using random resistor networks substantiate the efficiency of the present algorithm

a space quantization method for numerical integration abstract we propose a new method sqm for numerical integration of c functions 0 2 defined on a convex subset c of rd with respect to a continuous distribution it relies on a space quantization of c by a n tuplex x1 xn cn fd is approximated by a weighted sum of the f xi s the integration error bound depends on the distortion en x of the voronoi tessellation of x this notion comes from information theoretists its main properties existence of a minimizing n tuple in cn asymptotics of mincn en as n are presented for a wide class of measures a simple stochastic optimization procedure is proposed to compute in any dimension d x and the characteristics of its voronoi tessellation some new results on the competitive learning vector quantization algorithm when 2 are obtained as a by product some tests simulations and provisional remarks are proposed as a conclusion

on harmonic renewal measures let be a probability and nu h sum limits n 1 infty frac 1 n mu n the corresponding harmonic renewal measure complementing earlier results where is concentrated on a halfline we investigate the behaviour ofvh x x 1 and the harmonic renewal functiong x vh x asx ifm1 x dx 0 we also consider the casem1 0

uncertain convex programs randomized solutions and confidence levels many engineering problems can be cast as optimization problems subject to convex constraints that are parameterized by an uncertainty or instance parameter two main approaches are generally available to tackle constrained optimization problems in presence of uncertainty robust optimization and chance constrained optimization robust optimization is a deterministic paradigm where one seeks a solution which simultaneously satisfies all possible constraint instances in chance constrained optimization a probability distribution is instead assumed on the uncertain parameters and the constraints are enforced up to a pre specified level of probability unfortunately however both approaches lead to computationally intractable problem formulations

a branch reduce cut algorithm for the global optimization of probabilistically constrained linear programs we consider probabilistically constrained linear programs with general distributions for the uncertain parameters these problems involve non convex feasible sets we develop a branch and bound algorithm that searches for a global optimal solution to this problem by successively partitioning the non convex feasible region and by using bounds on the objective function to fathom inferior partition elements this basic algorithm is enhanced by domain reduction and cutting plane strategies to reduce the size of the partition elements and hence tighten bounds the proposed branch reduce cut algorithm exploits the monotonicity properties inherent in the problem and requires solving linear programming subproblems we provide convergence proofs for the algorithm some illustrative numerical results involving problems with discrete distributions are presented

on the rate of convergence of optimal solutions of monte carlo approximations of stochastic programs in this paper we discuss monte carlo simulation based approximations of a stochastic programming problem we show that if the corresponding random functions are convex piecewise linear and the distribution is discrete then an optimal solution of the approximating problem provides an exact optimal solution of the true problem with probability one for sufficiently large sample size moreover by using the theory of large deviations we show that the probability of such an event approaches one exponentially fast with increase of the sample size in particular this happens in the case of linear two or multi stage stochastic programming with recourse if the corresponding distributions are discrete the obtained results suggest that in such cases monte carlo simulation based methods could be very efficient we present some numerical examples to illustrate the ideas involved

mixing mixed integer inequalities mixed integer rounding mir inequalities play a central role in the development of strong cutting planes for mixed integer programs in this paper we investigate how known mir inequalities can be combined in order to generate new strong valid inequalities given a mixed integer region s and a collection of valid base mixed integer inequalities we develop a procedure for generating new valid inequalities for s the starting point of our procedure is to consider the mir inequalities related with the base inequalities for any subset of these mir inequalities we generate two new inequalities by combining or mixing them we show that the new inequalities are strong in the sense that they fully describe the convex hull of a special mixed integer region associated with the base inequalities we discuss how the mixing procedure can be used to obtain new classes of strong valid inequalities for various mixed integer programming problems in particular we present examples for production planning capacitated facility location capacitated network design and multiple knapsack problems we also present preliminary computational results using the mixing procedure to tighten the formulation of some difficult integer programs finally we study some extensions of this mixing procedure

an efficient algorithm for modelling progressive damage accumulation in disordered materials this paper presents an efficient algorithm for the simulation of progressive fracture in disordered quasi brittle materials using discrete lattice networks the main computational bottleneck involved in modelling the fracture simulations using large discrete lattice networks stems from the fact that a new large set of linear equations needs to be solved every time a lattice bond is broken using the present algorithm the computational complexity of solving the new set of linear equations after breaking a bond reduces to a simple triangular solves forward elimination and backward substitution using the already cholesky factored matrix this algorithm using the direct sparse solver is faster than the fourier accelerated iterative solvers such as the preconditioned conjugate gradient pcg solvers and eliminates the critical slowing down associated with the iterative solvers that is especially severe close to the percolation critical points numerical results using random resistor networks for modelling the fracture and damage evolution in disordered materials substantiate the efficiency of the present algorithm in particular the proposed algorithm is especially advantageous for fracture simulations wherein ensemble averaging of numerical results is necessary to obtain a realistic lattice system response

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

concavity and efficient points of discrete distributions in probabilistic programming we consider stochastic programming problems with probabilistic constraints involving integer valued random variables the concept of a p efficient point of a probability distribution is used to derive various equivalent problem formulations next we introduce the concept of r concave discrete probability distributions and analyse its relevance for problems under consideration these notions are used to derive lower and upper bounds for the optimal value of probabilistically constrained stochastic programming problems with discrete random variables the results are illustrated with numerical examples

backbone and elastic backbone of percolation clusters obtained by the new method of burning we introduce a method of obtaining information about percolation clusters in which the clusters are burned in different ways this method yields the backbone and the elastic backbone their number of loops shortest path and other quantities it works in any dimension and can be applied also to other clusters we obtain accurate estimates for the backbone exponents in two and three dimensions

probability densities for the displacement of random walks on percolation clusters the probability density of the displacement or end to end distance of a random walk on the incipient infinite percolation cluster in d 2 dimensions is studied by an exact enumeration method the numerical data suggest specific forms for the probability density both in the chemical distance variable l and the geometric distance r

robust convex optimization we study convex optimization problems for which the data is not specified exactly and it is only known to belong to a given uncertainty set u yet the constraints must hold for all possible values of the data from u the ensuing optimization problem is called robust optimization in this paper we lay the foundation of robust convex optimization in the main part of the paper we show that if u is an ellipsoidal uncertainty set then for some of the most important generic convex optimization problems linear programming quadratically constrained programming semidefinite programming and others the corresponding robust convex program is either exactly or approximately a tractable problem which lends itself to efficientalgorithms such as polynomial time interior point methods

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

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

crack cluster distributions in the random fuse model using large scale numerical simulations and extensive sampling we analyze the scaling properties of the crack cluster distribution and the largest crack cluster distribution at the peak load the simulations are performed using both two dimensional and three dimensional random fuse models the numerical results indicate that in contrast with the randomly diluted networks percolation disorder the crack cluster distribution in the random fuse model at the peak load follows neither a power law nor an exponential distribution the largest crack cluster distribution at the peak load follows a lognormal distribution and this is discussed in the context of whether there exists a relationship between the largest crack cluster size distribution at peak load and the fracture strength distribution contrary to popular belief we find that the fracture strength and the largest crack cluster size at the peak load are uncorrelated indeed quite often the final spanning crack is formed not due to the propagation of the largest crack at the peak load but instead due to coalescence of smaller cracks

on subordinated distributions and generalized renewal measures where an n eno is some sequence of nonnegative numbers sn neno is the sequence of partial sums s0 0 sn xflxk of another sequence xk ken of i i d random variables and a c r is a fixed borel set such as 0 1 or 0 oo examples of such convolution series are subordinated distributions f 0oan 1 which arise as distributions of random sums and harmonic and ordinary renewal measures a0 0 an 1 n for all n c n in the first an 1 for all n c no in the second case these examples are in turn essential for the analysis of the large time behaviour of diverse applied models such as branching and queueing processes they are also of interest in connection with representation theorems such as the levy representation of infinitely divisible distributions a traditional approach to such problems is via regular variation if the underlying random variables are nonnegative we can use laplace transforms and the related abelian and tauberian theorems see e g stam 1973 in the context of subordination and feller 1971 xiv 3 in connection with renewal theory embrechts maejima and omey 1984 is a recent treatment of generalized renewal measures along these lines the approach of the present paper is based on the wiener levy gel fand theorem and has occasionally been called the banach algebra method in gruibel 1983 we gave a new variant of this method for the special case of lattice distributions showing that by using the appropriate banach algebras of sequences arbitrarily fine expansions are possible under certain assumptions on the higher order differences of p x1 n fnen here we give a corresponding treatment of nonlattice distributions we restrict ourselves to an analogue of first order differences and obtain a number of theorems which perhaps are described best as next term results to explain this let us consider a special case in more detail

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

a strong renewal theorem for generalized renewal functions in the infinite mean case letf x be a nonarithmetic c d f on 0 such that 1 f x x l x wherel x is slowly varying and 0 1 leta x be regularly varying with exponent 1 a strong renewal theorem of blackwell type for generalized renewal functions of the form g t equiv sum limits n 0 infty a n f n t is proved here thus extending the recent work of embrechts maejima and omey 1 and that of erickson 4

renormalization group methods for flow in oil reservoirs abstract oil reservoirs are described as networks of bonds with random permeabilities which exhibit fractal features real space renormalization group methods give very accurate estimates of the reservoir s global effective permeability

the continuous mixing polyhedron we analyze the polyhedral structure of the setspcmix s r z r xr nxz n s r j z j f j j 1 n andp cmix pcmixn s 0 the setp cmix is a natural generalization of the mixing set studied by pochet and wolsey 15 16 and ga nla k and pochet 8 and recently has been introduced by miller and wolsey 12 we introduce a new class of valid inequalities that has proven to be sufficient for describing conv pcmix we give an extended formulation of sizeo n xo n2 variables and constraints and indicate how to separate over conv pcmix ino n3 time finally we show how the mixed integer rounding mir inequalities of nemhauser and wolsey 14 and the mixing inequalities of ga nla k and pochet 8 constitute special cases of the cycle inequalities

the mixed vertex packing problem we study a generalization of the vertex packing problem having both binary and bounded continuous variables called the mixed vertex packing problem mvpp the well known vertex packing model arises as a subproblem or relaxation of many 0 1 integer problems whereas the mixed vertex packing model arises as a natural counterpart of vertex packing in the context of mixed 0 1 integer programming we describe strong valid inequalities for the convex hull of solutions to the mvpp and separation algorithms for these inequalities we give a summary of computational results with a branch and cut algorithm for solving the mvpp and using it to solve general mixed integer problems

an efficient block circulant preconditioner for simulating fracture using large fuse networks critical slowing down associated with the iterative solvers close to the critical point often hinders large scale numerical simulation of fracture using discrete lattice networks this paper presents a block circulant preconditioner for iterative solvers for the simulation of progressive fracture in disordered quasi brittle materials using large discrete lattice networks the average computational cost of the present algorithm per iteration is o rslog s delops where the stiffness matrix a is partitioned into r r blocks such that each block is an s s matrix and delops represents the operational count associated with solving a block diagonal matrix with r r dense matrix blocks this algorithm using the block circulant preconditioner is faster than the fourier accelerated preconditioned conjugate gradient algorithm and alleviates the critical slowing down that is especially severe close to the critical point numerical results using random resistor networks substantiate the efficiency of the present algorithm

a polyhedral study on chance constrained program with random right hand side the essential structure of the mixed integer programming formulation for chance constrained program ccp is the intersection of multiple mixing sets with a 0 1 knapsack to improve our computational capacity on ccp an underlying substructure the single mixing set with a 0 1 knapsack has received substantial attentions recently in this study we consider a ccp problem with stochastic right hand side under a finite discrete distribution we first present a family of strong inequalities that subsumes known facet defining ones for that single mixing set due to the flexibility of our generalized inequalities we develop a new separation heuristic that has a complexity much less than existing one and guarantees generated cutting planes are facet defining for the polyhedron of ccp then we study lifting and superadditive lifting on knapsack cover inequalities and provide an implementable procedure on deriving another family of strong inequalities for the single mixing set finally different from the traditional approach that aggregates original constraints to investigate polyhedral implications due to their interactions we propose a novel blending procedure that produces strong valid inequalities for ccp by integrating those derived from individual mixing sets we show that under certain conditions they are the first type of facet defining inequalities describing intersection of multiple mixing sets and design an efficient separation heuristic for implementation in the computational experiments we perform a systematic study and illustrate the efficiency of the proposed inequalities on solving chance constrained static probabilistic lot sizing problems

an integer programming approach for linear programs with probabilistic constraints linear programs with joint probabilistic constraints pclp are difficult to solve because the feasible region is not convex we consider a special case of pclp in which only the right hand side is random and this random vector has a finite distribution we give a mixed integer programming formulation for this special case and study the relaxation corresponding to a single row of the probabilistic constraint we obtain two strengthened formulations as a byproduct of this analysis we obtain new results for the previously studied mixing set subject to an additional knapsack inequality we present computational results which indicate that by using our strengthened formulations instances that are considerably larger than have been considered before can be solved to optimality

crack roughness and avalanche precursors in the random fuse model we analyze the scaling of the crack roughness and of avalanche precursors in the two dimensional random fuse model by numerical simulations employing large system sizes and extensive sample averaging we find that the crack roughness exhibits anomalous scaling as recently observed in experiments the roughness exponents zeta zeta loc and the global width distributions are found to be universal with respect to the lattice geometry failure is preceded by avalanche precursors whose distribution follows a power law up to a cutoff size while the characteristic avalanche size scales as s 0 approximately l d with a universal fractal dimension d the distribution exponent tau differs slightly for triangular and diamond lattices and in both cases it is larger than the mean field fiber bundle value tau 5 2

harmonic renewal measures and bivariate domains of attraction in fluctuation theory each probability measure c on a first orthant is associated with a harmonic renewal measure g specifically we consider n s n the ladder time place of a random walk sn using bivariate g we show that when s1 is in a domain of attraction so is n sn this unifies and generalizes results of sinai rogosin

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

a sample approximation approach for optimization with probabilistic constraints we study approximations of optimization problems with probabilistic constraints in which the original distribution of the underlying random vector is replaced with an empirical distribution obtained from a random sample we show that such a sample approximation problem with a risk level larger than the required risk level will yield a lower bound to the true optimal value with probability approaching one exponentially fast this leads to an a priori estimate of the sample size required to have high confidence that the sample approximation will yield a lower bound we then provide conditions under which solving a sample approximation problem with a risk level smaller than the required risk level will yield feasible solutions to the original problem with high probability once again we obtain a priori estimates on the sample size required to obtain high confidence that the sample approximation problem will yield a feasible solution to the original problem finally we present numerical illustrations of how these results can be used to obtain feasible solutions and optimality bounds for optimization problems with probabilistic constraints

functions of discrete probability measures rates of convergence in the renewal theorem let and be finite measures on the set of integers such that the fourier transform of is an analytic function of the transform the central result shows how may be approximated by linear combinations of convolution powers of applications are given to renewal theory infinitely divisible measures and age dependent branching processes

fourier acceleration of iterative processes in disordered systems technical details are given on how to use fourier acceleration with iterative processes such as relaxation and conjugate gradient methods these methods are often used to solve large linear systems of equations but become hopelessly slow very rapidly as the size of the set of equations to be solved increases fourier acceleration is a method designed to alleviate these problems and result in a very fast algorithm the method is explained for the jacobi relaxation and conjugate gradient methods and is applied to two models the random resistor network and the random central force network in the first model acceleration works very well in the second little is gained we discuss reasons for this we also include a discussion of stopping criteria

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

optimization and plasticity in disordered media we study the plastic yielding of disordered media using the perfectly plastic random fuse model the yield surfaces are shown to be different from those obtained minimizing the sum of the local yield thresholds i e the so called minimum energy surfaces as a result the global yield stress is lower than expected from naive optimization and the difference persists as the sample size increases at variance with minimum energy surfaces height height fluctuations of yield surfaces exhibit multiscaling we provide a theoretical argument that explains how this behavior arises from the very different nature of the optimization problem in both cases

a new algorithm to extract the backbone in a random resistor network the authors present an efficient algorithm to determine the backbone the current carrying part of a lattice in the random resistor network problem it reduces mainly to the determination of the cluster structure on two lattices its efficiency is due to the use of the dual lattice and therefore is specific to the two dimensional case

asymptotic expansions for renewal measures in the plane let p be a distribution in the plane and define the renewal measure r p nwhere denotes convolution the main results of this paper are three term asymptotic expansions for r far from the origin as an application expansions are obtained for distributions in linear boundary crossing problems

probabilistic programming with discrete distributions and precedence constrained knapsack polyhedra we consider stochastic programming problems with probabilistic constraints involving random variables with discrete distributions they can be reformulated as large scale mixed integer programming problems with knapsack constraints using specific properties of stochastic programming problems and bounds on the probability of the union of events we develop new valid inequalities for these mixed integer programming problems we also develop methods for lifting these inequalities these procedures are used in a general iterative algorithm for solving probabilistically constrained problems the results are illustrated with a numerical example

generalized semi infinite programming a tutorial this tutorial presents an introduction to generalized semi infinite programming gsip which in recent years became a vivid field of active research in mathematical programming a gsip problem is characterized by an infinite number of inequality constraints and the corresponding index set depends additionally on the decision variables there exist a wide range of applications which give rise to gsip models some of them are discussed in the present paper furthermore geometric and topological properties of the feasible set and in particular the difference to the standard semi infinite case are analyzed by using first order approximations of the feasible set corresponding constraint qualifications are developed then necessary and sufficient first and second order optimality conditions are presented where directional differentiability properties of the optimal value function of the so called lower level problem are used finally an overview of numerical methods is given

probability bounds with cherry trees a third order upper bound is presented on the probability of the union of a finite number of events by means of graphs called cherry trees these are graphs that we construct recursively in such a way that every time we pick a new vertex connect it with two already existing vertices if the latters are always adjacent we call the cherry tree a t cherry tree a cherry tree has a weight that provides us with the upper bound on the union any hunter worsley bound can be majorized by a t cherry bound constructed by the use of the hunter worsley tree a cherry tree bound can be identified as a feasible solution to the dual of the boolean probability bounding problem a t cherry tree bound can be identified as the objective function value of the dual vector corresponding to a dual feasible basis in the boolean problem this enables us to improve on the bound algorithmically if we use the dual method of linear programming

hierarchy of effective sizes of dla clusters the authors discuss the possibility of having an infinite hierarchy of length scales characterising the moments of the laplacian field outside dla clusters in the entire space this hierarchy constitutes a multifractality for dla which is different from the usual one associated with the near field of these clusters the radius of gyration and the hydrodynamic radius are both members of this new family of length scales they present numerical simulations on the usual dla model and on the fracture growth model i e elastic dla testing this hypothesis

epi convergent discretizations of stochastic programs via integration quadratures the simplest and the best known method for numerical approximation of high dimensional integrals is the monte carlo method mc i e random sampling mc has also become the most popular method for constructing numerically solvable approximations of stochastic programs however certain modern integration quadratures are often superior to crude mc in high dimensional integration so it seems natural to try to use them also in discretization of stochastic programs this paper derives conditions that guarantee the epi convergence of the resulting objectives to the original one our epi convergence result is closely related to some of the existing ones but it is easier to apply to discretizations and it allows the feasible set to depend on the probability measure as examples we prove epi convergence of quadrature based discretizations of three different models of portfolio management and we study their behavior numerically besides mc our discretizations are the only existing ones with guaranteed epi convergence for these problem classes in our tests modern quadratures seem to result in faster convergence of optimal values than mc

holder and lipschitz stability of solution sets in programs with probabilistic constraints we study perturbations of a stochastic program with a probabilistic constraint and r concave original probability distribution first we improve our earlier results substantially and provide conditions implying holder continuity properties of the solution sets w r t the kolmogorov distance of probability distributions secondly we derive an upper lipschitz continuity property for solution sets under more restrictive conditions on the original program and on the perturbed probability measures the latter analysis applies to linear quadratic models and is based on work by bonnans and shapiro the stability results are illustrated by numerical tests showing the different asymptotic behaviour of parametric and nonparametric estimates in a program with a normal probabilistic constraint

fractal deterministic cracks abstract cracks are grown in an elastic medium submitted to external shear by solving the full equations of motion on a two dimensional lattice one finds that deterministic fracture patterns are in general branched and can be fractal this effect is due to the competition between the direction of global stress and the local growth direction imposed by the lattice anisotropy viscoelastic effects are also discussed

fracture patterns and scaling laws abstract fracture is determined by the growth of cracks and can be described as a moving boundary problem similar to dielectric breakdown or viscous fingering the vectorial nature of the elastic medium allows however for a much larger variety of boundary conditions and growth rules than in the scalar electrostatic analogue giving rise to a number of new effects one can model this problem on a square lattice and it is found that the patterns of cracks can be fractal even without including noise due only to the interplay of anisotropy and memory the shapes of the cracks can be compared to the ones found experimentally for stress corrosion an effect that can be mimicked by a damage memory on the surface of the crack a quantity of experimental interest in the breaking characteristics force vs displacement in a model with quenched disorder and the possibility that several cracks be present in the system one finds scaling laws in the system size for the breaking characteristics just before the system breaks fully apart the distribution of local strain is multifractal this means that the volume fraction responsible for the final cracking goes to zero in the thermodynamic limit the scaling and multiscaling laws seem to be quite universal

analytical solutions for diffusion on fractal objects a generalization of the euclidean diffusion equation is proposed for diffusion on fractals on the basis of a scaling argument a renormalization group theory for the green s function and numerical tests conjectures on the applicability to natural fractals such as porous media are presented

on generalized renewal measures and certain first passage times by relating it to the family x t infln 1 sn t t 0 one of the major results is that up t oo for all t e r iff 0 t u0 t 1 a 1 ua i1as t a oo iff e xj 2 x1 oo provided 4 is ultimately increasing a 2 0 a related result is proved for u t h u t and up t e n 2 o4 n p mn t where mn max0 i n s our results form extensions of earlier ones by heyde kalma gut and others who either considered more specific functions 4 or used stronger moment assumptions the proofs are based on a regeneration technique from renewal theory and two martingale inequalities by burkholder davis and gundy

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