robust two stage stochastic linear optimization with risk aversion abstract we study a two stage stochastic linear optimization problem where the recourse function is risk averse rather than risk neutral in particular we consider the mean conditional value at risk objective function in the second stage the model is robust in the sense that the distribution of the underlying random variable is assumed to belong to a certain family of distributions rather than to be exactly known we start from analyzing a simple case where uncertainty arises only in the objective function and then explore the general case where uncertainty also arises in the constraints we show that the former problem is equivalent to a semidefinite program and the latter problem is generally np hard applications to two stage portfolio optimization material order problems stochastic production transportation problem and single facility minimax distance problem are considered numerical results show that the proposed robust risk averse two stage stochastic programming model can effectively control the risk with solutions of acceptable good quality

the sample average approximation method for stochastic discrete optimization in this paper we study a monte carlo simulation based approach to stochastic discrete optimization problems the basic idea of such methods is that a random sample is generated and the expected value function is approximated by the corresponding sample average function the obtained sample average optimization problem is solved and the procedure is repeated several times until a stopping criterion is satisfied we discuss convergence rates stopping rules and computational complexity of this procedure and present a numerical example for the stochastic knapsack problem

a unified inverse scattering transformation for the local and nonlocal nonautonomous gross pitaevskii equations we present the inverse scattering transformation for a nonisospectral akns hierarchy in which the spectral parameter is determined by an ordinary differential equation with polynomial nonlinearity and thus we give a unified treatment for the local and nonlocal nonautonomous gross pitaevskii equations which possess the parity time pt symmetric invariance we find that unlike the local case the pt symmetry of the nonlocal gross pitaevskii equation allows two different choices of the symmetry relations of the eigenfunctions which guarantee two different kinds of inverse scattering solutions

on homogenization and scaling limit of some gradient perturbations of a massless free field we study the continuum scaling limit of some statistical mechanical models defined by convex hamiltonians which are gradient perturbations of a massless free field by proving a central limit theorem for these models we show that their long distance behavior is identical to a new homogenized continuum massless free field we shall also obtain some new bounds on the 2 point correlation functions of these models

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

distributionally robust convex optimization distributionally robust optimization is a paradigm for decision making under uncertainty where the uncertain problem data are governed by a probability distribution that is itself subject to uncertainty the distribution is then assumed to belong to an ambiguity set comprising all distributions that are compatible with the decision maker s prior information in this paper we propose a unifying framework for modeling and solving distributionally robust optimization problems we introduce standardized ambiguity sets that contain all distributions with prescribed conic representable confidence sets and with mean values residing on an affine manifold these ambiguity sets are highly expressive and encompass many ambiguity sets from the recent literature as special cases they also allow us to characterize distributional families in terms of several classical and or robust statistical indicators that have not yet been studied in the context of robust optimization we determine conditions under which distributionally robust optimization problems based on our standardized ambiguity sets are computationally tractable we also provide tractable conservative approximations for problems that violate these conditions

scheduling arrivals to a stochastic service delivery system using copositive cones in this paper we investigate a stochastic appointment scheduling problem in an outpatient clinic with a single doctor the number of patients and their sequence of arrivals are fixed and the scheduling problem is to determine an appointment time for each patient the service durations of the patients are stochastic and only the mean and covariance estimates are known we do not assume any exact distributional form of the service durations and we solve for distributionally robust schedules that minimize the expectation of the weighted sum of patients waiting time and the doctor s overtime we formulate this scheduling problem as a convex conic optimization problem with a tractable semidefinite relaxation our model can be extended to handle additional support constraints of the service durations using the primal dual optimality conditions we prove several interesting structural properties of the optimal schedules we develop an efficient semidefinite relaxation of the conic program and show that we c

rotational invariance and discrete analyticity in the 2d dimer model we exploit the discrete analytic structure present in the 2d dimer model to rigorously compute the exponents of a class of two point functions in all directions this is a dimer analogue of the critical 2d ising spin spin correlation function

pde with random coefficients and euclidean field theory in this paper a new proof of an identity of giacomin olla and spohn is given the identity relates the 2 point correlation function of a euclidean field theory to the expectation of the green s function for a pde with random coefficients the euclidean field theory is assumed to have convex potential an inequality of brascamp and lieb therefore implies gaussian bounds on the fourier transform of the 2 point correlation function by an application of results from random pde the previously mentioned identity implies pointwise gaussian bounds on the 2 point correlation function

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

continuous and discrete schrodinger systems with parity time symmetric nonlinearities odinger systems exhibiting parity time pt invariant nonlinearities we show that such equations behave in a fundamentally different fashion than their nonlinear schr odinger counterparts in particular the pt symmetric nonlinear schr odinger equation can simultaneously support both bright and dark soliton solutions in addition we study a discretized version of

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

distributionally robust multi item newsvendor problems with multimodal demand distributions we present a risk averse multi dimensional newsvendor model for a class of products whose demands are strongly correlated and subject to fashion trends that are not fully understood at the time when orders are placed the demand distribution is known to be multimodal in the sense that there are spatially separated clusters of probability mass but otherwise lacks a complete description we assume that the newsvendor hedges against distributional ambiguity by minimizing the worst case risk of the order portfolio over all distributions that are compatible with the given modality information we demonstrate that the resulting distributionally robust optimization problem is mathrm np np hard but admits an efficient numerical solution in quadratic decision rules this approximation is conservative and computationally tractable moreover it achieves a high level of accuracy in numerical tests we further demonstrate that disregarding ambiguity or multimodality can lead to unstable solutions that perform poorly in stress test experiments

distribution free inventory risk pooling in a multi location newsvendor we study a multilocation newsvendor network when the only information available on the joint distribution of demands are the values of its mean vector and covariance matrix we adopt a distribution

convexity and decomposition of mean risk stochastic programs traditional stochastic programming is risk neutral in the sense that it is concerned with the optimization of an expectation criterion a common approach to addressing risk in decision making problems is to consider a weighted mean risk objective where some dispersion statistic is used as a measure of risk we investigate the computational suitability of various mean risk objective functions in addressing risk in stochastic programming models we prove that the classical mean variance criterion leads to computational intractability even in the simplest stochastic programs on the other hand a number of alternative mean risk functions are shown to be computationally tractable using slight variants of existing stochastic programming decomposition algorithms we propose decomposition based parametric cutting plane algorithms to generate mean risk efficient frontiers for two particular classes of mean risk objectives

on the correlation for kac like models in the convex case the aim of this paper is to stu the behavior asm tends to of a family of measures exp m x dx m on m where m is a potential on m which is a perturbation in a suitable sense of the harmonic potential j x j 2

the value of stochastic modeling in two stage stochastic programs with cost uncertainty although stochastic programming is probably the most effective framework for handling decision problems that involve uncertain variables it is always a costly task to formulate the stochastic model that accurately embodies our knowledge of these variables in practice this might require one to collect a large amount of observations to consult with experts of the specialized field of practice or to make simplifying assumptions about the underlying system when none of these options seem feasible a common heuristic has been to simply seek the solution of a version of the problem where each uncertain variable takes on its expected value otherwise known as the solution of the mean value problem in this paper we show that when 1 the stochastic program takes the form of a two stage mixed integer stochastic linear programs and 2 the uncertainty is limited to the objective function the solution of the mean value problem is in fact robust with respect to the selection of a stochastic model we also propose tractable methods that will bound the actual value of stochastic modeling i e how much improvement can be achieved by investing more efforts in the resolution of the stochastic model our framework is applied to an airline fleet composition problem in the three cases that are considered our results indicate that resolving the stochastic model can not lead to more than a 7 improvement of expected profits thus providing arguments against the need to develop these more sophisticated models

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

motion by mean curvature from the ginzburg landau interface model we consider the scalar field t with a reversible stochastic dynamics which is defined by the standard dirichlet form relative to the gibbs measure with formal energy the potential v is even and strictly convex we prove that under a suitable large scale limit the t field becomes deterministic such that locally its normal velocity is proportional to its mean curvature except for some anisotropy effects as an essential input we prove that for every tilt there is a unique shift invariant ergodic gibbs measure for the field

integrable discrete pt symmetric model an exactly solvable discrete pt invariant nonlinear schr odinger like model is introduced it is an integrable hamiltonian system that exhibits a nontrivial nonlinear pt symmetry a discrete one soliton solution is constructed using a left right riemann hilbert formulation it is shown that this pure soliton exhibits unique features such as power oscillations and singularity formation the proposed model can be viewed as a discretization of a recently obtained integrable nonlocal nonlinear schr odinger equation

polyhedral risk measures in stochastic programming we consider stochastic programs with risk measures in the objective and study stability properties as well as decomposition structures thereby we place emphasis on dynamic models i e multistage stochastic programs with multiperiod risk measures in this context we define the class of polyhedral risk measures such that stochastic programs with risk measures taken from this class have favorable properties polyhedral risk measures are defined as optimal values of certain linear stochastic programs where the arguments of the risk measure appear on the right hand side of the dynamic constraints dual representations for polyhedral risk measures are derived and used to deduce criteria for convexity and coherence as examples of polyhedral risk measures we propose multiperiod extensions of the conditional value at risk

pt symmetric quantum mechanics this paper proposes to broaden the canonical formulation of quantum mechanics ordinarily one imposes the condition h h on the hamiltonian where represents the mathematical operation of complex conjugation and matrix transposition this conventional hermiticity condition is sufficient to ensure that the hamiltonian h has a real spectrum however replacing this mathematical condition by the weaker and more physical requirement h h where represents combined parity reflection and time reversal pt one obtains new classes of complex hamiltonians whose spectra are still real and positive this generalization of hermiticity is investigated using a complex deformation h p2 x2 ix e of the harmonic oscillator hamiltonian where e is a real parameter the system exhibits two phases when e 0 the energy spectrum of h is real and positive as a consequence of pt symmetry however when 1 e 0 the spectrum contains an infinite number of complex eigenvalues and a finite number of real positive eigenvalues b

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

applying the minimax criterion in stochastic recourse programs we consider an optimization problem in which some uncertain parameters are replaced by random variables the minimax approach to stochastic programming concerns the problem of minimizing the worst expected value of the objective function with respect to the set of probability measures that are consistent with the available information on the random data only very few practicable solution procedures have been proposed for this problem and the existing ones rely on simplifying assumptions in this paper we establish a number of stability results for the minimax stochastic program justifying in particular the approach of restricting attention to probability measures with support in some known finite set following this approach we elaborate solution procedures for the minimax problem in the setting of two stage stochastic recourse models considering the linear recourse case as well as the integer recourse case since the solution procedures are modifications of well known algorithms their efficacy is immediate from the computational testing of these procedures and we do not report results of any computational experiments

the exact solutions for the nonisospectral akns hierarchy through the inverse scattering transform abstract exact solutions for the nonisospectral akns hierarchy are obtained through the inverse scattering transform some explicit forms of these solutions are further given reductions are also considered exact solutions of some nonisospectral hierarchies such as the nonisospectral mkdv hierarchy the nonisospectral kdv hierarchy the nonisospectral nonlinear schrodinger hierarchy and the nonisospectral sine gordon hierarchy are obtained

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

a strong central limit theorem for a class of random surfaces this paper is concerned with d 2 dimensional lattice field models with action v nabla phi cdot where v mathbf r d rightarrow mathbf r is a uniformly convex function the fluctuations of the variable phi 0 phi x are studied for large x via the generating function given by g x mu ln langle e mu phi 0 phi x rangle a in two dimensions g x mu partial 2g x mu partial mu 2 is proportional to ln vert x vert the main result of this paper is a bound on g x mu partial 3 g x mu partial mu 3 which is uniform in vert x vert for a class of convex v the proof uses integration by parts following helffer sjostrand and witten and relies on estimates of singular integral operators on weighted hilbert spaces

lax representations and lax operator algebras of isospectral and nonisospectral hierarchies of evolution equations a method of constructing lax representations for isospectral and nonisospectral hierarchies of evolution equations is proposed it is shown that under some simple but essential conditions the corresponding lax operators may constitute infinite dimensional lie algebras with respect to the binary operation given in this paper furthermore the detailed analyses for the kdv couple the akns couple and a new couple of isospectral and nonisospectral hierarchies of integrable equations are presented as examples

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

on a class of minimax stochastic programs for a particular class of minimax stochastic programming models we show that the problem can be equivalently reformulated into a standard stochastic programming problem this permits the direct use of standard decomposition and sampling methods developed for stochastic programming we also show that this class of minimax stochastic programs is closely related to a large family of mean risk stochastic programs where risk is measured in terms of deviations from a quantile

interlacing adjacent levels of beta jacobi corners processes we study the asymptotics of the global fluctuations for the difference between two adjacent levels in the beta jacobi corners process multilevel and general beta extension of the classical jacobi ensemble of random matrices the limit is identified with the derivative of the 2d gaussian free field our main tools are integral forms for the macdonald type difference operators originating from the shuffle algebra

decomposition algorithms for two stage distributionally robust mixed binary programs in this paper we introduce and study a two stage distributionally robust mixed binary problem tsdr mbp in which the random parameters follow the worst case distribution belonging to an uncertain

robust sample average approximation sample average approximation saa is a widely popular approach to data driven decision making under uncertainty under mild assumptions saa is both tractable and enjoys strong asymptotic performance guarantees similar guarantees however do not typically hold in finite samples in this paper we propose a modification of saa which we term robust saa which retains saa s tractability and asymptotic properties and additionally enjoys strong finite sample performance guarantees the key to our method is linking saa distributionally robust optimization and hypothesis testing of goodness of fit beyond robust saa this connection provides a unified perspective enabling us to characterize the finite sample and asymptotic guarantees of various other data driven procedures that are based upon distributionally robust optimization this analysis provides insight into the practical performance of these various methods in real applications we present examples from inventory management and portfolio allocation and demonstrate numerically that our approach outperforms other data driven approaches in these applications

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

expected utility penalty functions and duality in stochastic nonlinear programming we consider nonlinear programming problem p with stochastic constraints the lagrangean corresponding to such problems has a stochastic part which in this work is replaced by its certainty equivalent in the sense of expected utility theory it is shown that the deterministic surrogate problem ce p thus obtained contains a penalty function which penalizes violation of the constraints in the mean the approach is related to several known methods in stochastic programming such as chance constraints stochastic goal programming reliability programming and mean variance analysis the dual problem of ce p is studied for problems with stochastic righthand sides in the constraints and a comprehensive duality theory is developed by introducing a new certainty equivalent nce concept motivation for the nce and its potential role in decision theory are discussed as well as mean variance approximations

ambiguity in risk preferences in robust stochastic optimization abstract we consider robust stochastic optimization problems for risk averse decision makers where there is ambiguity about both the decision maker s risk preferences and the underlying probability distribution we propose and analyze a robust optimization problem that accounts for both types of ambiguity first we derive a duality theory for this problem class and identify random utility functions as the lagrange multipliers second we turn to the computational aspects of this problem we show how to evaluate our robust optimization problem exactly in some special cases and then we consider some tractable relaxations for the general case finally we apply our model to both the newsvendor and portfolio optimization problems and discuss its implications

data driven distributionally robust optimization using the wasserstein metric performance guarantees and tractable reformulations we consider stochastic programs where the distribution of the uncertain parameters is only observable through a finite training dataset using the wasserstein metric we construct a ball in the space of multivariate and non discrete probability distributions centered at the uniform distribution on the training samples and we seek decisions that perform best in view of the worst case distribution within this wasserstein ball the state of the art methods for solving the resulting distributionally robust optimization problems rely on global optimization techniques which quickly become computationally excruciating in this paper we demonstrate that under mild assumptions the distributionally robust optimization problems over wasserstein balls can in fact be reformulated as finite convex programs in many interesting cases even as tractable linear programs leveraging recent measure concentration results we also show that their solutions enjoy powerful finite sample performance guarantees our theoretical results are exemplified in mean risk portfolio optimization as well as uncertainty quantification

toeplitz and wiener hopf determinants with piecewise continuous symbols abstract the asymptotics for determinants of toeplitz and wiener hopf operators with piecewise continuous symbols are obtained in this paper if w is the wiener hopf operator defined on l2 0 with piecewise continuous symbol having a finite number of discontinuities at r then under appropriate conditions it is shown that det w g r2k where g exp log 0 r 1 2 log r r and k is a completely determined constant an analogous result is obtained for toeplitz operators the main point of the paper is to obtain a result in the wiener hopf case since the toeplitz case had been treated earlier in the toeplitz case it was discovered that one could obtain asymptotics fairly easily for symbols with several singularities if for each singularity one could find a single example of a symbol with a singularity of that kind whose associated asymptotics were known fortunately in the toeplitz case such asymptotics were known the difficulty in the wiener hopf case is that there was not a single singular case where the determinant was explicitly known this problem was overcome by using the fact that wiener hopf determinants when discretized become toeplitz determinants whose entries depend on the size of the matrix no theorem on toeplitz matrices can be applied directly but these theorems are modified to obtain the desired results