ambiguous risk measures and optimal robust portfolios this paper deals with a problem of guaranteed robust financial decision making under model uncertainty an efficient method is proposed for determining optimal robust portfolios of risky financial instruments in the presence of ambiguity uncertainty on the probabilistic model of the returns specifically it is assumed that a nominal discrete return distribution is given while the true distribution is only known to lie within a distance d from the nominal one where the distance is measured according to the kullback leibler divergence the goal in this setting is to compute portfolios that are worst case optimal in the mean risk sense that is to determine portfolios that minimize the maximum with respect to all the allowable distributions of a weighted risk mean objective the analysis in the paper considers both the standard variance measure of risk and the absolute deviation measure

robust and adaptive network flows we study network flow problems in an uncertain environment from the viewpoint of robust optimization in contrast to previous work we consider the case that the network parameters e g capacities are known and deterministic but the network structure e g nodes and arcs is subject to uncertainty in this paper we study the robust and adaptive versions of the maximum flow problem and minimum cut problems in networks with node and arc failures and establish structural and computational results the adaptive two stage model adjusts the solution after the realization of the failures in the network this leads to a more flexible model and yields less conservative solutions compared to the robust model we show that the robust maximum flow problem can be solved in polynomial time but the robust minimum cut problem is np hard we also prove that the adaptive versions are np hard we further characterize the adaptive model as a two person zero sum game and prove the existence of an equilibrium in such g

solvers for mathcal o n electronic structure in the strong scaling limit we present a hybrid openmp charm tt framework for solving the mathcal o n self consistent field eigenvalue problem with parallelism in the strong scaling regime p gg n where p is the number of cores and n is a measure of system size i e the number of matrix rows columns basis functions atoms molecules etc this result is achieved with a nested approach to spectral projection and the sparse approximate matrix multiply bock and challacombe siam j sci comput 35 2013 pp c72 c98 and involves a recursive task parallel algorithm often employed by generalized n body solvers to occlusion and culling of negligible products in the case of matrices with decay employing classic technologies associated with generalized n body solvers including overdecomposition recursive task parallelism orderings that preserve locality and persistence based load balancing we obtain scaling beyond hundreds of cores per molecule for small water clusters h 2 o n n in 30

robust multiobjective optimization applications in portfolio optimization motivated by markowitz portfolio optimization problems under uncertainty in the problem data we consider general convex parametric multiobjective optimization problems under data uncertainty for the first time this uncertainty is treated by a robust multiobjective formulation in the gist of ben tal and nemirovski for this novel formulation we investigate its relationship to the original multiobjective formulation as well as to its scalarizations further we provide a characterization of the location of the robust pareto frontier with respect to the corresponding original pareto frontier and show that standard techniques from multiobjective optimization can be employed to characterize this robust efficient frontier we illustrate our results based on a standard mean variance problem

primal dual interior point methods for semidefinite programming convergence rates stability and numerical results primal dual interior point path following methods for semidefinite programming are considered several variants are discussed based on newton s method applied to three equations primal feasibility dual feasibility and some form of centering condition the focus is on three such algorithms called the xz xz zx and q methods for the xz zx and q algorithms the newton system is well defined and its jacobian is nonsingular at the solution under nondegeneracy assumptions the associated schur complement matrix has an unbounded condition number on the central path under the nondegeneracy assumptions and an additional rank assumption practical aspects are discussed including mehrotra predictor corrector variants and issues of numerical stability compared to the other methods considered the xz zx method is more robust with respect to its ability to step close to the boundary converges more rapidly and achieves higher accuracy

parameter estimation with expected and residual at risk criteria abstract in this paper we study a class of uncertain linear estimation problems in which the data are affected by random uncertainty we consider two estimation criteria one based on minimization of the expected l 1 or l 2 norm residual and one based on minimization of the level within which the l 1 or l 2 norm residual is guaranteed to lie with an a priori fixed probability residual at risk the random uncertainty affecting the data is characterized by means of its first two statistical moments and the above criteria are intended in a worst case probabilistic sense that is worst case expectations and probabilities over all possible distribution having the specified moments are considered the ensuing estimation problems can be solved efficiently via convex programming yielding exact solutions in the l 2 norm case and upper bounds on the optimal solutions in the l 1 case

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

a probabilistic analytic center cutting plane method for feasibility of uncertain lmis many robust control problems can be formulated in abstract form as convex feasibility programs where one seeks a solution x that satisfies a set of inequalities of the form f f x d 0 d d this set typically contains an infinite and uncountable number of inequalities and it has been proved that the related robust feasibility problem is numerically hard to solve in general in this paper we discuss a family of cutting plane methods that solve efficiently a probabilistically relaxed version of the problem specifically under suitable hypotheses we show that an analytic center cutting plane scheme based on a probabilistic oracle returns in a finite and pre specified number of iterations a solution x which is feasible for most of the members of f except possibly for a subset having arbitrarily small probability measure

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

decay rates for inverses of band matrices spectral theory and classical approximation theory are used to give a new proof of the exponential decay of the entries of the inverse of band matrices the rate of decay of a 1 can be bounded in terms of the essential spectrum of aa for general a and in terms of the essential spectrum of a for positive definite a in the positive definite case the bound can be attained these results are used to establish the exponential decay for a class of generalized eigenvalue problems and to establish exponential decay for certain sparse but nonbanded matrices we also establish decay rates for certain generalized inverses

robust performance analysis of uncertain lti systems dual lmi approach and verifications for exactness this paper addresses robust performance analysis problems of linear time invariant lti systems affected by real parametric uncertainties these problems known also as a special class of structured singular value computation problems are inherently intractable np hard problems as such intensive research effort has been made to obtain computationally tractable and less conservative analysis conditions where linear matrix inequality lmi plays an important nevertheless since lmi based conditions are expected to be conservative in general it is often the case that we cannot conclude anything if the lmi at hand turns out to be infeasible this motivates us to consider the dual of the lmi and examine the structure of the dual solution by pursuing this direction in this paper we provide rank conditions on the dual solution matrix under which we can conclude that the underlying robust performance is never attained in particular a set of uncertain parameters that violates the specified performance can be computed these results come from block moment matrix structure of the dual variable which is consistent with the recent results on polynomial optimization this particular structure enables us to make good use of simultaneous diagonalizability property of commuting diagonalizable matrices so that the sound rank conditions for the exactness verification can be obtained

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

an interpretation of membership functions and the properties of general probabilistic operators as fuzzy set operators part i case of type 1 fuzzy sets abstract the grade of membership is assumed to be determined as the relative size of evidence grounds supporting a given element s membership compared to all the necessary reference grounds relevant to the judgement of membership grade and is interpreted as conditional probability conditioned by the given element by assigning the dependence relations of the evidence grounds to those of the fuzzy subsets the set operators are defined as probabilistic operators with dependence relations the properties of the operators are studied and it is found that with type 1 fuzzy subsets they satisfy all the basic properties satisfied by the crisp set operators illustrative examples are given describing the construction of membership functions and the selection of set operators in relation to the situations to be modelled the proposed operators a kind of parametrized operators with an index value of the dependence relations as a parameter cover the entire operators ranging from the so called max min operators to the bold union intersection operators an extension of the above operator range is also discussed

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

a kinship function approach to robust and probabilistic optimization under polynomial uncertainty in this paper we study a class of robust design problems with polynomial dependence on the uncertainty one of the main motivations for considering these problems comes from robust controller design where one often encounters systems that depend polynomially on the uncertain parameters this paper can be seen as integrated in the emerging area of probabilistic robustness where a probabilistic relaxation of the original robust problem is adopted thus requiring the satisfaction of the constraints not for all possible values of the uncertainty but for most of them different from the randomized approach for tackling probabilistic relaxations which is only guaranteed to provide soft bounds on the probability of satisfaction we present a deterministic approach based on the novel concept of kinship function introduced in this paper this allows the development of an original framework which leads to easily computable deterministic convex relaxations of the probabilistic problem in particular optimal polynomial kinship functions are introduced which can be computed a priori and once for all and provide the best convex bound on the probability of constraint violation more importantly it is proven that the solution of the relaxed problem converges to that of the original robust optimization problem as the degree of the optimal polynomial kinship function increases furthermore by relying on quadrature formulas for computation of integrals of polynomials it is shown that the computational complexity of the proposed approach is polynomial in the number of uncertain parameters finally unlike other deterministic approaches to robust polynomial optimization the number of variables in the ensuing optimization problem is not increased by the proposed approximation an important feature of this approach is that a significant amount of the computational burden is shifted to a one time offline computation whose results can be stored and provided to the end user

design of spl lscr sub 1 optimal controllers with flexible disturbance rejection level this note presents a new design methodology that allows for flexible management of the tradeoff between the ability of a system to attenuate disturbance signals versus its expected worst peak to peak amplification the proposed strategy applies to linear time invariant systems which are subject to persistent disturbance signals and combines a recently developed quasi robust linear programming concept with a well known spl lscr sub 1 optimal controller synthesis approach the benefit of the resulting technique is demonstrated using an example

robustness in the presence of mixed parametric uncertainty and unmodeled dynamics continuing the development of the structured singular value approach to robust control design the authors investigate the problem of computing mu in the case of mixed real parametric and complex uncertainty the problem is shown to be equivalent to a smooth constrained finite dimensional optimization problem in view of the fact that the functional to be maximized may have several local extrema an upper bound on mu whose computation is numerically tractable is established this leads to a sufficient condition of robust stability and performance a historical perspective on the development of the mu theory is included

two stage robust network flow and design under demand uncertainty we describe a two stage robust optimization approach for solving network flow and design problems with uncertain demand in two stage network optimization one defers a subset of the flow decisions until after the realization of the uncertain demand availability of such a recourse action allows one to come up with less conservative solutions compared to single stage optimization however this advantage often comes at a price two stage optimization is in general significantly harder than single stage optimization for network flow and design under demand uncertainty we give a characterization of the first stage robust decisions with an exponential number of constraints and prove that the corresponding separation problem is np hard even for a network flow problem on a bipartite graph we show however that if the second stage network topology is totally ordered or an arborescence then the separation problem is tractable unlike single stage robust optimization under demand uncertainty two stage robust optimization allows one to control conservatism of the solutions by means of an allowed budget for demand uncertainty using a budget of uncertainty we provide an upper bound on the probability of infeasibility of a robust solution for a random demand vector we generalize the approach to multicommodity network flow and design and give applications to lot sizing and location transportation problems by projecting out second stage flow variables we define an upper bounding problem for the two stage min max min optimization problem finally we present computational results comparing the proposed two stage robust optimization approach with single stage robust optimization as well as scenario based two stage stochastic optimization

a model for the measurement of membership and the consequences of its empirical implementation representation and uniqueness theorems are presented for the fundamental measurement of the membership of a fuzzy set when the domain of discourse is order dense the conclusion that membership is on an interval scale is justified by the inapplicability of extensive measurement to fuzziness and the lack of a natural origin for membership some descriptive statistics are also presented from an empirical study currently in progress to construct membership functions they suggest that subjects differ in their perception of fuzziness and that the models in the literature for certain linguistic operators such as not very and antonym are not sufficiently general to account for the range of individual perceptions of vague attributes a more general mathematical model is proposed the question of the meaningfulness of statements about membership functions is also discussed in the light of the assertion that membership can be fundamentally measured at best on an interval scale some of the difficulties which arise in fuzzy set theory due to this property are illustrated and a solution is suggested involving the use of a function which is derived from membership but is on an absolute scale

robust nonlinear optimization with conic representable uncertainty set the robust optimization methodology is known as a popular method dealing with optimization problems with uncertain data and hard constraints this methodology has been applied so far to various convex conic optimization problems where only their inequality constraints are subject to uncertainty in this paper the robust optimization methodology is applied to the general nonlinear programming nlp problem involving both uncertain inequality and equality constraints the uncertainty set is defined by conic representable sets the proposed uncertainty set is general enough to include many uncertainty sets which have been used in literature as special cases the robust counterpart rc of the general nlp problem is approximated under this uncertainty set it is shown that the resulting approximate rc of the general nlp problem is valid in a small neighborhood of the nominal value furthermore a rather general class of programming problems is posed that the robust counterparts of its problems can be derived exactly under the proposed uncertainty set our results show the applicability of robust optimization to a wider area of real applications and theoretical problems with more general uncertainty sets than those considered so far the resulting robust counterparts which are traditional optimization problems make it possible to use existing algorithms of mathematical optimization to solve more complicated and general robust optimization problems

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

on eigenvalue optimization in this paper we study optimization problems involving eigenvalues of symmetric matrices one of the difficulties with numerical analysis of such problems is that the eigenvalues considered as functions of a symmetric matrix are not differentiable at those points where they coalesce we present a general framework for a smooth differentiable approach to such problems it is based on the concept of transversality borrowed from differential geometry in that framework we discuss first and second order optimality conditions and rates of convergence of the corresponding second order algorithms finally we present some results on the sensitivity analysis of such problems

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

the scenario approach to robust control design this paper proposes a new probabilistic solution framework for robust control analysis and synthesis problems that can be expressed in the form of minimization of a linear objective subject to convex constraints parameterized by uncertainty terms this includes the wide class of np hard control problems representable by means of parameter dependent linear matrix inequalities lmis it is shown in this paper that by appropriate sampling of the constraints one obtains a standard convex optimization problem the scenario problem whose solution is approximately feasible for the original usually infinite set of constraints i e the measure of the set of original constraints that are violated by the scenario solution rapidly decreases to zero as the number of samples is increased we provide an explicit and efficient bound on the number of samples required to attain a priori specified levels of probabilistic guarantee of robustness a rich family of control problems which are in general hard to solve in a deterministically robust sense is therefore amenable to polynomial time solution if robustness is intended in the proposed risk adjusted sense

orderings for factorized sparse approximate inverse preconditioners the influence of reorderings on the performance of factorized sparse approximate inverse preconditioners is considered some theoretical results on the effect of orderings on the fill in and decay behavior of the inverse factors of a sparse matrix are presented it is shown experimentally that certain reorderings like minimum degree and nested dissection can be very beneficial the benefit consists of a reduction in the storage and time required for constructing the preconditioner and of faster convergence of the preconditioned iteration in many cases of practical interest

first and second order analysis of nonlinear semidefinite programs in this paper we study nonlinear semidefinite programming problems convexity duality and first order optimality conditions for such problems are presented a second order analysis is also given second order necessary and sufficient optimality conditions are derived finally sensitivity analysis of such programs is discussed

robust semidefinite programming problems with general nonlinear parameter dependence two conservative approaches are proposed to a semidefinite programming problem nonlinearly dependent on uncertain parameters these approaches are applicable to general nonlinear parameter dependence not necessarily polynomial or rational they are based on the mild assumption that the parameter dependence is expressed as the difference of two convex functions the first approach uses constant bounds on the parameter dependence optimization of the bounds is reduced to convex nonsmooth minimization the second approach uses parameter dependent bounds for a less conservative result optimization of the bounds is immediate when the centroid is computable for the parameter set numerical examples are presented for illustration of the approaches

a statistical learning theory approach for uncertain linear and bilinear matrix inequalities abstract in this paper we consider the problem of minimizing a linear functional subject to uncertain linear and bilinear matrix inequalities which depend in a possibly nonlinear way on a vector of uncertain parameters motivated by recent results in statistical learning theory we show that probabilistic guaranteed solutions can be obtained by means of randomized algorithms in particular we show that the vapnik chervonenkis dimension vc dimension of the two problems is finite and we compute upper bounds on it in turn these bounds allow us to derive explicitly the sample complexity of these problems using these bounds in the second part of the paper we derive a sequential scheme based on a sequence of optimization and validation steps the algorithm is on the same lines of recent schemes proposed for similar problems but improves both in terms of complexity and generality the effectiveness of this approach is shown using a linear model of a robot manipulator subject to uncertain parameters

the u lagrangian of the maximum eigenvalue function in this paper we apply the u lagrangian theory to the maximum eigenvalue function la and to its precomposition with affine matrix valued mappings we first give geometrical interpretations of the u objects that we introduce we also show that the u lagrangian of la has a hessian which can be explicitly computed the second order development of the u lagrangian provides a second order development of la along a characteristic smooth manifold the set of symmetric matrices whose maximal eigenvalues have a fixed multiplicity the same results can be obtained when we precompose la with an affine matrix valued mapping a provided that this mapping satisfies a regularity condition em transversality condition we show that the hessian of the u lagrangian of la circ a coincides with the reduced hessian encountered in sequential quadratic programming finally we use the u lagrangian to derive second order algorithms for minimizing la circ a

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

robust truss topology design via semidefinite programming we present and motivate a new model of the truss topology design problem where the rigidity of the resulting truss with respect both to given loading scenarios and small occasional loads is optimized it is shown that the resulting optimization problem is a semidefinite program we derive and analyze several equivalent reformulations of the problem and present illustrative numerical examples

semidefinite programming for uncertain linear equations in static analysis of structures abstract this paper presents a method for computing a minimal bound that contains the solution set to the uncertain linear equations particularly we aim at finding a bounding ellipsoid for static response of structures where both external forces and elastic moduli of members are assumed to be imprecisely known and bounded by using the s lemma we formulate a semidefinite programming sdp problem which provides an outer approximation of the minimal bounding ellipsoid bounding ellipsoids are computed for nodal displacements of uncertain braced frames as the solutions of the presented sdp problems by using the primal dual interior point method

applications of convex optimization in signal processing and digital communication in the last two decades the mathematical programming community has witnessed some spectacular advances in interior point methods and robust optimization these advances have recently started to significantly impact various fields of applied sciences and engineering where computational efficiency is essential this paper focuses on two such fields digital signal processing and communication in the past the widely used optimization methods in both fields had been the gradient descent or least squares methods both of which are known to suffer from the usual headaches of stepsize selection algorithm initialization and local minima with the recent advances in conic and robust optimization the opportunity is ripe to use the newly developed interior point optimization techniques and highly efficient software tools to help advance the fields of signal processing and digital communication this paper surveys recent successes of applying interior point and robust optimization to solve some core problems in these two fields the successful applications considered in this paper include adaptive filtering robust beamforming design and analysis of multi user communication system channel equalization decoding and detection throughout our emphasis is on how to exploit the hidden convexity convex reformulation of semi infinite constraints analysis of convergence complexity and performance as well as efficient practical implementation

a linear decision based approximation approach to stochastic programming stochastic optimization especially multistage models is well known to be computationally excruciating moreover such models require exact specifications of the probability distributions of the underlying uncertainties which are often unavailable in this paper we propose tractable methods of addressing a general class of multistage stochastic optimization problems which assume only limited information of the distributions of the underlying uncertainties such as known mean support and covariance one basic idea of our methods is to approximate the recourse decisions via decision rules we first examine linear decision rules in detail and show that even for problems with complete recourse linear decision rules can be inadequate and even lead to infeasible instances hence we propose several new decision rules that improve upon linear decision rules while keeping the approximate models computationally tractable specifically our approximate models are in the forms of the so called second order cone soc programs which could be solved efficiently both in theory and in practice we also present computational evidence indicating that our approach is a viable alternative and possibly advantageous to existing stochastic optimization solution techniques in solving a two stage stochastic optimization problem with complete recourse

extending scope of robust optimization comprehensive robust counterparts of uncertain problems in this paper we propose a new methodology for handling optimization problems with uncertain data with the usual robust optimization paradigm one looks for the decisions ensuring a required performance for all realizations of the data from a given bounded uncertainty set whereas with the proposed approach we require also a controlled deterioration in performance when the data is outside the uncertainty set the extension of robust optimization methodology developed in this paper opens up new possibilities to solve efficiently multi stage finite horizon uncertain optimization problems in particular to analyze and to synthesize linear controllers for discrete time dynamical systems

robust transmission network expansion planning in energy systems improving computational performance abstract in recent advances in solving the problem of transmission network expansion planning the use of robust optimization techniques has been put forward as an alternative to stochastic mathematical programming methods to make the problem tractable in realistic systems different sources of uncertainty have been considered mainly related to the capacity and availability of generation facilities and demand and making use of adaptive robust optimization models the mathematical formulations for these models give rise to three level mixed integer optimization problems which are solved using different strategies although it is true that these robust methods are more efficient than their stochastic counterparts it is also correct that solution times for mixed integer linear programming problems increase exponentially with respect to the size of the problem because of this practitioners and system operators need to use computationally efficient methods when solving this type of problem in this paper the issue of improving computational performance by taking different features from existing algorithms is addressed in particular we replace the lower level problem with a dual one and solve the resulting bi level problem using a primal cutting plane algorithm within a decomposition scheme by using this alternative and simple approach the computing time for solving transmission expansion planning problems has been reduced drastically numerical results in an illustrative example the ieee 24 and ieee 118 bus test systems demonstrate that the algorithm is superior in terms of computational performance with respect to the existing methods

ellipsoidal load domain shakedown analysis with von mises yield criterion a robust optimization approach summary the static formulation of elastic shakedown analysis based upon melan s lower bound theorem can essentially be viewed as a robust optimization problem this paper discusses an advantage that is enjoyed by taking this perspective specifically we assume the von mises yield criterion and an ellipsoidal load domain in this setting the shakedown analysis problem which is viewed as robust second order cone programming can be recast as semidefinite programming copyright 2016 john wiley sons ltd

a region dividing technique for constructing the sum of squares approximations to robust semidefinite programs in this technical note we present a novel approach to robust semidefinite programs of which coefficient matrices depend polynomially on uncertain parameters the approach is based on approximation with the sum of squares polynomials but in contrast to the conventional sum of squares approach the quality of approximation is improved by dividing the parameter region into several subregions the optimal value of the approximate problem converges to that of the original problem as the resolution of the division becomes finer an advantage of this approach is that an upper bound on the approximation error can be explicitly obtained in terms of the resolution of the division a numerical example on polynomial optimization is presented to show usefulness of the present approach

interval robust multi objective algorithm abstract this paper introduces a method for solving multi objective optimization problems in uncertain environment when the uncertainty factors of the optimization problem can be included into the mathematical model through bounded intervals i rmoa interval robust multi objective algorithm can find an enclosure of the robust pareto frontier in this approach the robust pareto solutions are the ones that have the best performance when the worst case scenario characterized by the uncertainty parameters is considered i rmoa has some positive aspects it does not require the calculus of derivatives it has only two input parameters it is a reliable tool for solving different robust optimization problems which can be nonlinear and discontinuous or have nonconvex pareto frontier for instance the success of the method depends only on the quality of the objective inclusion functions and the precision parameters its disadvantage lies in the fact that it requires high computational effort when high dimensional problems are considered or when a very accurate enclosure is needed two analytical robust test functions are proposed to be treated by i rmoa and to validate the results provided by a stochastic multi objective optimization method the results are satisfactory

applications of second order cone programming in a second order cone program socp a linear function is minimized over the intersection of an affine set and the product of second order quadratic cones socps are nonlinear convex problems that include linear and convex quadratic programs as special cases but are less general than semidefinite programs sdps several efficient primaldual interior point methods for socp have been developed in the last few years after reviewing the basic theory of socps we describe general families of problems that tan be recast as socps these include robust linear programming and robust leastsquares problems problems involving sums or maxima of norms or with convex hyperbolic constraints we discuss a variety of engineering applications such as filter design antenna array weight design truss design and grasping forte optimization in robotics we describe an efficient primaldual interior point method for solving socps which shares many of the features of primaldual interior point methods for linear program

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

multivariate nonnegative quadratic mappings in this paper we study several issues related to the characterization of specific classes of multivariate quadratic mappings that are nonnegative over a given domain with nonnegativity defined by a prespecified conic order in particular we consider the set cone of nonnegative quadratic mappings defined with respect to the positive semidefinite matrix cone and study when it can be represented by linear matrix inequalities we also discuss the applications of the results in robust optimization especially the robust quadratic matrix inequalities and the robust linear programming models in the latter application the implementational errors of the solution are taken into account and the problem is formulated as a semidefinite program

robust solutions to uncertain semidefinite programs in this paper we consider semidefinite programs sdps whose data depend on some unknown but bounded perturbation parameters we seek robust solutions to such programs that is solutions which minimize the worst case objective while satisfying the constraints for every possible value of parameters within the given bounds assuming the data matrices are rational functions of the perturbation parameters we show how to formulate sufficient conditions for a robust solution to exist as sdps when the perturbation is full our conditions are necessary and sufficient in this case we provide sufficient conditions which guarantee that the robust solution is unique and continuous holder stable with respect to the unperturbed problem s data the approach can thus be used to regularize ill conditioned sdps we illustrate our results with examples taken from linear programming maximum norm minimization polynomial interpolation and integer programming

tractable approximations to robust conic optimization problems in earlier proposals the robust counterpart of conic optimization problems exhibits a lateral increase in complexity i e robust linear programming problems lps become second order cone problems socps robust socps become semidefinite programming problems sdps and robust sdps become np hard we propose a relaxed robust counterpart for general conic optimization problems that a preserves the computational tractability of the nominal problem specifically the robust conic optimization problem retains its original structure i e robust lps remain lps robust socps remain socps and robust sdps remain sdps and b allows us to provide a guarantee on the probability that the robust solution is feasible when the uncertain coefficients obey independent and identically distributed normal distributions

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

a novel robust fuzzy stochastic programming for closed loop supply chain network design under hybrid uncertainty abstract in today s business environments the high importance of economic benefits and environmental impacts of using scrapped products has caused most companies to move to designing the closed loop supply chain network this paper considers the closed loop supply chain network design problem under hybrid uncertainty while there are two sources of uncertainty for most parameters thus require fortifying of the robustness of the decision the first source is that some uncertain parameters may be based on the future scenarios which are considered according to the probability of their occurrence the second source is that the values of these parameters in each scenario are usually imprecise and can be specified by possibilistic distributions in this case the best robust decision has some additional properties in terms of mean value and variability of the objective function we introduced two types of the variability named scenario variability and possibilistic variability possibility theory is used to choose a solution in such a problem and a novel robust fuzzy stochastic programming approach is proposed that has significant advantages the performance of the proposed model is also compared with that of other models in term of the mean cost and variability by simulation

method of centers for minimizing generalized eigenvalues abstract we consider the problem of minimizing the largest generalized eigenvalue of a pair of symmetric matrices each of which depends affinely on the decision variables although this problem may appear specialized it is in fact quite general and includes for example all linear quadratic and linear fractional programs many problems arising in control theory can be cast in this form the problem is nondifferentiable but quasiconvex so methods such as kelley s cutting plane algorithm or the ellipsoid algorithm of shor nemirovsky and yudin are guaranteed to minimize it in this paper we describe relevant background material and a simple interior point method that solves such problems more efficiently the algorithm is a variation on huard s method of centers using a self concordant barrier for matrix inequalities developed by nesterov and nemirovsky nesterov and nemirovsky have also extended their potential reduction methods to handle the same problem since the problem is quasiconvex but not convex devising a nonheuristic stopping criterion i e one that guarantees a given accuracy is more difficult than in the convex case we describe several nonheuristic stopping criteria that are based on the dual of a related convex problem and a new ellipsoidal approximation that is slightly sharper in some cases than a more general result to nesterov and nemirovsky the algorithm is demonstrated on an example determining the quadratic lyapunov function that optimizes a decay rate estimate for a differential inclusion

cuts for mixed 0 1 conic programming in this we paper we study techniques for generating valid convex constraints for mixed 0 1 conic programs we show that many of the techniques developed for generating linear cuts for mixed 0 1 linear programs such as the gomory cuts the lift and project cuts and cuts from other hierarchies of tighter relaxations extend in a straightforward manner to mixed 0 1 conic programs we also show that simple extensions of these techniques lead to methods for generating convex quadratic cuts gomory cuts for mixed 0 1 conic programs have interesting implications for comparing the semidefinite programming and the linear programming relaxations of combinatorial optimization problems e g we show that all the subtour elimination inequalities for the traveling salesman problem are rank 1 gomory cuts with respect to a single semidefinite constraint we also include results from our preliminary computational experiments with these cuts

a soft robust model for optimization under ambiguity in this paper we propose a framework for robust optimization that relaxes the standard notion of robustness by allowing the decision maker to vary the protection level in a smooth way across the uncertainty set we apply our approach to the problem of maximizing the expected value of a payoff function when the underlying distribution is ambiguous and therefore robustness is relevant our primary objective is to develop this framework and relate it to the standard notion of robustness which deals with only a single guarantee across one uncertainty set first we show that our approach connects closely to the theory of convex risk measures we show that the complexity of this approach is equivalent to that of solving a small number of standard robust problems we then investigate the conservatism benefits and downside probability guarantees implied by this approach and compare to the standard robust approach finally we illustrate the methodology on an asset allocation example consisting of historical market data over a 25 year investment horizon and find in every case we explore that relaxing standard robustness with soft robustness yields a seemingly favorable risk return trade off each case results in a higher out of sample expected return for a relatively minor degradation of out of sample downside performance

low complexity polytopic invariant sets for linear systems subject to norm bounded uncertainty we propose a novel algorithm to compute low complexity polytopic robust control invariant rci sets along with the corresponding state feedback gain for linear discrete time systems subject to norm bounded uncertainty additive disturbances and state input constraints using a slack variable approach we propose new results to transform the original nonlinear problem into a convex lmi problem whilst introducing only minor conservatism in the formulation through numerical examples we illustrate that the proposed algorithm can yield improved maximal minimal volume rci set approximations in comparison with the schemes given in the literature

an interpretation of membership functions and the properties of general probabilistic operators as fuzzy set operators ii extension to three valued and interval valued fuzzy sets abstract previously in part i of the present paper and its supplement an approach was made to an interpretation of membership functions as probability and a proposal of the probabilistic operators with dependence relation of the sets this paper is an extension to three valued and interval valued fuzzy sets by taking into consideration uncertainty and contradiction in the judgment whether or not an element of universe of discourse belongs to a fuzzy subset upper lower and intermediate grades of membership are defined this definition leads to a notion of three valued fuzzy set and when only uncertainty is considered interval valued fuzzy set results under the proposed set operation rules the probabilistic operators with dependency various basic set operations and their properties are studied of the properties satisfied by the crisp set operators many of them are found to be valid some properties e g excluded middle laws are found to be invalid due to the uncertainty and contradiction involved also discussed are a representation of set operations of type 2 fuzzy sets by the three valued fuzzy set system and the connection of the proposed operators with other typical set operators

operations on type 2 fuzzy sets in this paper we discuss set operations on type 2 fuzzy sets including join and meet under minimum product t norm algebraic operations properties of membership grades of type 2 sets and type 2 relations and their compositions all this is needed to implement a type 2 fuzzy logic system fls

sensitivity analysis of optimization problems under second order regular constraints we present a perturbation theory for finite dimensional optimization problems subject to abstract constraints satisfying a second order regularity condition this is a technical condition that is always satisfied in the case of semi definite optimization we derive lipschitz and holder expansions of approximate optimal solutions under a directional constraint qualification hypothesis and various second order sufficient conditions that take into account the curvature of the set defining the constraints of the problem we show how the theory applies to semi infinite programs in which the contact set is a smooth manifold and the quadratic growth condition in the constraint space holds and discuss the differentiability of metric projections as well as the moreau yosida regularization finally we show how the theory applies to semi definite optimization

from a gaussian mixture model to nonadditive fuzzy systems this work explores how a kind of probabilistic system namely the gaussian mixture model gmm can be translated to an additive fuzzy system we will prove the mathematical equivalence between the conditional mean of a gmm and the defuzzified output of a generalized fuzzy model gfm the relationship between a gmm and a gfm and the conditions for gmm to gfm translation will be made explicit in the form of theorems the work will then extend to special cases of the gfm specifically the mamdani larsen and takagi sugeno fuzzy models the possibility of reverse translation that is from a gfm to a gmm will also be discussed finally we will consider the generality of a gmm specifically how it can approximate other distribution functions

principles for constructing three way approximations of fuzzy sets a comparative evaluation based on unsupervised learning abstract three way approximations of fuzzy sets are an important scheme of granular computing by abstracting a fuzzy set to its discrete three option alternatives which adhere to human cognitive behaviors and reduce the computational burden the key point of such three way approximations of fuzzy sets is how to choose a suitable design leading to their realization undesired three way approximations might be produced if the selected mechanism is unsuitable to data distribution in this study the principles for constructing three way approximations of fuzzy sets are summarized the following taxonomy of these principles is provided namely i uncertainty balance based principle ii prototype based principle and iii model based invoking the tradeoff between classification error and the number of data that have to be classified a number of detailed optimization models are discussed in detail to evaluate the performance of different construction principles a general unsupervised learning framework based on three way approximations of fuzzy sets is exhibited some synthetic data sets along with sixteen data sets from uci repository are involved for experiments friedman testing followed by holm bonferroni testing are exploited to test the performance significance of the proposed criteria which can provide insights and deliver guidance when choosing a principle for constructing three way approximations of fuzzy sets in the real world scenarios the research methods in this paper can also be extended to supervised and semi supervised learning areas

competitive multi period pricing for perishable products a robust optimization approach we study a multi period oligopolistic market for a single perishable product with fixed inventory our goal is to address the competitive aspect of the problem together with demand uncertainty using ideas from robust optimization and variational inequalities the demand function for each seller has some associated uncertainty and we assume that the sellers would like to adopt a policy that is robust to adverse uncertain circumstances we believe this is the first paper that uses robust optimization for dynamic pricing under competition in particular starting with a given fixed inventory each seller competes over a multi period time horizon in the market by setting prices and protection levels for each period at the beginning of the time horizon any unsold inventory at the end of the horizon is worthless the sellers do not have the option of periodically reviewing and replenishing their inventory we study non cooperative nash equilibrium policies for sellers under such a model this kind of a setup can be used to model pricing of air fares hotel reservations bandwidth in communication networks etc in this paper we demonstrate our results through some numerical examples

extended matrix cube theorems with applications to µ theory in control in this paper we study semi infinite systems of linear matrix inequalities which are generically np hard for these systems we introduce computationally tractable approximations and derive quantitative guarantees of their quality as applications we discuss the problem of maximizing a hermitian quadratic form over the complex unit cube and the problem of bounding the complex structured singular value with the help of our complex matrix cube theorem we demonstrate that the standard scaling upper bound on m is a tight upper bound on the largest level of structured perturbations of the matrixm for which all perturbed matrices share a common lyapunov certificate for the discrete time stability

an optimized sparse approximate matrix multiply for matrices with decay we present an optimized single precision implementation of the sparse approximate matrix multiply spamm m challacombe and n bock arxiv 1011 3534 2010 a fast algorithm for matrix matrix multiplication for matrices with decay that achieves an mathcal o n log n computational complexity with respect to matrix dimension n we find that the max norm of the error achieved with a spamm tolerance below 2 times 10 8 is lower than that of the single precision general matrix matrix multiply sgemm for dense quantum chemical matrices while outperforming sgemm with a cross over already for small matrices n sim 1000 relative to naive implementations of spamm using intel s math kernel library or amd s core math library our optimized version is found to be significantly faster detailed performance comparisons are made for quantum chemical matrices with differently structured sub blocks finally we discuss the potential of improved hardware prefetch to yield 2x to 3x speedups

verifying exactness of relaxations for robust semi definite programs by solving polynomial systems abstract in this paper robust semi definite programs are considered with the goal of verifying whether a particular lmi relaxation is exact a procedure is presented showing that verifying exactness amounts to solving a polynomial system the main contribution of the paper is a new algorithm to compute all isolated solutions of a system of polynomials standard techniques in computational algebra often referred to as stetter s method h j stetter numerical polynomial algebra siam 2004 involve the computation of a grobner basis of the ideal generated by the polynomials and further require joint eigenvector computations in order to arrive at the zeros of the polynomial system our algorithm does neither require structural knowledge on the polynomial system nor does it rely on the computation of joint eigenvectors

optimal design for multi item auctions a robust optimization approach in this paper we revisit the auction design problem for multi item auctions with budget constrained buyers by introducing a robust optimization approach to model a concepts such as incentive compatibility and individual rationality that are naturally expressed in the language of robust optimization and b the auctioneer s beliefs on the buyers valuations of the items rather than using probability distributions the classical probabilistic approach or an adversarial model to model valuations we introduce an uncertainty set based model for these valuations we construct these uncertainty sets to incorporate historical information available to the auctioneer in a way that is consistent with limit theorems of probability theory or knowledge of the probability distribution in this setting we formulate the auction design problem as a robust optimization problem and provide a characterization of the optimal solution as an auction with reservation prices thus extending the work of myerson myerson rb

fuzzy capital rationing model in this paper we study the fuzzification of weingartner s pure capital rationing model and its analysis we develop a primal dual pair based on t norm t conorm relation for the constraints and objective function for a fully fuzzified pure capital rationing problem except project selection variables we define the a interval under which the weak duality is proved we perform sensitivity analysis for a change in a budget level or in a cash flow level of a non basic as well as a basic variable we analyze the problem based on duality and complementary slackness results we illustrate the proposed model by computational analysis and interpret the results

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

on tractable approximations of uncertain linear matrix inequalities affected by interval uncertainty we present efficiently verifiable sufficient conditions for the validity of specific np hard semi infinite systems of linear matrix inequalities lmis arising from lmis with uncertain data and demonstrate that these conditions are tight up to an absolute constant factor in particular we prove that given an n times n interval matrix cal u rho a mid a ij a ij leq rho c ij one can build a computable lower bound accurate within the factor pi over 2 on the supremum of those rho for which all instances of cal u rho share a common quadratic lyapunov function we then obtain a similar result for the problem of quadratic lyapunov stability synthesis finally we apply our techniques to the problem of maximizing a homogeneous polynomial of degree 3 over the unit cube

selected topics in robust convex optimization robust optimization is a rapidly developing methodology for handling optimization problems affected by non stochastic uncertain but bounded data perturbations in this paper we overview several selected topics in this popular area specifically 1 recent extensions of the basic concept of robust counterpart of an optimization problem with uncertain data 2 tractability of robust counterparts 3 links between ro and traditional chance constrained settings of problems with stochastic data and 4 a novel generic application of the ro methodology in robust linear control

symmetric hankel operators minimal norm extensions and eigenstructures abstract the minimal norm extension problem for real partial hankel matrices is studied let x i i n 1 n be given real numbers find x i i n such that the finite hankel matrix a h x x 1 x 2 x n x 2 x 3 0 x n 0 0 has lowest possible norm as an operator on the eucledian space r n this min max problem is reduced to an unconstrained maximization problem it is close to a nonlinear eigenvalue problem the results suggest a new class of computer algorithms

robust convex quadratically constrained programs in this paper we study robust convex quadratically constrained programs a subset of the class of robust convex programs introduced by ben tal and nemirovski 4 in contrast to 4 where it is shown that such robust problems can be formulated as semidefinite programs our focus in this paper is to identify uncertainty sets that allow this class of problems to be formulated as second order cone programs socp we propose three classes of uncertainty sets for which the robust problem can be reformulated as an explicit socp and present examples where these classes of uncertainty sets are natural

supplement to an interpretation of membership functions and the properties of general probabilistic operators as fuzzy set operators part i abstract basic idea of part i of an interpretation of membership functions and the properties of general probabilistic operators as fuzzy set operators 1992 mainly on the conceptual construction of membership functions is straightened out and illustrated by detailed examples the purpose of this article is to clarify unclear points in notations and concepts of mabuchi 1992 as a step toward the extension of the theory from type 1 to three valued fuzzy sets

on the road between robust optimization and the scenario approach for chance constrained optimization problems we propose a new method for solving chance constrained optimization problems that lies between robust optimization and scenario based methods our method does not require prior knowledge of the underlying probability distribution as in robust optimization methods nor is it based entirely on randomization as in the scenario approach it instead involves solving a robust optimization problem with bounded uncertainty where the uncertainty bounds are randomized and are computed using the scenario approach to guarantee that the resulting robust problem is solvable we impose certain assumptions on the dependency of the constraint functions with respect to the uncertainty and show that tractability is ensured for a wide class of systems our results lead immediately to guidelines under which the proposed methodology or the scenario approach is preferable in terms of providing less conservative guarantees or reducing the computational cost