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

the geometry of the universal teichmuller space and the euler weil petersson equation on the identity component of the universal teichmuller space endowed with the takhtajan teo topology the geodesics of the weil petersson metric are shown to exist for all time this component is naturally a subgroup of the quasisymmetric homeomorphisms of the circle viewed this way the regularity of its elements is shown to be h3 2 epsilon for all epsilon 0 the evolutionary pde associated to the spatial representation of the geodesics of the weil petersson metric is derived using multiplication and composition below the critical sobolev index 3 2 geodesic completeness is used to introduce special classes of solutions of this pde analogous to peakons our setting is used to prove that there exists a unique geodesic between each two shapes in the plane in the context of the application of the weil petersson metric in imaging c 2015 elsevier inc all rights reserved

distributionally robust optimization and its tractable approximations in this paper we focus on a linear optimization problem with uncertainties having expectations in the objective and in the set of constraints we present a modular framework to obtain an approximate solution to the problem that is distributionally robust and more flexible than the standard technique of using linear rules our framework begins by first affinely extending the set of primitive uncertainties to generate new linear decision rules of larger dimensions and is therefore more flexible next we develop new piecewise linear decision rules that allow a more flexible reformulation of the original problem the reformulated problem will generally contain terms with expectations on the positive parts of the recourse variables finally we convert the uncertain linear program into a deterministic convex program by constructing distributionally robust bounds on these expectations these bounds are constructed by first using different pieces of information on the distribution of the underlying uncertainties to develop separate bounds and next integrating them into a combined bound that is better than each of the individual bounds

an integral operator induced by a zygmund function abstract we introduce a kernel function induced by a continuous function f on the unit circle and show that the corresponding integral operator on the standard bergman space is a bounded compact or hilbert schmidt operator precisely when f belongs to the big zygmund class the little zygmund class or the sobolev class h 3 2 respectively this may be considered as the infinitesimal version of the main result obtained in hu and shen 2012 9 by two of the authors

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

hasse principle for hermitian spaces over semi global fields abstract in a recent paper colliot thelene parimala and suresh conjectured that a local global principle holds for projective homogeneous spaces under connected linear algebraic groups over function fields of p adic curves in this paper we show that the conjecture is true for any linear algebraic group whose almost simple factors of its semisimple part are isogenous to unitary groups or special unitary groups of hermitian or skew hermitian spaces over central simple algebras with involutions the proof implements patching techniques of harbater hartmann and krashen as an application we obtain a springer type theorem for isotropy of hermitian spaces over odd degree extensions of function fields of p adic curves

the henselization of a valued division algebra the main result of this paper is the following if f v is a valued field fh its henselization d a division algebra with center f and d f then v extends to d iff d ffh is a division algebra it is also shown that if p char f p 1if char f 0 then d f d f d f pb for some nonnegative integer b

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

weil petersson teichmuller space revisited abstract in our previous paper 23 it was proved that a sense preserving homeomorphism g on the unit circle s 1 belongs to the weil petersson class wp s 1 namely g can be extended to a quasiconformal mapping to the unit disk whose beltrami coefficient is square integrable in the poincare metric if and only if g is absolutely continuous such that log g belongs to the sobolev class h 1 2 in this sequel to 23 we show that the smooth hilbert manifold structure on wp s 1 inherited from h 1 2 by the pullback g log g is compatible with the standard hilbert manifold structure introduced by takhtajan teo 27 this enables us to give a fast approach to some results in our previous papers 23 and 24

convergence analysis of a lasserre hierarchy of upper bounds for polynomial minimization on the sphere we study the convergence rate of a hierarchy of upper bounds for polynomial minimization problems proposed by lasserre siam j optim 21 3 864 885 2011 for the special case when the feasible set is the unit hyper sphere the upper bound at level r n of the hierarchy is defined as the minimal expected value of the polynomial over all probability distributions on the sphere when the probability density function is a sum of squares polynomial of degree at most 2r with respect to the surface measure we show that the rate of convergence is o 1 r2 and we give a class of polynomials of any positive degree for which this rate is tight in addition we explore the implications for the related rate of convergence for the generalized problem of moments on the sphere

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

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

improved convergence analysis of lasserre s measure based upper bounds for polynomial minimization on compact sets we consider the problem of computing the minimum value fmin k of a polynomial f over a compact set k rn which can be reformulated as finding a probability measure on k minimizing kfd lasserre showed that it suffices to consider such measures of the form q where q is a sum of squares polynomial and is a given borel measure supported on k by bounding the degree of q by 2r one gets a converging hierarchy of upper bounds f r for fmin k when k is the hypercube 1 1 n equipped with the chebyshev measure the parameters f r are known to converge to fmin k at a rate in o 1 r2 we extend this error estimate to a wider class of convex bodies while also allowing for a broader class of reference measures including the lebesgue measure our analysis applies to simplices balls and convex bodies that locally look like a ball in addition we show an error estimate in o logr r when k satisfies a minor geometrical condition and in o log2r r2 when k is a convex body equipped with the lebesgue measure this improves upon the currently best known error estimates in o 1 r and o 1 r for these two respective cases

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

on divergences and informations in statistics and information theory the paper deals with the f divergences of csiszar generalizing the discrimination information of kullback the total variation distance the hellinger divergence and the pearson divergence all basic properties of f divergences including relations to the decision errors are proved in a new manner replacing the classical jensen inequality by a new generalized taylor expansion of convex functions some new properties are proved too e g relations to the statistical sufficiency and deficiency the generalized taylor expansion also shows very easily that all f divergences are average statistical informations differences between prior and posterior bayes errors mutually differing only in the weights imposed on various prior distributions the statistical information introduced by de groot and the classical information of shannon are shown to be extremal cases corresponding to alpha 0 and alpha 1 in the class of the so called arimoto alpha informations introduced in this paper for 0 alpha 1 by means of the arimoto alpha entropies some new examples of f divergences are introduced as well namely the shannon divergences and the arimoto alpha divergences leading for alphauarr1 to the shannon divergences square roots of all these divergences are shown to be metrics satisfying the triangle inequality the last section introduces statistical tests and estimators based on the minimal f divergence with the empirical distribution achieved in the families of hypothetic distributions for the kullback divergence this leads to the classical likelihood ratio test and estimator

information divergence and risk for binary experiments we unify f divergences bregman divergences surrogate regret bounds proper scoring rules cost curves roc curves and statistical information we do this by systematically studying integral and variational representations of these objects and in so doing identify their representation primitives which all are related to cost sensitive binary classification as well as developing relationships between generative and discriminative views of learning the new machinery leads to tight and more general surrogate regret bounds and generalised pinsker inequalities relating f divergences to variational divergence the new viewpoint also illuminates existing algorithms it provides a new derivation of support vector machines in terms of divergences and relates maximum mean discrepancy to fisher linear discriminants

certainty equivalents and information measures duality and extremal principles given a convex function ir ir the csiszar divergence csiszar 1978 is a function i ir n ir n ir

weil petersson teichmuller space iii dependence of riemann mappings for weil petersson curves the classical riemann mapping theorem implies that there exists a so called riemann mapping which takes the upper half plane onto the left domain bounded by a jordan curve in the extended complex plane the primary purpose of the paper is to study the basic problem how does a riemann mapping depend on the corresponding jordan curve we are mainly concerned with those jordan curves in the weil petersson class namely the corresponding riemann mappings can be quasiconformally extended to the whole plane with beltrami coefficients being square integrable under the poincare metric after giving a geometric characterization of a weil petersson curve we endow the space of all normalized weil petersson curves with a new real hilbert manifold structure in a geometric manner and show that this new structure is topologically equivalent to the standard complex hilbert manifold structure which implies that an appropriately chosen riemann mapping depends continuously on a weil petersson curve and vice versa this can be considered as the first result about the continuous dependence of riemann mappings on non smooth jordan curves

ambiguous joint chance constraints under mean and dispersion information we study joint chance constraints where the distribution of the uncertain parameters is only known to belong to an ambiguity set characterized by the mean and support of the uncertainties and by an upper bound on their dispersion this setting gives rise to pessimistic optimistic ambiguous chance constraints which require the corresponding classical chance constraints to be satisfied for every for at least one distribution in the ambiguity set we demonstrate that the pessimistic joint chance constraints are conic representable if i the constraint coefficients of the decisions are deterministic ii the support set of the uncertain parameters is a cone and iii the dispersion function is of first order that is it is positively homogeneous we also show that pessimistic joint chance constrained programs become intractable as soon as any of the conditions i ii or iii is relaxed in the mildest possible way we further prove that the optimistic joint chance constraints are conic representab

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

comparison of lasserre s measure based bounds for polynomial optimization to bounds obtained by simulated annealing we consider the problem of minimizing a continuous function f over a compact set k we compare the hierarchy of upper bounds proposed by lasserre lasserre jb 2011 a new look at nonnegativity on closed sets and polynomial optimization siam j optim 21 3 864 885 to bounds that may be obtained from simulated annealing we show that when f is a polynomial and k a convex body this comparison yields a faster rate of convergence of the lasserre hierarchy than what was previously known in the literature

a semidefinite programming approach to optimal moment bounds for convex classes of distributions we provide an optimization framework for computing optimal upper and lower bounds on functional expectations of distributions with special properties given moment constraints bertsimas and popescu optimal inequalities in probability theory a convex optimization approach siam j optim 2004 forthcoming have already shown how to obtain optimal moment inequalities for arbitrary distributions via semidefinite programming these bounds are not sharp if the underlying distributions possess additional structural properties including symmetry unimodality convexity or smoothness for convex distribution classes that are in some sense generated by an appropriate parametric family we use conic duality to show how optimal moment bounds can be efficiently computed as semidefinite programs in particular we obtain generalizations of chebyshev s inequality for symmetric and unimodal distributions and provide numerical calculations to compare these bounds given higher order moments we also extend these results for multivariate distributions

integrable teichmuller space abstract this paper deals with the p integrable teichmuller space and gives an intrinsic characterization of a p integrable asymptotic affine homeomorphism for p 2 more precisely it is proved that a sense preserving homeomorphism h on the unit circle is p integrable asymptotic affine namely h can be extended to a quasiconformal mapping to the unit disk whose beltrami coefficient is p integrable in the poincare metric if and only if h is absolutely continuous such that log h belongs to the besov class b p s 1

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

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

applications of patching to quadratic forms and central simple algebras this paper provides applications of patching to quadratic forms and central simple algebras over function fields of curves over henselian valued fields in particular we use a patching approach to reprove and generalize a recent result of parimala and suresh in preprint arxiv 0708 3128 2007 on the u invariant of p adic function fields p 2 the strategy relies on a local global principle for homogeneous spaces for rational algebraic groups combined with local computations

ambiguous risk constraints with moment and unimodality information optimization problems face random constraint violations when uncertainty arises in constraint parameters effective ways of controlling such violations include risk constraints e g chance constraints and conditional value at risk constraints this paper studies these two types of risk constraints when the probability distribution of the uncertain parameters is ambiguous in particular we assume that the distributional information consists of the first two moments of the uncertainty and a generalized notion of unimodality we find that the ambiguous risk constraints in this setting can be recast as a set of second order cone soc constraints in order to facilitate the algorithmic implementation we also derive efficient ways of finding violated soc constraints finally we demonstrate the theoretical results via computational case studies on power system operations

admissibility of groups over function fields of p adic curves let k be a field and g a finite group the question of admissibility of g over k was originally posed by schacher who gave partial results in the case k q in this paper we give necessary conditions for admissibility of a finite group g over function fields of curves over complete discretely valued fields using this criterion we give an example of a finite group which is not admissible over qp t we also prove a certain hasse principle for division algebras over such fields

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

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

fully distribution free profit maximization the inventory management case we study profit maximization in inventory control problems where demands are unknown neither probabilistic distributions nor sets are available to characterize the unknown demand parameters therefore we adopt an online optimization perspective for our analysis the usual competitive ratio is not well defined for the problems we analyze consequently we introduce a new worst case metric that is suitable we consider two inventory management frameworks 1 perishable products with lost sales and 2 durable products with backlogged demand we consider both finite and infinite planning horizons we design best possible online procurement strategies for all cases

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

a new look at nonnegativity on closed sets and polynomial optimization we first show that a continuous function f is nonnegative on a closed set k subseteq r n if and only if countably many moment matrices of some signed measure d nu fd mu with support equal to k are all positive semidefinite if k is compact mu is an arbitrary finite borel measure with support equal to k in particular we obtain a convergent explicit hierarchy of semidefinite outer approximations with it no lifting of the cone of nonnegative polynomials of degree at most d wen used in polynomial optimization on certain simple closed sets k like e g the whole space r n the positive orthant a box a simplex or the vertices of the hypercube it provides a nonincreasing sequence of upper bounds which converges to the global minimum by solving a hierarchy of semidefinite programs with only one variable this convergent sequence of upper bounds complements the convergent sequence of lower bounds obtained by solving a hierarchy of semidefinite relaxations

universal teichmuller space and bmo abstract we first give some new characterizations on bmoa teichmuller space and various characterizations on vmoa teichmuller space as well in particular we prove that a quasisymmetric conformal welding h corresponds to an asymptotically smooth curve in the sense of pommerenke 1978 32 precisely when h is absolutely continuous with log h vmo we then show that these bmo teichmuller spaces have natural complex structures

diff s 1 and the teichmuller spaces precisely two of the homogeneous spaces that appear as coadjoint orbits of the group of string reparametrizations widehat diff s 1 1 carry in a natural way the structure of infinite dimensional holomorphically homogeneous complex analytic kahler manifolds these aren diff s1 rot s1 andm diff s1 mob s1 note thatn is a holomorphic disc fiber space overm now m can be naturally considered as embedded in the classical universal teichmuller spacet 1 simply by noting that a diffeomorphism ofs1 is a quasisymmetric homeomorphism t 1 is itself a homomorphically homogeneous complex banach manifold we prove in the first part of the paper that the inclusion ofm int 1 iscomplex analytic

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

the mathbf k moment problem for continuous linear functionals given a closed and non necessarily compact basic semi algebraic set k subseteq r n we solve the k moment problem for continuous linear functionals namely we introduce a weighted ell 1 norm ell w on r x and show that the ell w closures of the preordering p and quadratic module q associated with the generators of k is the cone psd k of polynomials nonnegative on k we also prove that p an q solve the k moment problem for ell w continuous linear functionals and completely characterize those ell w continuous linear functionals nonnegative on p and q hence on psd k when k has a nonempty interior we also provide in explicit form a canonical ell w projection g w f for any polynomial f on the degree truncated preordering or quadratic module remarkably the support of g w f is very sparse and does not depend on k this enables us to provide an explicit positivstellensatz on k at last but not least we provide a simple characterization of polynomials nonnegative on k which is crucial in proving the above results

constructing uncertainty sets for robust linear optimization in this paper we propose a methodology for constructing uncertainty sets within the framework of robust optimization for linear optimization problems with uncertain parameters our approach relies on decision maker risk preferences specifically we utilize the theory of coherent risk measures initiated by artzner et al 1999 artzner p f delbaen j eber d heath 1999 coherent measures of risk math finance9 203 228 and show that such risk measures in conjunction with the support of the uncertain parameters are equivalent to explicit uncertainty sets for robust optimization we explore the structure of these sets in detail in particular we study a class of coherent risk measures called distortion risk measures which give rise to polyhedral uncertainty sets of a special structure that is tractable in the context of robust optimization in the case of discrete distributions with rational probabilities which is useful in practical settings when we are sampling from data we show that the class of all distortion risk measures and their corresponding polyhedral sets are generated by a finite number of conditional value at risk cvar measures a subclass of the distortion risk measures corresponds to polyhedral uncertainty sets symmetric through the sample mean we show that this subclass is also finitely generated and can be used to find inner approximations to arbitrary polyhedral uncertainty sets

weil petersson teichmuller space ii smoothness of flow curves of h32 vector fields abstract given a continuous vector field t of sobolev class h 3 2 on the unit circle s 1 the flow maps g t of the differential equation d d t t 0 are known to be quasisymmetric homeomorphisms very recently gay balmaz ratiu 15 conjectured that the flow curve g t is in the weil petersson class wp s 1 and is continuously differentiable with respect to the hilbert manifold structure of wp s 1 introduced by takhtajan teo 40 the first assertion had already been demonstrated in our previous paper 36 in this sequel to 36 we will continue to deal with the weil petersson class wp s 1 and completely solve this conjecture in the affirmative

cyclic algebras over p adic curves abstract in this paper we study division algebras over the function fields of curves over q p the first and main tool is to view these fields as function fields over nonsingular s which are projective of relative dimension 1 over the p adic ring z p a previous paper showed such division algebras had index bounded by n 2 assuming the exponent was n and n was prime to p in this paper we consider algebras of prime degree and hence exponent q p and show these algebras are cyclic we also find a geometric criterion for a brauer class to have index q

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

hermitian analogue of a theorem of springer with involutions and in fact in a much more general setupse for algebraic groups the weak form of springer s theorem forhermitian forms was proved by bayer fluckiger and lenstra in bl andthe analogue of springer s theorem for hermitian forms was raised as anopen question in bst the aim of this paper is to settle this question inthe affirmative for hermitian forms over quaternion algebras with respectto involutions of the first kind we prove that if

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

on flows of h 3 2 vector fields on the circle in this paper we study the regularity of flow maps of h 3 2 vector fields on the circle in terms of fractional sobolev spaces this problem is motivated by the understanding of the geometry of bers s universal teichmuller space

some notes on quasisymmetric flows of zygmund vector fields abstract in an important paper 24 reimann showed that the flow mappings of a continuous vector field of zygmund class are quasisymmetric homeomorphisms in this paper we will discuss the flow mappings when the vector field belongs to the smooth zygmund class or the sobolev class h 3 2

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

regret in the newsvendor model with partial information traditional stochastic inventory models assume full knowledge of the demand probability distribution however in practice it is often difficult to completely characterize the demand distribution especially in fast changing markets in this paper we study the newsvendor problem with partial information about the demand distribution e g mean variance symmetry unimodality in particular we derive the order quantities that minimize the newsvendor s maximum regret of not acting optimally most of our solutions are tractable which makes them attractive for practical application our analysis also generates insights into the choice of the demand distribution as an input to the newsvendor model in particular the distributions that maximize the entropy perform well under the regret criterion our approach can be extended to a variety of problems that require a robust but not conservative solution

goodness of fit tests via phi divergences a unified family of goodness of fit tests based on o divergences is introduced and studied the new family of test statistics s n s includes both the supremum version of the anderson darling statistic and the test statistic of berk and jones z wahrsch verw gebiete 47 1979 47 59 as special cases s 2 and s 1 resp we also introduce integral versions of the new statistics we show that the asymptotic null distribution theory of berk and jones z wahrsch verw gebiete 47 1979 47 59 and wellner and koltchinskii high dimensional probability iii 2003 321 332 birkhauser basel for the berk jones statistic applies to the whole family of statistics s n s with s 1 2 on the side of power behavior we study the test statistics under fixed alternatives and give extensions of the poisson boundary phenomena noted by berk and jones for their statistic we also extend the results of donoho and jin ann statist 32 2004 962 994 by showing that all our new tests for s 1 2 have the same optimal detection boundary for normal shift mixture alternatives as tukey s higher criticism statistic and the berk jones statistic

a springer theorem for hermitian forms throughout this paper we let d be a central f division algebra with involution such that f d f d d is a henselian valued field by 11 the valuation on f extends uniquely to a valuation on d we denote this valuation by v moreover we assume that the characteristic of the residue field open image in new window is not 2 if the valuation on f is discrete then any quadratic form q can be written as q q1 q2 where is a uniformizer and qi are unit forms springer s theorem states that q is isotropic if and only if at least one of the residue forms open image in new window and open image in new window is isotropic in this paper we generalize this result to hermitian forms in section 4 we use the connection between involutions on algebras and hermitian forms to prove an analog of the springer theorem for involutions

a conic programming approach to generalized tchebycheff inequalities consider the problem of finding optimal bounds on the expected value of piecewise polynomials over all measures with a given set of moments this is a special class ofgeneralized tchebycheff inequalities in probability theory we study this problem within the framework of conic programming relying on a general approximation scheme for conic programming we show that these bounds can be numerically computed or approximated via semidefinite programming we also describe some applications of this class of generalized tchebycheff inequalities in probability finance and inventory theory

convergence analysis for lasserre s measure based hierarchy of upper bounds for polynomial optimization we consider the problem of minimizing a continuous function f over a compact set mathbf k k we analyze a hierarchy of upper bounds proposed by lasserre siam j optim 21 3 864 885 2011 obtained by searching for an optimal probability density function h on mathbf k k which is a sum of squares of polynomials so that the expectation int mathbf k f x h x dx kf x h x dx is minimized we show that the rate of convergence is no worse than o 1 sqrt r o 1 r where 2r is the degree bound on the density function this analysis applies to the case when f is lipschitz continuous and mathbf k k is a full dimensional compact set satisfying some boundary condition which is satisfied e g for convex bodies the rth upper bound in the hierarchy may be computed using semidefinite programming if f is a polynomial of degree d and if all moments of order up to 2r d 2r d of the lebesgue measure on mathbf k k are known which holds for example if mathbf k k is a simplex hypercube or a euclidean ball

period index and u invariant questions for function fields over complete discretely valued fields let k be a complete discretely valued field with residue field and f the function field of a curve over k let p be the characteristic of and l a prime not equal to p if the brauer l dimensions of all finite extensions of are bounded by d and the brauer l dimensions of all extensions of of transcendence degree at most 1 are bounded by d 1 then it is known that the brauer l dimension of f is at most d 2 lieblich in j reine angew math 659 1 41 2011 saltman in j ramanujan math soc 12 25 47 1997 harbater et al in invent math 178 231 263 2009 in this paper we give a bound for the brauer p dimension of f in terms of the p rank of as an application we show that if is a perfect field of characteristic 2 then any quadratic form over f in at least 9 variables is isotropic this leads to the fact that every element in h 3 f mathbb z 2 mathbb z is a symbol if is a finite field of characteristic 2 u f 8 is a result of heath brown leep heath brown in compos math 146 271 287 2010 leep in j reine angew math 2013 to appear

a semidefinite programming approach to the generalized problem of moments we consider the generalized problem of moments gpm from a computational point of view and provide a hierarchy of semidefinite programming relaxations whose sequence of optimal values converges to the optimal value of the gpm we then investigate in detail various examples of applications in optimization probability financial economics and optimal control which all can be viewed as particular instances of the gpm

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

constructing risk measures from uncertainty sets we illustrate the correspondence between uncertainty sets in robust optimization and some popular risk measures in finance and show how robust optimization can be used to generalize the concepts of these risk measures we also show that by using properly defined uncertainty sets in robust optimization models one can construct coherent risk measures and address the issue of the computational tractability of the resulting formulations our results have implications for efficient portfolio optimization under different measures of risk