multiple objectives satisficing under uncertainty we propose a class of functions called multiple objective satisficing mos criteria for evaluating the level of compliance of a set of objectives in meeting their targets collectively under uncertainty the mos criteria include the joint targets achievement probability joint success probability criterion as a special case and also extend to situations when the probability distributions are not fully characterized we focus on a class of mos criteria that favors diversification which has the potential to mitigate severe shortfalls in scenarios when any objective fails to achieve its target naturally this class excludes joint success probability we further propose the shortfall aware mos criterion s mos which is inspired by the probability measure and is diversification favoring we also show how to build tractable approximations of the s mos criterion because the s mos criterion maximization is not a convex optimization problem we propose improvement algorithms via solving sequences of convex optimization problems we report encouraging computational results on a blending problem in meeting specification targets even in the absence of full probability distribution description

multiple conversions of measurements for nonlinear estimation a multiple conversion approach mca to nonlinear estimation is proposed in this paper it jointly considers multiple hypotheses on the joint distribution of the quantity to be estimated and its measurement the overall mca estimate is a probabilistically weighted sum of the hypothesis conditional estimates to describe the hypothesized joint distributions used to match the truth a general distribution form characterized by a linear or nonlinear measurement conversion is found this form is more general than gaussian and includes gaussian as a special case moreover the minimum mean square error mmse optimal estimate given a hypothesized distribution in this form is simply the linear mmse lmmse estimate using the converted measurement lmmse based estimators including the original lmmse estimator and its generalization the recently proposed uncorrelated conversion based filter can all be incorporated into the mca framework given a nonlinear problem a specific form of the hypothesized distribution can be optimally obtained by quadratic programming using the information in the nonlinear measurement function and the measurement conversion then the mca estimate can be obtained easily for dynamic problems an interacting multiple conversion algorithm is proposed for recursive estimation the mca approach has a simple and flexible structure and takes advantage of multiple lmmse based nonlinear estimators the overall estimates are obtained adaptively depending on the performance of the candidate estimators simulation results demonstrate the effectiveness of the proposed approach compared with other nonlinear filters

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

discontinuous discretization for least squares formulation of singularly perturbed reaction diffusion problems in one and two dimensions in this paper we consider the singularly perturbed reaction diffusion problem in one and two dimensions the boundary value problem is decomposed into a first order system to which a suitable weighted least squares formulation is proposed a robust stable and efficient approach is developed based on local discontinuous galerkin ldg discretization for the weak form uniform error estimates are derived numerical examples are presented to illustrate the method and the theoretical results comparison studies are made between the proposed method and other methods

robust solutions of linear programming problems contaminated with uncertain data optimal solutions of linear programming problems may become severely infeasible if the nominal data is slightly perturbed we demonstrate this phenomenon by studying 90 lps from the well known netlib collection we then apply the robust optimization methodology ben tal and nemirovski 1 3 el ghaoui et al 5 6 to produce robust solutions of the above lps which are in a sense immuned against uncertainty surprisingly for the netlib problems these robust solutions nearly lose nothing in optimality

distributionally robust optimization under moment uncertainty with application to data driven problems stochastic programming can effectively describe many decision making problems in uncertain environments unfortunately such programs are often computationally demanding to solve in addition their solution can be misleading when there is ambiguity in the choice of a distribution for the random parameters in this paper we propose a model that describes uncertainty in both the distribution form discrete gaussian exponential etc and moments mean and covariance matrix we demonstrate that for a wide range of cost functions the associated distributionally robust or min max stochastic program can be solved efficiently furthermore by deriving a new confidence region for the mean and the covariance matrix of a random vector we provide probabilistic arguments for using our model in problems that rely heavily on historical data these arguments are confirmed in a practical example of portfolio selection where our framework leads to better performing policies on the true distribution underlying the daily returns of financial assets

on the complexity of nonoverlapping multivariate marginal bounds for probabilistic combinatorial optimization problems given a combinatorial optimization problem with an arbitrary partition of the set of random objective coefficients we evaluate the tightest possible bound on the expected optimal value for joint distributions consistent with the given multivariate marginals of the subsets in the partition for univariate marginals this bound was first proposed by meilijson and nadas meilijson i a nadas 1979 convex majorization with an application to the length of critical path j appl probab 16 3 671 677 we generalize the bound to nonoverlapping multivariate marginals using multiple choice integer programming new instances of polynomial time computable bounds are identified for discrete distributions for the problem of selecting up to m items out of a set of n items of maximum total weight the multivariate marginal bound is shown to be computable in polynomial time when the size of each subset in the partition is o log n for an activity on arc pert network the partition is naturally defined by subsets of incoming arcs into nodes the multivariate marginal bound on expected project duration is shown to be computable in time polynomial in the maximum number of scenarios for any subset and the size of the network as an application a polynomial time solvable two stage stochastic program for project crashing is identified an important feature of the bound developed in this paper is that it is exactly achievable by a joint distribution unlike many of the existing bounds

a distributionally robust perspective on uncertainty quantification and chance constrained programming the objective of uncertainty quantification is to certify that a given physical engineering or economic system satisfies multiple safety conditions with high probability a more ambitious goal is to actively influence the system so as to guarantee and maintain its safety a scenario which can be modeled through a chance constrained program in this paper we assume that the parameters of the system are governed by an ambiguous distribution that is only known to belong to an ambiguity set characterized through generalized moment bounds and structural properties such as symmetry unimodality or independence patterns we delineate the watershed between tractability and intractability in ambiguity averse uncertainty quantification and chance constrained programming using tools from distributionally robust optimization we derive explicit conic reformulations for tractable problem classes and suggest efficiently computable conservative approximations for intractable ones

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

data driven chance constrained stochastic program in this paper we study data driven chance constrained stochastic programs or more specifically stochastic programs with distributionally robust chance constraints dccs in a data driven setting to provide robust solutions for the classical chance constrained stochastic program facing ambiguous probability distributions of random parameters we consider a family of density based confidence sets based on a general phi divergence measure and formulate dcc from the perspective of robust feasibility by allowing the ambiguous distribution to run adversely within its confidence set we derive an equivalent reformulation for dcc and show that it is equivalent to a classical chance constraint with a perturbed risk level we also show how to evaluate the perturbed risk level by using a bisection line search algorithm for general phi divergence measures in several special cases our results can be strengthened such that we can derive closed form expressions for the perturbed risk levels in addition we show that the conservatism of dcc vanishes as the size of historical data goes to infinity furthermore we analyze the relationship between the conservatism of dcc and the size of historical data which can help indicate the value of data finally we conduct extensive computational experiments to test the performance of the proposed dcc model and compare various phi divergence measures based on a capacitated lot sizing problem with a quality of service requirement

generalized gauss inequalities via semidefinite programming a sharp upper bound on the probability of a random vector falling outside a polytope based solely on the first and second moments of its distribution can be computed efficiently using semidefinite programming however this chebyshev type bound tends to be overly conservative since it is determined by a discrete worst case distribution in this paper we obtain a less pessimistic gauss type bound by imposing the additional requirement that the random vector s distribution must be unimodal we prove that this generalized gauss bound still admits an exact and tractable semidefinite representation moreover we demonstrate that both the chebyshev and gauss bounds can be obtained within a unified framework using a generalized notion of unimodality we also offer new perspectives on the computational solution of generalized moment problems since we use concepts from choquet theory instead of traditional duality arguments to derive semidefinite representations for worst case probability bounds

robustness to dependency in portfolio optimization using overlapping marginals in this paper we develop a distributionally robust portfolio optimization model where the robustness is across different dependency structures among the random losses for a fr echet class of discrete distributions with overlapping marginals we show that the distributionally robust portfolio optimization problem is efficiently solvable with linear programming to guarantee the existence of a joint multivariate distribution consistent with the overlapping marginal information we make use of a graph theoretic property known as the running intersection property building on this property we develop a tight linear programming formulation to find the optimal portfolio that minimizes the worst case conditional value at risk measure lastly we use a data driven approach with financial return data to identify the fr echet class of distributions satisfying the running intersection property and then optimize the portfolio over this class of distributions numerical results in two different datasets show that the distributionally robust portfolio optimization model improves on the sample based approach

from cvar to uncertainty set implications in joint chance constrained optimization we review and develop different tractable approximations to individual chance constrained problems in robust optimization on a variety of uncertainty sets and show their interesting connections with bounds on the conditional value at risk cvar measure we extend the idea to joint chance constrained problems and provide a new formulation that improves upon the standard approach our approach builds on a classical worst case bound for order statistics problems and is applicable even if the constraints are correlated we provide an application of the model on a network resource allocation problem with uncertain demand

distributionally robust joint chance constraints with second order moment information we develop tractable semidefinite programming based approximations for distributionally robust individual and joint chance constraints assuming that only the first and second order moments as well as the support of the uncertain parameters are given it is known that robust chance constraints can be conservatively approximated by worst case conditional value at risk cvar constraints we first prove that this approximation is exact for robust individual chance constraints with concave or not necessarily concave quadratic constraint functions and we demonstrate that the worst case cvar can be computed efficiently for these classes of constraint functions next we study the worst case cvar approximation for joint chance constraints this approximation affords intuitive dual interpretations and is provably tighter than two popular benchmark approximations the tightness depends on a set of scaling parameters which can be tuned via a sequential convex optimization algorithm we show that the approximation becomes essentially exact when the scaling parameters are chosen optimally and that the worst case cvar can be evaluated efficiently if the scaling parameters are kept constant we evaluate our joint chance constraint approximation in the context of a dynamic water reservoir control problem and numerically demonstrate its superiority over the two benchmark approximations

robust solutions of optimization problems affected by uncertain probabilities in this paper we focus on robust linear optimization problems with uncertainty regions defined by divergences for example chi squared hellinger kullback leibler we show how uncertainty regions based on divergences arise in a natural way as confidence sets if the uncertain parameters contain elements of a probability vector such problems frequently occur in for example optimization problems in inventory control or finance that involve terms containing moments of random variables expected utility etc we show that the robust counterpart of a linear optimization problem with divergence uncertainty is tractable for most of the choices of typically considered in the literature we extend the results to problems that are nonlinear in the optimization variables several applications including an asset pricing example and a numerical multi item newsvendor example illustrate the relevance of the proposed approach this paper was accepted by gerard p cachon optimization

convex optimal uncertainty quantification optimal uncertainty quantification ouq is a framework for numerical extreme case analysis of stochastic systems with imperfect knowledge of the underlying probability distribution this paper presents sufficient conditions under which an ouq problem can be reformulated as a finite dimensional convex optimization problem for which efficient numerical solutions can be obtained the sufficient conditions include that the objective function is piecewise concave and the constraints are piecewise convex in particular we show that piecewise concave objective functions may appear in applications where the objective is defined by the optimal value of a parameterized linear program

optimal inequalities in probability theory a convex optimization approach we propose a semidefinite optimization approach to the problem of deriving tight moment inequalities for p x in s for a set s defined by polynomial inequalities and a random vector x defined on omega subseteq cal r n that has a given collection of up to kth order moments in the univariate case we provide optimal bounds on p x in s when the first k moments of x are given as the solution of a semidefinite optimization problem in k 1 dimensions in the multivariate case if the sets s and omega are given by polynomial inequalities we obtain an improving sequence of bounds by solving semidefinite optimization problems of polynomial size in n for fixed k we characterize the complexity of the problem of deriving tight moment inequalities we show that it is np hard to find tight bounds for k geq 4 and omega cal r n and for k geq 2 and omega cal r n when the data in the problem is rational for k 1 and omega cal r n we show that we can find tight upper bounds by solving n convex optimization problems when the set s is convex and we provide a polynomial time algorithm when s and omega are unions of convex sets over which linear functions can be optimized efficiently for the case k 2 and omega cal r n we present an efficient algorithm for finding tight bounds when s is a union of convex sets over which convex quadratic functions can be optimized efficiently

convex approximations of chance constrained programs we consider a chance constrained problem where one seeks to minimize a convex objective over solutions satisfying with a given close to one probability a system of randomly perturbed convex constraints this problem may happen to be computationally intractable our goal is to build its computationally tractable approximation i e an efficiently solvable deterministic optimization program with the feasible set contained in the chance constrained problem we construct a general class of such convex conservative approximations of the corresponding chance constrained problem moreover under the assumptions that the constraints are affine in the perturbations and the entries in the perturbation vector are independent of each other random variables we build a large deviation type approximation referred to as bernstein approximation of the chance constrained problem this approximation is convex and efficiently solvable we propose a simulation based scheme for bounding the optimal value in the chance constrained problem and report numerical experiments aimed at comparing the bernstein and well known scenario approximation approaches finally we extend our construction to the case of ambiguous chance constrained problems where the random perturbations are independent with the collection of distributions known to belong to a given convex compact set rather than to be known exactly while the chance constraint should be satisfied for every distribution given by this set

generalized chebyshev bounds via semidefinite programming a sharp lower bound on the probability of a set defined by quadratic inequalities given the first two moments of the distribution can be efficiently computed using convex optimization this result generalizes chebyshev s inequality for scalar random variables two semidefinite programming formulations are presented with a constructive proof based on convex optimization duality and elementary linear algebra

a robust optimization perspective on stochastic programming in this paper we introduce an approach for constructing uncertainty sets for robust optimization using new deviation measures for random variables termed the forward and backward deviations these deviation measures capture distributional asymmetry and lead to better approximations of chance constraints using a linear decision rule we also propose a tractable approximation approach for solving a class of multistage chance constrained stochastic linear optimization problems an attractive feature of the framework is that we convert the original model into a second order cone program which is computationally tractable both in theory and in practice we demonstrate the framework through an application of a project management problem with uncertain activity completion time

optimization under probabilistic envelope constraints chance constraints are an important modeling tool in stochastic optimization providing probabilistic guarantees that a solution succeeds in satisfying a given constraint although they control the probability of success they provide no control whatsoever in the event of a failure that is they do not distinguish between a slight overshoot or undershoot of the bounds and more catastrophic violation in short they do not capture the magnitude of violation of the bounds this paper addresses precisely this topic focusing on linear constraints and ellipsoidal gaussian like uncertainties we show that the problem of requiring different probabilistic guarantees at each level of constraint violation can be reformulated as a semi infinite optimization problem we provide conditions that guarantee polynomial time solvability of the resulting semi infinite formulation we show further that this resulting problem is what has been called a comprehensive robust optimization problem in the literature as a byproduct we provide tight probabilistic bounds for comprehensive robust optimization thus analogously to the connection between chance constraints and robust optimization we provide a broader connection between probabilistic envelope constraints and comprehensive robust optimization

ambiguous chance constrained problems and robust optimization in this paper we study ambiguous chance constrained problems where the distributions of the random parameters in the problem are themselves uncertain we focus primarily on the special case where the uncertainty set of the distributions is of the form where p denotes the prohorov metric the ambiguous chance constrained problem is approximated by a robust sampled problem where each constraint is a robust constraint centered at a sample drawn according to the central measure the main contribution of this paper is to show that the robust sampled problem is a good approximation for the ambiguous chance constrained problem with a high probability this result is established using the strassen dudley representation theorem that states that when the distributions of two random variables are close in the prohorov metric one can construct a coupling of the random variables such that the samples are close with a high probability we also show that the robust sampled problem can be solved efficiently both in theory and in practice