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

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 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

on maximal weight solutions of two matricial moment problems in the nondegenerate case abstract we first revisit the question of existence and uniqueness of maximal weight solutions of the truncated trigonometric matrix moment problem problem ttm in the nondegenerate case the starting point of the consideration is a specialized representation of the matrix weight assigned at each point on the unit circle for the solutions of nondegenerate problem ttm our approach is essentially self contained and different from the existing methods moreover based on this approach we also derive some important weight features of the so called n extremal solutions within the whole solution set of nondegenerate problem ttm and the discrete structure of them in the second part these observations inspire us to look for the corresponding weight features of the solutions of a nondegenerate moment problem for rational matrix valued functions problem r where only the theory of orthogonal matrix polynomials on the unit circle is involved

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

moment problems and orthogonal functions abstract the caratheodory coefficient problem for an infinite sequence c n can be formulated as follows find a caratheodory function f z i e an analytic function mapping the unit disk d z z into the right half plane p w re w 0 whose taylor coefficients at a fixed points in d for convenience the origin are the c n this problem is equivalent to the trigonometric moment problem find a measure on such that e i n d c n for n 0 1 2 these problems are closely related to the theory of szego polynomials i e orthogonal polynomials on the unit circle t z z 1 let n be an arbitrary given sequence of not necessarily distinct points in d and let w n be a given sequence of numbers the nevanlinna pick problem for this situation is to find a caratheodory function f z with the interpolation property f n w n for all n when points n are repeated the interpolation property involves the appropriate number of taylor coefficients also this problem is connected with the problem of finding a measure generating moments the nevanlinna pick problem is related to a theory of rational functions which are orthogonal on t with poles at the points 1 n this relationship is analogous to and generalizes the relationship between the caratheodory coefficient problem or trigonometric moment problem and polynomials orthogonal on t the situation when points n are given on the boundary t or d is also briefly discussed

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

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

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

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

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

checking the coherence and redundancy of fuzzy knowledge bases checking the coherence of a set of rules is an important step in knowledge base validation coherence is also needed in the field of fuzzy systems indeed rules are often used regardless of their semantics and it sometimes leads to sets of rules that make no sense avoiding redundancy is also of interest in real time systems for which the inference engine is time consuming a knowledge base is potentially inconsistent or incoherent if there exists a piece of input data that respects integrity constraints and that leads to logical inconsistency when added to the knowledge base we more particularly consider knowledge bases composed of parallel fuzzy rules then coherence means that the projection on the input variables of the conjunctive combination of the possibility distributions representing the fuzzy rules leaves these variables completely unrestricted i e any value for these variables is possible or at least not more restrictive than integrity constraints fuzzy rule representations can be implication based or conjunction based we show that only implication based models may lead to coherence problems however unlike conjunction based models they allow to design coherence checking processes some conditions that a set of parallel rules has to satisfy in order to avoid inconsistency problems are given for certainty or gradual rules the problem of redundancy which is also of interest for fuzzy knowledge bases validation is addressed for these two kinds of rules

on a class of extremal solutions of a moment problem for rational matrix valued functions in the nondegenerate case ii the main theme of this paper is the discussion of a family of extremal solutions of a finite moment problem for rational matrix functions in the nondegenerate case we will point out that each member of this family is extremal in several directions thereby the investigations below continue the studies in fritzsche et al in press 1 in doing so an application of the theory of orthogonal rational matrix functions with respect to a nonnegative hermitian matrix borel measure on the unit circle is used to get some insights into the structure of the extremal solutions in question in particular we explain characterizations of these solutions in the whole solution set in terms of orthogonal rational matrix functions we will also show that the associated riesz herglotz transform of such a particular solution admits specific representations where orthogonal rational matrix functions are involved

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

a synthesis of fuzzy rule based system verification abstract the verification of fuzzy rule bases for anomalies has received increasing attention these last few years many different approaches have been suggested and many are still under investigation in this paper we give a synthesis of methods proposed in literature that try to extend the verification of classical rule bases to the case of fuzzy knowledge modeling without needing a set of representative input within this area of fuzzy validation and verification v v we identify two dual lines of thought leading to what is identified as static and dynamic anomaly detection methods static anomaly detection essentially tries to use similarity affinity or matching measures to identify anomalies within a fuzzy rule base it is assumed that the detection methods can be the same as those used in a non fuzzy environment except that the former measures indicate the degree of matching of two fuzzy expressions dynamic anomaly detection starts from the basic idea that any anomaly within a knowledge representation formalism i e fuzzy if then rules can be identified by performing a dynamic analysis of the knowledge system even without providing special input to the system by imposing a constraint on the results of inference for an anomaly not to occur one creates definitions of the anomalies that can only be verified if the inference process and thereby the fuzzy inference operator is involved in the analysis the major outcome of the confrontation between both approaches is that their results stated in terms of necessary and or sufficient conditions for anomaly detection within a particular situation are difficult to reconcile the duality between approaches seems to have translated into a duality in results this article addresses precisely this issue by presenting a theoretical framework which enables us to effectively evaluate the results of both static and dynamic verification theories

a lift and project cutting plane algorithm for mixed 0 1 programs we propose a cutting plane algorithm for mixed 0 1 programs based on a family of polyhedra which strengthen the usual lp relaxation we show how to generate a facet of a polyhedron in this family which is most violated by the current fractional point this cut is found through the solution of a linear program that has about twice the size of the usual lp relaxation a lifting step is used to reduce the size of the lp s needed to generate the cuts an additional strengthening step suggested by balas and jeroslow is then applied we report our computational experience with a preliminary version of the algorithm this approach is related to the work of balas on disjunctive programming the matrix cone relaxations of lovasz and schrijver and the hierarchy of relaxations of sherali and adams

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

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

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

on maximum mass measures in truncated matricial hamburger moment problems and related interpolation problems abstract a significant extremal question is considered within the solution sets of two different nondegenerate truncated matricial hamburger moment problems taking an arbitrary r whether there exists one and only one solution of each of those two moment problems which has a concentrated matrix mass at the point equal to the maximum mass the main concern of this paper is to indicate that for the first moment problem problem thm 2 n the answer is yes however for the second moment problem problem thm m the existence is not generally fulfilled and it holds if and only if r has to satisfy some additional condition these observations further serve as starting point to look for the corresponding extremal feature for three nondegenerate matrix interpolation problems of nevanlinna pick type based on the so called block hankel vector approach

fuzzy sets as a basis for a theory of possibility the theory of possibility described in this paper is related to the theory of fuzzy sets by defining the concept of a possibility distribution as a fuzzy restriction which acts as an elastic constraint on the values that may be assigned to a variable more specifically if f is a fuzzy subset of a universe of discourse u u which is characterized by its membership function f then a proposition of the form x is f where x is a variable taking values in u induces a possibility distribution tx which equates the possibility of x taking the value u to f u the compatibility of u with f in this way x becomes a fuzzy variable which is associated with the possibility distribution tx in much the same way as a random variable is associated with a probability distribution in general a variable may be associated both with a possibility distribution and a probability distribution with the weak connection between the two expressed as the possibility probability consistency principle a thesis advanced in this paper is that the imprecision that is intrinsic in natural languages is in the main possibilistic rather than probabilistic in nature thus by employing the concept of a possibility distribution a proposition p in a natural language may be translated into a procedure which computes the probability distribution of a set of attributes which are implied by p several types of conditional translation rules are discussed and in particular a translation rule for propositions of the form x is f is possible where is a number in the interval 0 1 is formulated and illustrated by examples

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

on the multiple nevanlinna pick matrix interpolation in the class cp and the caratheodory matrix coefficient problem abstract in the present paper an intrinsic one to one correspondence between the finite multiple nevanlinna pick matrix interpolation np problem in the caratheodory class and the caratheodory matrix coefficient cc problem or equivalently the truncated trigonometric matrix moment tm problem is established thanks to this correspondence the np problem is reduced to what amounts to solving the cc problem or the tm problem associated with the so called toeplitz block vector of the former in both the nondegenerate and degenerate cases simultaneously

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

n extremal matrices of measures for an indeterminate matrix moment problem in this paper we study the n extremal matrices of measures associated to a completely indeterminate matrix moment problem i e those matrices of measure w solutions of a completely indeterminate matrix moment problem for which the linear space of matrix polynomials is dense in the corresponding l2 w

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

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

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

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

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

radial fuzzy systems the class of radial fuzzy systems is introduced the fuzzy systems in this class use radial functions to implement membership functions of fuzzy sets and exhibit a shape preservation property in antecedents of their rules the property is called the radial property it enables the radial fuzzy systems to have their computational model mathematically tractable under both conjunctive and implicative representations of their rule bases coherence of radial implicative fuzzy systems is discussed and a sufficient condition for coherence is stated

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

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

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

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

on a moment problem for rational matrix valued functions the papers deals with a finite moment problem for rational matrix valued functions we present a necessary and sufficient condition for the solvability of the problem in the nondegenerate case we construct a particular solution which has interesting extremal properties

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