on reduced semidefinite programs for second order moment bounds with applications we show that the complexity of computing the second order moment bound on the expected optimal value of a mixed integer linear program with a random objective coefficient vector is closely related to the complexity of characterizing the convex hull of the points 1 atopwithdelims varvec x 1 atopwithdelims varvec x varvec x in mathcal x 1x1x x x where mathcal x x is the feasible region in fact we can replace the completely positive programming formulation for the moment bound on mathcal x x with an associated semidefinite program provided we have a linear or a semidefinite representation of this convex hull as an application of the result we identify a new polynomial time solvable semidefinite relaxation of the distributionally robust multi item newsvendor problem by exploiting results from the boolean quadric polytope for mathcal x x described explicitly by a finite set of points our formulation leads to a reduction in the size of the semidefinite program we illustrate the usefulness of the reduced semidefinite programming bounds in estimating the expected range of random variables with two applications arising in random walks and best worst choice models

probabilistic combinatorial optimization moments semidefinite programming and asymptotic bounds we address the problem of evaluating the expected optimal objective value of a 0 1 optimization problem under uncertainty in the objective coefficients the probabilistic model we consider prescribes limited marginal distribution information for the objective coefficients in the form of moments we show that for a fairly general class of marginal information a tight upper lower bound on the expected optimal objective value of a 0 1 maximization minimization problem can be computed in polynomial time if the corresponding deterministic problem is solvable in polynomial time we provide an efficiently solvable semidefinite programming formulation to compute this tight bound we also analyze the asymptotic behavior of a general class of combinatorial problems that includes the linear assignment spanning tree and traveling salesman problems under knowledge of complete marginal distributions with and without independence we calculate the limiting constants exactly

an iterated radau method for time dependent pdes this paper is concerned with the time integration of semi discretized multi dimensional pdes of advection diffusion reaction type to cope with the stiffness of these odes an implicit method has been selected viz the two stage third order radau iia method the main topic of this paper is the efficient solution of the resulting implicit relations first a modified newton process has been transformed into an iteration process in which the 2 stages are decoupled and moreover can exploit the same lu factorization of the iteration matrix next we apply a so called approximate matrix factorization amf technique to solve the linear systems in each newton iteration this amf approach is very efficient since it reduces the multi dimensional system to a series of one dimensional systems the total amount of linear algebra work involved is reduced enormously by this approach the idea of applying amf to two dimensional problems is quite old and goes back to peaceman and rachford in the early fifties the situation in three space dimensions is less favourable and will be analyzed here in more detail both theoretically and experimentally furthermore we analyze a variant in which the amf technique has been used to really solve until convergence the underlying radau iia method so that we can rely on its excellent stability and accuracy characteristics finally the method has been tested on several examples also a comparison has been made with the existing codes vodpk and imexrkc and the efficiency cpu time versus accuracy is shown to be at least competitive with the efficiency of these solvers

persistence in discrete optimization under data uncertainty an important question in discrete optimization under uncertainty is to understand the persistency of a decision variable i e the probability that it is part of an optimal solution for instance in project management when the task activity times are random the challenge is to determine a set of critical activities that will potentially lie on the longest path in the spanning tree and shortest path network problems when the arc lengths are random the challenge is to pre process the network and determine a smaller set of arcs that will most probably be a part of the optimal solution under different realizations of the arc lengths building on a characterization of moment cones for single variate problems and its associated semidefinite constraint representation we develop a limited marginal moment model to compute the persistency of a decision variable under this model we show that finding the persistency is tractable for zero one optimization problems with a polynomial sized representation of the convex hull of the feasible region through extensive experiments we show that the persistency computed under the limited marginal moment model is often close to the simulated persistency value under various distributions that satisfy the prescribed marginal moments and are generated independently

irkc an imex solver for stiff diffusion reaction pdes the fortran 90 code irkc is intended for the time integration of systems of partial differential equations pdes of diffusion reaction type for which the reaction jacobian has real negative eigenvalues it is based on a family of implicit explicit runge kutta chebyshev methods which are unconditionally stable for reaction terms and which impose a stability constraint associated with the diffusion terms that is quadratic in the number of stages special properties of the family make it possible for the code to select at each step the most efficient stable method as well as the most efficient step size moreover they make it possible to apply the methods using just a few vectors of storage a further step towards minimal storage requirements and optimal efficiency is achieved by exploiting the fact that the implicit terms originating from the stiff reactions are not coupled over the spatial grid points hence the systems to be solved have a small dimension viz equal to the number of pdes these characteristics of the code make it especially attractive for problems in several spatial variables irkc is a successor to the rkc code b p sommeijer l f shampine j g verwer rkc an explicit solver for parabolic pdes j comput appl math 88 1997 315 326 that solves similar problems without stiff reaction terms

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 theoretical and empirical aspects of marginal distribution choice models in this paper we study the properties of a recently proposed class of semiparametric discrete choice models referred to as the marginal distribution model mdm by optimizing over a family of joint error distributions with prescribed marginal distributions surprisingly the choice probabilities arising from the family of generalized extreme value models of which the multinomial logit model is a special case can be obtained from this approach despite the difference in assumptions on the underlying probability distributions we use this connection to develop flexible and general choice models to incorporate consumer and product level heterogeneity in both partworths and scale parameters in the choice model furthermore the extremal distributions obtained from the mdm can be used to approximate the fisher s information matrix to obtain reliable standard error estimates of the partworth parameters without having to bootstrap the method we use simulated and empirical data sets to test the performance of this approach we evaluate the performance against the classical multinomial logit mixed logit and a machine learning approach that accounts for partworth heterogeneity our numerical results indicate that mdm provides a practical semiparametric alternative to choice modeling this paper was accepted by eric bradlow special issue on business analytics

error estimators for stiff differential equations abstract when solving ordinary differential equations numerically the local error is estimated at each step in the classical situation of small step sizes it is clear what is required of the error estimators stiff problems are solved with large step sizes the quality of error estimators is studied in this situatuion and it is shown how to modify unsatisfactory estimators so as to improve them greatly several formulas from the literature are treated as examples

quadratically constrained attitude control via semidefinite programming we consider the attitude control problem augmented with an arbitrary number of nonconvex quadratic constraints by utilizing the implicit magnitude constraint of the state vector i e quaternion we are able to reduce this problem to a convex quadratically constrained quadratic program or a semidefinite program this result leads to an efficient solution strategy for general quadratically constrained attitude control problems an example demonstrates the proposed methodology

a class of high order runge kutta chebyshev stability polynomials the analytic form of a new class of factorized runge kutta chebyshev frkc stability polynomials of arbitrary order n is presented roots of frkc stability polynomials of degree l m n are used to construct explicit schemes comprising l forward euler stages with internal stability ensured through a sequencing algorithm which limits the internal amplification factors to l 2 the associated stability domain scales as m 2 along the real axis marginally stable real valued points on the interior of the stability domain are removed via a prescribed damping procedure by construction frkc schemes meet all linear order conditions for nonlinear problems at orders above 2 complex splitting or butcher series composition methods are required linear order conditions of the frkc stability polynomials are verified at orders 2 4 and 6 in numerical experiments comparative studies with existing methods show the second order unsplit frkc2 scheme and higher order 4 and 6 split frkcs schemes are efficient for large moderately stiff problems

models for minimax stochastic linear optimization problems with risk aversion we propose a semidefinite optimization sdp model for the class of minimax two stage stochastic linear optimization problems with risk aversion the distribution of second stage random variables belongs to a set of multivariate distributions with known first and second moments for the minimax stochastic problem with random objective we provide a tight sdp formulation the problem with random right hand side is np hard in general in a special case the problem can be solved in polynomial time explicit constructions of the worst case distributions are provided applications in a productiontransportation problem and a single facility minimax distance problem are provided to demonstrate our approach in our experiments the performance of minimax solutions is close to that of data driven solutions under the multivariate normal distribution and better under extremal distributions the minimax solutions thus guarantee to hedge against these worst possible distributions and provide a natural distribution to stress test stochastic optimization problems under distributional ambiguity

the lr cholesky algorithm for symmetric hierarchical matrices abstract we investigate the application of the lr cholesky algorithm to symmetric hierarchical matrices symmetric simple structured hierarchical matrices and symmetric hierarchically semiseparable hss matrices the data sparsity of these matrices make the otherwise expensive lr cholesky algorithm applicable as long as the data sparsity is preserved we will see in an example that the ranks of the low rank blocks grow and the data sparsity gets lost we will explain this behavior by applying a theorem on the structure preservation of diagonal plus semiseparable matrices under lr cholesky transformations therefore we have to give a new more constructive proof for the theorem we will show that the structure of h l matrices is almost preserved and so the lr cholesky algorithm is of almost quadratic complexity for h l matrices

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

computing bounds for stochastic programming problems by means of a generalized moment problem bounds on the expected value of a convex function are obtained by means of an approximating generalized moment problem numerical implementation is discussed in the context of stochastic programming problems

two classes of implicit explicit multistep methods for nonlinear stiff initial value problems the initial value problems of nonlinear ordinary differential equations which contain stiff and nonstiff terms often arise from many applications in order to reduce the computation cost implicit explicit imex methods are often applied to these problems i e the stiff and non stiff terms are discretized by using implicit and explicit methods respectively in this paper we mainly consider the nonlinear stiff initial value problems satisfying the one sided lipschitz condition and a class of singularly perturbed initial value problems and present two classes of the imex multistep methods by combining implicit one leg methods with explicit linear multistep methods and explicit one leg methods respectively the order conditions and the convergence results of these methods are obtained some efficient methods are constructed some numerical examples are given to verify the validity of the obtained theoretical results

distributionally robust optimization with polynomial densities theory models and algorithms in distributionally robust optimization the probability distribution of the uncertain problem parameters is itself uncertain and a fictitious adversary e g nature chooses the worst distribution from within a known ambiguity set a common shortcoming of most existing distributionally robust optimization models is that their ambiguity sets contain pathological discrete distributions that give nature too much freedom to inflict damage we thus introduce a new class of ambiguity sets that contain only distributions with sum of squares sos polynomial density functions of known degrees we show that these ambiguity sets are highly expressive as they conveniently accommodate distributional information about higher order moments conditional probabilities conditional moments or marginal distributions exploiting the theoretical properties of a measure based hierarchy for polynomial optimization due to lasserre siam j optim 21 3 864 885 2011 we prove that certain worst case expectation constraints are polynomial time solvable under these new ambiguity sets we also show how sos densities can be used to approximately solve the general problem of moments we showcase the applicability of the proposed approach in the context of a stylized portfolio optimization problem and a risk aggregation problem of an insurance company

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

an implicit explicit runge kutta chebyshev scheme for diffusion reaction equations an implicit explicit imex extension of the explicit runge kutta chebyshev rkc scheme designed for parabolic pdes is proposed for diffusion reaction problems with severely stiff reaction terms the imex scheme treats these reaction terms implicitly and diffusion terms explicitly within the setting of linear stability theory the new imex scheme is unconditionally stable for reaction terms having a jacobian matrix with a real spectrum for diffusion terms the stability characteristics remain unchanged a numerical comparison for a stiff nonlinear radiation diffusion problem between an rkc solver an imex rkc solver and the popular implicit bdf solver vodpk using the krylov solver gmres illustrates the excellent performance of the new scheme

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

on constraint sampling in the linear programming approach to approximate dynamic programming in the linear programming approach to approximate dynamic programming one tries to solve a certain linear program the alp that has a relatively small numberk of variables but an intractable numberm of constraints in this paper we study a scheme that samples and imposes a subset ofm

optimization with reference based robust preference constraints we propose an optimization model using a novel robust preference relationship reference based almost stochastic dominance rsd the concept of rsd addresses the two problems in utility based dec

stochastic optimization problems with incomplete information on distribution functions the main purpose of this paper is to discuss numerical optimization procedures based on duality theory for stochastic extremal problems in which the distribution function is only partially known we formulate such problems as minimax problems in which the inner problem involves optimization with respect to probability measures the latter problem is solved using generalized linear programming techniques then we state the dual problem to the initial stochastic optimization problem numerical procedures that avoid the difficulties associated with solving the inner problem are proposed

robust portfolio selection problems in this paper we show how to formulate and solve robust portfolio selection problems the objective of these robust formulations is to systematically combat the sensitivity of the optimal portfolio to statistical and modeling errors in the estimates of the relevant market parameters we introduce uncertainty structures for the market parameters and show that the robust portfolio selection problems corresponding to these uncertainty structures can be reformulated as second order cone programs and therefore the computational effort required to solve them is comparable to that required for solving convex quadratic programs moreover we show that these uncertainty structures correspond to confidence regions associated with the statistical procedures employed to estimate the market parameters finally we demonstrate a simple recipe for efficiently computing robust portfolios given raw market data and a desired level of confidence

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

mixed 0 1 linear programs under objective uncertainty a completely positive representation in this paper we analyze mixed 0 1 linear programs under objective uncertainty the mean vector and the second moment matrix of the nonnegative objective coefficients are assumed to be known but the exact form of the distribution is unknown our main result shows that computing a tight upper bound on the expected value of a mixed 0 1 linear program in maximization form with random objective is a completely positive program this naturally leads to semidefinite programming relaxations that are solvable in polynomial time but provide weaker bounds the result can be extended to deal with uncertainty in the moments and more complicated objective functions examples from order statistics and project networks highlight the applications of the model our belief is that the model will open an interesting direction for future research in discrete and linear optimization under uncertainty

numerical approximation of turing patterns in electrodeposition by adi methods in this paper we study the numerical approximation of turing patterns corresponding to steady state solutions of a pde system of reaction diffusion equations modeling an electrodeposition process we apply the method of lines mol and describe the semi discretization by high order finite differences in space given by the extended central difference formulas ecdfs that approximate neumann boundary conditions bcs with the same accuracy we introduce a test equation to describe the interplay between the diffusion and the reaction time scales we present a stability analysis of a selection of time integrators imex 2 sbdf method crank nicolson cn alternating direction implicit adi method for the test equation as well as for the schnakenberg model prototype of nonlinear reaction diffusion systems with turing patterns eventually we apply the adi ecdf schemes to solve the electrodeposition model until the stationary patterns spots worms and only spots are reached we validate the model by comparison with experiments on cu film growth by electrodeposition

worst case distribution analysis of stochastic programs we show that for even quasi concave objective functions the worst case distribution with respect to a family of unimodal distributions of a stochastic programming problem is a uniform distribution this extends the so called uniformity principle of barmish and lagoa 1997 where the objective function is the indicator function of a convex symmetric set

complexity analysis of the analytic center cutting plane method that uses multiple cuts we analyze the complexity of the analytic center cutting plane or column generation algorithm for solving general convex problems defined by a separation oracle the oracle is called at the analytic center of a polytope which contains a solution set and is given by the intersection of the linear inequalities previously generated from the oracle if the center is not in the solution set separating hyperplanes will be placed through the center to shrink the containing polytope while the complexity result has been recently established for the algorithm when one cutting plane is placed in each iteration the result remains open when multiple cuts are added moreover adding multiple cuts actually is a key to practical effectiveness in solving many problems and it presents theoretical difficulties in analyzing cutting plane methods in this paper we show that the analytic center cutting plane algorithm with multiple cuts added in each iteration still is a fully polynomial approximation algorithm

robust optimization of sums of piecewise linear functions with application to inventory problems robust optimization is a methodology that has gained a lot of attention in the recent years this is mainly due to the simplicity of the modeling process and ease of resolution even for large scale models unfortunately the second property is usually lost when the cost function that needs to be robustified is not concave or linear with respect to the perturbing parameters in this paper we study robust optimization of sums of piecewise linear functions over polyhedral uncertainty set given that these problems are known to be intractable we propose a new scheme for constructing conservative approximations based on the relaxation of an embedded mixed integer linear program and relate this scheme to methods that are based on exploiting affine decision rules our new scheme gives rise to two tractable models that respectively take the shape of a linear program and a semidefinite program with the latter having the potential to provide solutions of better quality than the former at the price of heavie

amf runge kutta formulas and error estimates for the time integration of advection diffusion reaction pdes the convergence of a family of amf runge kutta methods in short amf rk for the time integration of evolutionary partial differential equations pdes of advection diffusion reaction type semi discretized in space is considered the methods are based on very few inexact newton iterations applied to implicit runge kutta formulas by combining the use of a natural splitting for the underlying jacobians and the approximate matrix factorization amf technique this approach allows a very cheap implementation of the runge kutta formula under consideration particular amf rk methods based on radau iia formulas are considered these methods have given very competitive results when compared with important formulas in the literature for multidimensional systems of non linear parabolic pde problems uniform bounds for the global time space errors on semi linear pdes when simultaneously the time step size and the spatial grid resolution tend to zero are derived numerical illustrations supporting the theory are presented

persistency model and its applications in choice modeling given a discrete maximization problem with a linear objective function where the coefficients are chosen randomly from a distribution we would like to evaluate the expected optimal value and the marginal distribution of the optimal solution we call this the persistency problem for a discrete optimization problem under uncertain objective and the marginal probability mass function of the optimal solution is named the persistence value in general this is a difficult problem to solve even if the distribution of the objective coefficient is well specified in this paper we solve a subclass of this problem when the distribution is assumed to belong to the class of distributions defined by given marginal distributions or given marginal moment conditions under this model we show that the persistency problem maximizing the expected objective value over the set of distributions can be solved via a concave maximization model the persistency model solved using this formulation can be used to obtain important qualitative insights to the behavior of stochastic discrete optimization problems we demonstrate how the approach can be used to obtain insights to problems in discrete choice modeling using a set of survey data from a transport choice modeling study we calibrate the random utility model with choice probabilities obtained from the persistency model numerical results suggest that our persistency model is capable of obtaining estimates that perform as well if not better than classical methods such as logit and cross nested logit models we can also use the persistency model to obtain choice probability estimates for more complex choice problems we illustrate this on a stochastic knapsack problem which is essentially a discrete choice problem under budget constraint

robust and stochastically weighted multiobjective optimization models and reformulations we introduce and study a family of models for multiexpert multiobjective criteria decision making these models use a concept of weight robustness to generate a risk averse decision in particular the multiexpert multicriteria robust weighted sum approach mcrow introduced in this paper identifies a robust pareto decision that minimizes the worst case weighted sum of objectives over a given weight region the corresponding objective value called the robust value of a decision is shown to be increasing and concave in the weight set we study compact reformulations of the mcrow model with polyhedral and conic descriptions of the weight regions the mcrow model is developed further for stochastic multiexpert multicriteria decision making by allowing ambiguity or randomness in the weight region as well as the objective functions the properties of the proposed approach are illustrated with a few textbook examples the usefulness of the stochastic mcrow model is demonstrated using a disaster planning example and an agriculture revenue management example

robust filtering for discrete time systems with bounded noise and parametric uncertainty this note presents a new approach to finite horizon guaranteed state prediction for discrete time systems affected by bounded noise and unknown but bounded parameter uncertainty our framework handles possibly nonlinear dependence of the state space matrices on the uncertain parameters the main result is that a minimal confidence ellipsoid for the state consistent with the measured output and the uncertainty description may be recursively computed in polynomial time using interior point methods for convex optimization with n states l uncertain parameters appearing linearly in the state space matrices with rank one matrix coefficients the worst case complexity grows as o l n l sup 3 5 with unstructured uncertainty in all system matrices the worst case complexity reduces to o n sup 3 5

solving convex programs by random walks minimizing a convex function over a convex set in n dimensional space is a basic general problem with many interesting special cases here we present a simple new algorithm for convex optimization based on sampling by a random walk it extends naturally to minimizing quasi convex functions and to other generalizations

computing all or some eigenvalues of symmetric mathcal h _ ell matrices we use a bisection method b parlett the symmetric eigenvalue problem prentice hall englewood cliffs nj 1980 p 51 to compute the eigenvalues of a symmetric mathcal h ell matrix m the number of negative eigenvalues of m mu i is computed via the ldl t factorization of m mu i for dense matrices the ldl t factorization is too expensive to yield an efficient eigenvalue algorithm in general but not so for mathcal h ell matrices in the special structure of mathcal h ell matrices there is an ldl t factorization with linear polylogarithmic complexity the bisection method requires only matrix size independent many iterations to find an eigenvalue up to the desired accuracy so that an eigenvalue can be found in linear polylogarithmic time for all n eigenvalues mathcal o n 2 log n 4 log vert m vert 2 epsilon text ev flops are needed to compute all eigenvalues with an accuracy epsilon text ev it is also possible to compute only eigenvalues in a specific interval or the j th smallest one numerical experiments demonstrate the efficiency of the algorithm in particular for the case when some interior eigenvalues are required a corrected pdf is attached to this article

parallelizing the qr algorithm for the unsymmetric algebraic eigenvalue problem myths and reality over the last few years it has been suggested that the popular qr algorithm for the unsymmetric schur decomposition does not parallelize in this paper we present both positive and negative results on this subject in theory asymptotically perfect speedup can be obtained in practice reasonable speedup can be obtained on an mimd distributed memory computer for a relatively small number of processors however we also show theoretically that it is impossible for the standard qr algorithm to be scalable performance of a parallel implementation of the lapack dlahqr routine on the intel text paragon text tm system is reported

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

choice prediction with semidefinite optimization when utilities are correlated we consider the problem of making choice prediction by optimizing the expectation of maximum utility in discrete choice situations and propose a discrete choice model which generates choice probabilities through a semidefinite program the choice model termed as the cross moment model cmm is parsimonious in that it uses only the mean and covariance information of the utilities it is encouraging that cmm generates reasonable choice estimates when utilities are correlated even though no distributional assumptions on random utilities are made we present a few examples in route choice setting and random walk to test the quality of choice prediction using cmm by capturing correlations among utilities cmm avoids some of the common behavioral limitations such as the independence of irrelevant alternatives and the invariate proportion of substitution present in several discrete choice models being a convex optimization problem it obviates the use of exhaustive simulation in computing choice probabilities for which the multinomial probit mnp model is often criticized cmm can be easily used in design problems such as product line selection and assortment planning we exploit cmm to solve a flexible packaging design problem faced by online retailers and warehouses we use the data provided by a local service part supplier for this design problem and compare the results with multinomial logit and mnp models we find that cmm not only captures utility correlations in this problem and provides good designs but also has computational advantages over mnp which uses simulation

a survey of the s lemma in this survey we review the many faces of the s lemma a result about the correctness of the s procedure the basic idea of this widely used method came from control theory but it has important consequences in quadratic and semidefinite optimization convex geometry and linear algebra as well these were all active research areas but as there was little interaction between researchers in these different areas their results remained mainly isolated here we give a unified analysis of the theory by providing three different proofs for the s lemma and revealing hidden connections with various areas of mathematics we prove some new duality results and present applications from control theory error estimation and computational geometry

least squares approximation to the distribution of project completion times with gaussian uncertainty this paper is motivated by the following question how to construct good approximation for the distribution of the solution value to linear optimization problem when the random objective coefficients follow a multivariate normal distribution using stein s identity we show that the least squares normal approximation of the random optimal value can be computed by estimating the persistency values of the corresponding optimization problem we further extend our method to construct a least squares quadratic estimator to improve the accuracy of the approximation in particular to capture the skewness of the objective computational studies show that the new approach provides more accurate estimates of the distributions of project completion times compared to existing methods

a convex optimization approach for computing correlated choice probabilities with many alternatives a popular discrete choice model that incorporates correlation information is the multinomial probit mnp model where the random utilities of the alternatives are chosen from a multivariate normal distribution computing the choice probabilities is challenging in the mnp model when the number of alternatives is large mishra et al ieee transactions on automatic control 2012 have proposed a semidefinite optimization approach to compute choice probabilities for the distribution of the random utilities that maximizes expected agent utility given only the mean variance and covariance information their model is referred to as the cross moment cmm model computing the choice probabilities with many alternatives is challenging in the cmm model since one needs to solve large scale semidefinite programs we develop a simpler formulation as a representative agent model by maximizing over the choice probabilities in the unit simplex where the objective function is the sum of the expected utilities and a strongly concave perturbation function by characterizing the perturbation function for the cmm model and its gradient we develop a simple first order gradient method with inexact line search to compute choice probabilities we establish local linear convergence of this algorithm under mild assumptions on the choice probabilities an implication of our results is that inverting the choice probabilities to compute the mean utilities is straightforward given any positive definite covariance matrix numerical experiments show that this method can compute choice probabilities for a large number of alternatives within a reasonable amount of time while explicitly capturing the correlation information comparisons with simulation methods for mnp and semidefinite programming methods for cmm indicate the efficacy of the method

worst case conditional value at risk with application to robust portfolio management this paper considers the worst case conditional value at risk cvar in the situation where only partial information on the underlying probability distribution is available the minimization of the worst case cvar under mixture distribution uncertainty box uncertainty and ellipsoidal uncertainty are investigated the application of the worst case cvar to robust portfolio optimization is proposed and the corresponding problems are cast as linear programs and second order cone programs that can be solved efficiently market data simulation and monte carlo simulation examples are presented to illustrate the proposed approach

solution of the k th eigenvalue problem in large scale electronic structure calculations abstract we consider computing the k th eigenvalue and its corresponding eigenvector of a generalized hermitian eigenvalue problem of n n large sparse matrices in electronic structure calculations several properties of materials such as those of optoelectronic device materials are governed by the eigenpair with a material specific index k we present a three stage algorithm for computing the k th eigenpair with validation of its index in the first stage of the algorithm we propose an efficient way of finding an interval containing the k th eigenvalue 1 k n with a non standard application of the lanczos method in the second stage spectral bisection for large scale problems is realized using a sparse direct linear solver to narrow down the interval of the k th eigenvalue in the third stage we switch to a modified shift and invert lanczos method to reduce bisection iterations and compute the k th eigenpair with validation numerical results with problem sizes up to 1 5 million are reported and the results demonstrate the accuracy and efficiency of the three stage algorithm

on stabilized integration for time dependent pdes an integration method is discussed which has been designed to treat parabolic and hyperbolic terms explicitly and stiff reaction terms implicitly the method is a special two step form of the one step imex implicit explicit rkc runge kutta chebyshev method the special two step form is introduced with the aim of getting a non zero imaginary stability boundary which is zero for the one step method having a non zero imaginary stability boundary allows for example the integration of pure advection equations space discretized with centered schemes the integration of damped or viscous wave equations the integration of coupled sound and heat flow equations etc for our class of methods it also simplifies the choice of temporal step sizes satisfying the von neumann stability criterion by embedding a thin long rectangle inside the stability region embedding rectangles or other tractable domains with this purpose is an idea of wesseling

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

a column generation algorithm for choice based network revenue management during the past few years there has been a trend to enrich traditional revenue management models built upon the independent demand paradigm by accounting for customer choice behavior this extension involves both modeling and computational challenges one way to describe choice behavior is to assume that each customer belongs to a segment which is characterized by a consideration set i e a subset of the products provided by the firm that a customer views as options customers choose a particular product according to a multinomial logit criterion a model widely used in the marketing literature in this paper we consider the choice based deterministic linear programming model cdlp of gallego et al 2004 gallego g g iyengar r phillips a dubey 2004 managing flexible products on a network technical report corc tr 2004 01 department of industrial engineering and operations research columbia university new york and the follow up dynamic programming decomposition heuristic of van ryzin and liu 2008 van ryzin g j q liu 2008 on the choice based linear programming model for network revenue management manufacturing service oper management10 2 288 310 we focus on the more general version of these models where customers belong to overlapping segments to solve the cdlp for real size networks we need to develop a column generation algorithm we prove that the associated column generation subproblem is indeed np hard and propose a simple greedy heuristic to overcome the complexity of an exact algorithm our computational results show that the heuristic is quite effective and that the overall approach leads to high quality practical solutions

on the computation of weighted analytic centers and dual ellipsoids with the projective algorithm the primal projective algorithm for linear programs with unknown optimal objective function value is extended to the case where one uses a weighted karmarkar potential function this potential is defined with respect to a strict lower bound to the optimum the minimization of this potential when the lower bound is kept fixed yields a primal and a dual feasible solution the dual solution is the weighted analytic center of a certain dual polytope finally one exhibits a pair of homothetic dual ellipsoids that extends results obtained by sonnevend todd ye freund and anstreicher

robust mean covariance solutions for stochastic optimization we provide a method for deriving robust solutions to certain stochastic optimization problems based on mean covariance information about the distributions underlying the uncertain vector of returns we prove that for a general class of objective functions the robust solutions amount to solving a certain deterministic parametric quadratic program we first prove a general projection property for multivariate distributions with given means and covariances which reduces our problem to optimizing a univariate mean variance robust objective this allows us to use known univariate results in the multidimensional setting and to add new results in this direction in particular we characterize a general class of objective functions the so called one or two point support functions for which the robust objective is reduced to a deterministic optimization problem in one variable finally we adapt a result from geoffrion 1967a to reduce the main problem to a parametric quadratic program in particular our results are true for increasing concave utilities with convex or concave convex derivatives closed form solutions are obtained for special discontinuous criteria motivated by bonus and commission based incentive schemes for portfolio management we also investigate a multiproduct pricing application which motivates extensions of our results for the case of nonnegative and decision dependent returns

a cutting surface algorithm for semi infinite convex programming with an application to moment robust optimization we present and analyze a central cutting surface algorithm for general semi infinite convex optimization problems and use it to develop a novel algorithm for distributionally robust optimization problems in which the uncertainty set consists of probability distributions with given bounds on their moments moments of arbitrary order as well as nonpolynomial moments can be included in the formulation we show that this gives rise to a hierarchy of optimization problems with decreasing levels of risk aversion with classic robust optimization at one end of the spectrum and stochastic programming at the other although our primary motivation is to solve distributionally robust optimization problems with moment uncertainty the cutting surface method for general semi infinite convex programs is also of independent interest the proposed method is applicable to problems with nondifferentiable semi infinite constraints indexed by an infinite dimensional index set examples comparing the cutting surface algorit

a second order exponential time differencing scheme for non linear reaction diffusion systems with dimensional splitting abstract a second order l stable exponential time differencing etd method is developed by combining an etd scheme with approximating the matrix exponentials by rational functions having real distinct poles rdp together with a dimensional splitting integrating factor technique a variety of non linear reaction diffusion equations in two and three dimensions with either dirichlet neumann or periodic boundary conditions are solved with this scheme and shown to outperform a variety of other second order implicit explicit schemes an additional performance boost is gained through further use of basic parallelization techniques

rkc an explicit solver for parabolic pdes the fortran program rkc is intended for the time integration of parabolic partial differential equations discretized by the method of lines it is based on a family of runge kutta chebyshev formulas with a stability bound that is quadratic in the number of stages remarkable properties of the family make it possible for the program to select at each step the most efficient stable formula as well as the most efficient step size moreover they make it possible to evaluate the explicit formulas in just a few vectors of storage these characteristics of the program make it especially attractive for problems in several spatial variables rkc is compared to the bdf solver vodpk on two test problems in three spatial variables