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 scattering map in two coupled piecewise smooth systems with numerical application to rocking blocks abstract we consider a non autonomous dynamical system formed by coupling two piecewise smooth systems in r 2 through a non autonomous periodic perturbation we study the dynamics around one of the heteroclinic orbits of one of the piecewise smooth systems in the unperturbed case the system possesses two c 0 normally hyperbolic invariant manifolds of dimension two with a couple of three dimensional heteroclinic manifolds between them these heteroclinic manifolds are foliated by heteroclinic connections between c 0 tori located at the same energy levels by means of the impact map we prove the persistence of these objects under perturbation in addition we provide sufficient conditions of the existence of transversal heteroclinic intersections through the existence of simple zeros of melnikov like functions the heteroclinic manifolds allow us to define the scattering map which links asymptotic dynamics in the invariant manifolds through heteroclinic connections first order properties of this map provide sufficient conditions for the asymptotic dynamics to be located in different energy levels in the perturbed invariant manifolds hence we have an essential tool for the construction of a heteroclinic skeleton which when followed can lead to the existence of arnold diffusion trajectories that on large time scales destabilize the system by further accumulating energy we validate all the theoretical results with detailed numerical computations of a mechanical system with impacts formed by the linkage of two rocking blocks with a spring

covering relations for multidimensional dynamical systems abstract we present a topological technique for analyzing dynamical systems with complex behavior based on the general notion of covering relations our method can be used to study multidimensional dynamical systems with an arbitrary number of topologically expanding directions

renormalization and transition to chaos in area preserving nontwist maps abstract the problem of transition to chaos i e the destruction of invariant circles or kam kolmogorov arnold moser curves in area preserving nontwist maps is studied within the renormalization group framework nontwist maps are maps for which the twist condition is violated along a curve known as the shearless curve in renormalization language this problem is that of finding and studying the fixed points of the renormalization group operator r that acts on the space of maps a simple period two fixed point of r whose basin of attraction contains the nontwist maps for which the shearless curve exists is found also a critical period 12 fixed point of r with two unstable eigenvalues is found the basin of attraction of this critical fixed point contains the nontwist maps for which the shearless curve is at the threshold of destruction this basin defines a new universality class for the transition to chaos in area preserving maps

instability of high dimensional hamiltonian systems multiple resonances do not impede diffusion abstract we consider models given by hamiltonians of the form h i p q t e h i j 1 n 1 2 p j 2 v j q j e q i p q t e where i i r d t d p q r n t t 1 these are higher dimensional analogues both in the center and hyperbolic directions of the models studied in 28 29 43 and are usually called a priori unstable hamiltonian systems all these models present the large gap problem we show that for 0 e 1 under regularity and explicit non degeneracy conditions on the model there are orbits whose action variables i perform rather arbitrary excursions in a domain of size o 1 this domain includes resonance lines and hence large gaps among d dimensional kam tori this phenomenon is known as arnold diffusion the method of proof follows closely the strategy of 28 29 the main new phenomenon that appears when the dimension d of the center directions is larger than one is the existence of multiple resonances in the space of actions i i r d we show that since these multiple resonances happen in sets of codimension greater than one in the space of actions i they can be contoured this corresponds to the mechanism called diffusion across resonances in the physics literature the present paper however differs substantially from 28 29 on the technical details of the proofs we have taken advantage of the theory of the scattering map developed in 31 notably the symplectic properties which were not available when the above papers were written we have analyzed the conditions imposed on the resonances in more detail more precisely we have found that there is a simple condition on the melnikov potential which allows us to conclude that the resonances are crossed in particular this condition does not depend on the resonances so that the results are new even when applied to the models in 28 29

on box totally dual integral polyhedra edmonds and giles introduced the class of box totally dual integral polyhedra as a generalization of submodular flow polyhedra in this paper a geometric characterization of these polyhedra is given this geometric result is used to show that each tdi defining system for a box tdi polyhedron is in fact a box tdi system that the class of box tdi polyhedra is in co np and is closed under taking projections and dominants that the class of box perfect graphs is in co np and a result of edmonds and giles which is related to the facets of box tdi polyhdera

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

technical note deriving robust and globalized robust solutions of uncertain linear programs with general convex uncertainty sets we propose a new way to derive tractable robust counterparts of a linear program based on the duality between the robust pessimistic primal problem and its optimistic dual first we obtain a new convex reformulation of the dual problem of a robust linear program and then show how to construct the primal robust solution from the dual optimal solution our result allows many new uncertainty regions to be considered we give examples of tractable uncertainty regions that were previously intractable the results are illustrated by solving a multi item newsvendor problem we also apply the new method to the globalized robust counterpart scheme and show its tractability

computation of minimum volume covering ellipsoids we present a practical algorithm for computing the minimum volume n dimensional ellipsoid that must contain m given points a1 am n this convex constrained problem arises in a variety of applied computational settings particularly in data mining and robust statistics its structure makes it particularly amenable to solution by interior point methods and it has been the subject of much theoretical complexity analysis here we focus on computation we present a combined interior point and active set method for solving this problem our computational results demonstrate that our method solves very large problem instances m 30 000 and n 30 to a high degree of accuracy in under 30 seconds on a personal computer

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

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

the dynamics of pseudographs in convex hamiltonian systems we study the evolution under convex hamiltonian flows on cotangent bundles of compact manifolds of certain distinguished subsets of the phase space these subsets are generalizations of lagrangian graphs we call them pseudographs they emerge in a natural way from fathi s weak kam theory by this method we find various orbits which connect prescribed regions of the phase space our study is inspired by works of john mather as an application we obtain the existence of diffusion in a large class of a priori unstable systems and provide a solution to the large gap problem we hope that our method will have applications to more examples

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

arnold diffusion in arbitrary degrees of freedom and normally hyperbolic invariant cylinders we prove a form of arnold diffusion in the a priori stable case let h 0 p epsilon h 1 theta p t quad theta in mathbb t n p in b n t in mathbb t mathbb r mathbb t be a nearly integrable system of arbitrary degrees of freedom n geqslant 2 with a strictly convex h0 we show that for a generic epsilon h 1 there exists an orbit theta p satisfying p t p 0 l h 1 0 where l h 1 is independent of epsilon the diffusion orbit travels along a codimension 1 resonance and the only obstruction to our construction is a finite set of additional resonances

on total dual integrality abstract we prove that each rational polyhedron of full dimension is determined by a unique minimal total dual integral system of linear inequalities with integral left hand sides thus extending a result of giles and pulleyblank and we give a characterization of total dual integrality

a hierarchy of near optimal policies for multistage adaptive optimization in this paper we propose a new tractable framework for dealing with linear dynamical systems affected by uncertainty applicable to multistage robust optimization and stochastic programming we introduce a hierarchy of near optimal polynomial disturbance feedback control policies and show how these can be computed by solving a single semidefinite programming problem the approach yields a hierarchy parameterized by a single variable the degree of the polynomial policies which controls the trade off between the optimality gap and the computational requirements we evaluate our framework in the context of three classical applications two in inventory management and one in robust regulation of an active suspension system in which very strong numerical performance is exhibited at relatively modest computational expense

a multiproduct risk averse newsvendor with law invariant coherent measures of risk we consider a multiproduct risk averse newsvendor under the law invariant coherent measures of risk we first establish several fundamental properties of the model regarding the convexity of the problem the symmetry of the solution and the impact of risk aversion specifically we show that for identical products with independent demands increased risk aversion leads to decreased orders for a large but finite number of heterogeneous products with independent demands we derive closed form approximations for the optimal order quantities the approximations are as simple to compute as the classical risk neutral solutions we also show that the risk neutral solution is asymptotically optimal as the number of products tends to be infinity and thus risk aversion has no impact in the limit for a risk averse newsvendor with dependent demands we show that positively negatively dependent demands lead to lower higher optimal order quantities than independent demands using a numerical study we examine the convergence rates of the approximations and develop additional insights into the interplay between dependent demands and risk aversion

sensitivity theorems in integer linear programming we consider integer linear programming problems with a fixed coefficient matrix and varying objective function and right hand side vector among our results we show that for any optimal solution to a linear program max wx ax b the distance to the nearest optimal solution to the corresponding integer program is at most the dimension of the problem multiplied by the largest subdeterminant of the integral matrixa using this we strengthen several integer programming proximity results of blair and jeroslow graver and wolsey we also show that the chvatal rank of a polyhedron x ax b can be bounded above by a function of the matrixa independent of the vectorb a result which as blair observed is equivalent to blair and jeroslow s theorem that each integer programming value function is a gomory function

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

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

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

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

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

a geometric approach to the existence of orbits with unbounded energy in generic periodic perturbations by a potential of generic geodesic flows of 2 we give a proof based in geometric perturbation theory of a result proved by j n mather using variational methods namely the existence of orbits with unbounded energy in perturbations of a generic geodesic flow in 2 by a generic periodic potential

tractable stochastic analysis in high dimensions via robust optimization modern probability theory whose foundation is based on the axioms set forth by kolmogorov is currently the major tool for performance analysis in stochastic systems while it offers insights in understanding such systems probability theory in contrast to optimization has not been developed with computational tractability as an objective when the dimension increases correspondingly some of its major areas of application remain unsolved when the underlying systems become multidimensional queueing networks auction design in multi item multi bidder auctions network information theory pricing multi dimensional options among others we propose a new approach to analyze stochastic systems based on robust optimization the key idea is to replace the kolmogorov axioms and the concept of random variables as primitives of probability theory with uncertainty sets that are derived from some of the asymptotic implications of probability theory like the central limit theorem in addition we observe that several desired system properties such as incentive compatibility and individual rationality in auction design are naturally expressed in the language of robust optimization in this way the performance analysis questions become highly structured optimization problems linear semidefinite mixed integer for which there exist efficient practical algorithms that are capable of solving problems in high dimensions we demonstrate that the proposed approach achieves computationally tractable methods for a analyzing queueing networks b designing multi item multi bidder auctions with budget constraints and c pricing multi dimensional options

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

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

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

topologically crossing heteroclinic connections to invariant tori we consider transition tori of arnold which have topologically crossing heteroclinic connections we prove the existence of shadowing orbits to a bi inflnite sequence of tori and of symbolic dynamics near a flnite collec tion of tori topological crossing intersections of stable and unstable manifolds of tori can be found as non trivial zeroes of certain melnikov functions our treatment relies on an extension of easton s method of correctly aligned win dows due to zgliczy

total dual integrality implies local strong unimodularity we prove that any totally dual integral description of a full dimensional polyhedron is locally strongly unimodular in every vertex

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

a dynamic game approach to distributionally robust safety specifications for stochastic systems abstract this paper presents a new safety specification method that is robust against errors in the probability distribution of disturbances our proposed distributionally robust safe policy maximizes the probability of a system remaining in a desired set for all times subject to the worst possible disturbance distribution in an ambiguity set we propose a dynamic game formulation of constructing such policies and identify conditions under which a non randomized markov policy is optimal based on this existence result we develop a practical design approach to safety oriented stochastic controllers with limited information about disturbance distributions however an associated bellman equation involves infinite dimensional minimax optimization problems since the disturbance distribution may have a continuous density to alleviate computational issues we propose a duality based reformulation method that converts the infinite dimensional minimax problem into a semi infinite program that can be solved using existing convergent algorithms we prove that there is no duality gap and that this approach thus preserves optimality the results of numerical tests confirm that the proposed method is robust against distributional errors in disturbances while a standard stochastic safety verification tool is not

hyperbolicity versus partial hyperbolicity and the transversality torsion phenomenon abstract in this paper we describe a process to create hyperbolicity in the neighbourhood of a homoclinic orbit to a partially hyperbolic torus for three degrees of freedom hamiltonian systems the transversality torsion phenomenon

exponentially small splitting of separatrices beyond melnikov analysis rigorous results in this paper we study the problem of exponentially small splitting of separatrices of one degree of freedom classical hamiltonian systems with a non autonomous perturbation which is fast and periodic in time we provide a result valid for general systems which are polynomials or trigonometric polynomials in the state variables our result consists in obtaining a rigorous proof of the asymptotic formula for the measure of the splitting we have obtained that the splitting has the asymptotic behavior k e a identifying the constants k and a in terms of the features of the system the study of our problem leads us to consider several cases in some cases assuming the per turbation is small enough it turns out that the values of k coincides with the classical melnikov approach we have identified the limit size of the perturbation for which this classical theory holds true however for the limit cases which appear naturally both in averaging theory and bifurcation theory we encounter that generically neither k nor are actually well predicted by melnikov theory

universal homoclinic bifurcations and chaos near double resonances we study the dynamics near the intersection of a weaker and a stronger resonance inn degree of freedom nearly integrable hamiltonian systems for a truncated normal form we show the existence of n 2 dimensional hyperbolic invariant tori whose whiskers intersect inmultipulse homoclinic orbits with large splitting angles the homoclinic obits are doubly asymptotic to solutions that diffuse across the weak resonance along the strong resonance we derive a universalhomoclinic tree that describes the bifurcations of these orbits which are shown to survive in the full normal form we illustrate our results on a three degree of freedom mechanical system

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

identifying effective scenarios in distributionally robust stochastic programs with total variation distance traditional stochastic programs assume that the probability distribution of uncertainty is known however in practice the probability distribution oftentimes is not known or cannot be accurately approximated one way to address such distributional ambiguity is to work with distributionally robust convex stochastic programs drsps which minimize the worst case expected cost with respect to a set of probability distributions in this paper we analyze the case where there is a finite number of possible scenarios and study the question of how to identify the critical scenarios resulting from solving a drsp we illustrate that not all but only some scenarios might have effect on the optimal value and we formally define this notion for our general class of problems in particular we examine problems where the distributional ambiguity is modeled by the so called total variation distance we propose easy to check conditions to identify effective and ineffective scenarios for that class of problems computational results show that identifying effective scenarios provides useful insight on the underlying uncertainties of the problem

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

arnold diffusion in a priori chaotic symplectic maps we assume that a symplectic real analytic map has an invariant normally hyperbolic cylinder and an associated transverse homoclinic cylinder we prove that generically in the real analytic category the boundaries of the invariant cylinder are connected by trajectories of the map

a general mechanism of diffusion in hamiltonian systems qualitative results we present a general mechanism to establish the existence of diffusing orbits in a large class of nearly integrable hamiltonian systems our approach relies on the scattering map outer dynamics and on the recurrence property of the inner dynamics restricted to a normally hyperbolic invariant manifold we apply topological methods to find trajectories that follow these two dynamics this method differs in several crucial aspects from earlier works there are virtually no assumptions on the inner dynamics as the method does not use at all the invariant objects for the inner dynamics e g primary and secondary tori lower dimensional hyperbolic tori and their stable unstable manifolds aubry mather sets the method applies when the unperturbed hamiltonian is not necessarily convex and of arbitrary degrees of freedom in addition this mechanism is easy to verify analytically or numerically in concrete examples as well as to establish diffusion in generic systems we include several applications such as bridging large gaps in a priori unstable models in any dimension and establishing diffusion in cases when the inner dynamics in a non twist map

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

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

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

a min max theorem on feedback vertex sets we establish a necessary and sufficient condition for the linear system x hx e x 0 associated with a bipartite tournament to be totally dual integral whereh is the cycle vertex incidence matrix ande is the all one vector the consequence is a min max relation on packing and covering cycles together with strongly polynomial time algorithms for the feedback vertex set problem and the cycle packing problem on the corresponding bipartite tournaments in addition we show that the feedback vertex set problem on general bipartite tournaments isnp complete and approximable within 3 5 based on the min max theorem

robust counterparts of inequalities containing sums of maxima of linear functions abstract this paper addresses the robust counterparts of optimization problems containing sums of maxima of linear functions these problems include many practical problems e g problems with sums of absolute values and arise when taking the robust counterpart of a linear inequality that is affine in the decision variables affine in a parameter with box uncertainty and affine in a parameter with general uncertainty in the literature often the reformulation is used that is exact when there is no uncertainty however in robust optimization this reformulation gives an inferior solution and provides a pessimistic view we observe that in many papers this conservatism is not mentioned some papers have recognized this problem but existing solutions are either conservative or their performance for different uncertainty regions is not known a comparison between them is not available and they are restricted to specific problems we describe techniques for general problems and compare them with numerical examples in inventory management regression and brachytherapy based on these examples we give recommendations for reducing the conservatism

speed of arnold diffusion for analytic hamiltonian systems for a convex real analytic e close to integrable hamiltonian system with n 5 degrees of freedom we construct an orbit exhibiting arnold diffusion with the diffusion time bounded by exp c epsilon frac 1 2 n 2 this upper bound of the diffusion time almost matches the lower bound of order exp epsilon frac 1 2 n 1 predicted by the nekhoroshev type stability results our method is based on the variational approach of bessi and mather and includes a new construction on the space of frequencies

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

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

a 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

data uncertainty in markov chains application to cost effectiveness analyses of medical innovations cost effectiveness studies of medical innovations often suffer from data inadequacy when markov chains are used as a modeling framework for such studies this data inadequacy can manifest itself as imprecision in the elements of the transition matrix in this paper we study how to compute maximal and minimal values for the discounted value of the chain with respect to a vector of state wise costs or rewards as these uncertain transition parameters jointly vary within a given uncertainty set we show that these problems are computationally tractable if the uncertainty set has a row wise structure conversely we prove that the row wise structure is necessary for tractability without it the problems become computationally intractable strongly np hard we apply our model to assess the cost effectiveness of fecal immunochemical testing fit a new screening method for colorectal cancer our results show that despite the large uncertainty in fit s performance it could be cost effective relative to the prevailing screening method of colonoscopy the online appendix is available at https doi org 10 1287 opre 2017 1685

compromise solutions for robust combinatorial optimization with variable sized uncertainty in classic robust optimization it is assumed that a set of possible parameter realizations the uncertainty set is modeled in a previous step and part of the input as recent work has shown finding the most suitable uncertainty set is in itself already a difficult task we consider robust problems where the uncertainty set is not completely defined only the shape is known but not its size such a setting is known as variable sized uncertainty in this work we present an approach how to find a single robust solution that performs well on average over all possible uncertainty set sizes we demonstrate that this approach can be solved efficiently for min max robust optimization but is more involved in the case of min max regret where positive and negative complexity results for the selection problem the minimum spanning tree problem and the shortest path problem are provided we introduce an iterative solution procedure and evaluate its performance in an experimental comparison

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

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

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

dual stochastic dominance and related mean risk models we consider the problem of constructing mean risk models which are consistent with the second degree stochastic dominance relation by exploiting duality relations of convex analysis we develop the quantile model of stochastic dominance for general distributions this allows us to show that several models using quantiles and tail characteristics of the distribution are in harmony with the stochastic dominance relation we also provide stochastic linear programming formulations of these models

invariant foliations near normally hyperbolic invariant manifolds for semiflows let m be a compact c1 manifold which is invariant and normally hyperbolic with respect to a c1 semiflow in a banach space then in an neighborhood of m there exist local c1 center stable and center unstable manifolds w cs and w cu respectively here we show that w cs and w cu may each be decomposed into the disjoint union of c1 submanifolds leaves in such a way that the semiflow takes leaves into leaves of the same collection furthermore each leaf intersects m in a single point which determines the asymptotic behavior of all points of that leaf in either forward or backward time

generalized polymatroids and submodular flows polyhedra related to matroids and sub or supermodular functions play a central role in combinatorial optimization the purpose of this paper is to present a unified treatment of the subject the structure of generalized polymatroids and submodular flow systems is discussed in detail along with their close interrelation in addition to providing several applications we summarize many known results within this general framework

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

orbits of unbounded energy in quasi periodic perturbations of geodesic flows abstract we show that certain mechanical systems including a geodesic flow in any dimension plus a quasi periodic perturbation by a potential have orbits of unbounded energy the assumptions we make in the case of geodesic flows are a the metric and the external perturbation are smooth enough b the geodesic flow has a hyperbolic periodic orbit such that its stable and unstable manifolds have a tranverse homoclinic intersection c the frequency of the external perturbation is diophantine d the external potential satisfies a generic condition depending on the periodic orbit considered in b the assumptions on the metric are c 2 open and are known to be dense on many manifolds the assumptions on the potential fail only in infinite codimension spaces of potentials the proof is based on geometric considerations of invariant manifolds and their intersections the main tools include the scattering map of normally hyperbolic invariant manifolds as well as standard perturbation theories averaging kam and melnikov techniques we do not need to assume that the metric is riemannian and we obtain results for finsler or lorentz metrics indeed there is a formulation for hamiltonian systems satisfying scaling hypotheses we do not need to make assumptions on the global topology of the manifold nor on its dimension

geometric properties of the scattering map of a normally hyperbolic invariant manifold given a normally hyperbolic invariant manifold for a map f whose sta ble and unstable invariant manifolds intersect transversally we consider its associated scattering map that is the map that given an asymptotic orbit in the past gives the asymptotic orbit in the future we show that when f and are symplectic resp exact symplectic then the scattering map is symplectic resp exact symplectic furthermore we show that in the exact symplectic case there are extremely easy formulas for the primitive function which have a variational interpretation as dierence of actions we use this geometric information to obtain ecien t perturbative calculations of the scattering map using deformation theory this perturbation theory generalizes and extends several results already obtained using the melnikov method analogous results are true for hamiltonian o ws the proofs are obtained by geometrically natural methods and do not involve the use of particular coordinate systems hence the results can be used to obtain intersection properties of objects of any type we also reexamine the calculation of the scattering map in a geodesic o w perturbed by a quasi periodic potential we show that the geometric theory reproduces the results obtained in dls06b using methods of fast slow systems moreover the geometric theory allows to compute perturbatively the dependence on the slow variables which does not seem to be accessible to the previous methods

unbounded energy growth in hamiltonian systems with a slowly varying parameter we study hamiltonian systems which depend slowly on time we show that if the corresponding frozen system has a uniformly hyperbolic invariant set with chaotic behaviour then the full system has orbits with unbounded energy growth under very mild genericity assumptions we also provide formulas for the calculation of the rate of the fastest energy growth we apply our general theory to non autonomous perturbations of geodesic flows and hamiltonian systems with billiard like and homogeneous potentials in these examples we show the existence of orbits with the rates of energy growth that range depending on the type of perturbation from linear to exponential in time our theory also applies to non hamiltonian systems with a first integral

homoclinic points in symplectic and volume preserving diffeomorphisms letmn be a compactn dimensional manifold and be a symplectic or volume form onmn let be ac1 diffeomorphism onmn that preserves and letp be a hyperbolic periodic point of we show that genericallyp has a homoclinic point and moreover the homoclinic points ofp is dense on both stable manifold and unstable manifold ofp takens 11 obtained the same result forn 2

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

combination laws for scaling exponents and relation to the geometry of renormalization operators the principle of approximate combination of scaling exponents renormalization group has become a standard tool for describing universal properties of different routes to chaos period doubling in unimodal maps quasiperiodic transitions in circle maps dynamics on the boundaries of siegel disks destruction of invariant circles of area preserving twist maps and others the universal scaling exponents for each route are related to the properties of the corresponding renormalization operators

invariant manifolds and the parameterization method in coupled energy harvesting piezoelectric oscillators abstract energy harvesting systems based on oscillators aim to capture energy from mechanical oscillations and convert it into electrical energy widely extended are those based on piezoelectric materials whose dynamics are hamiltonian submitted to different sources of dissipation damping and coupling these dissipations bring the system to low energy regimes which is not desired in long term as it diminishes the absorbed energy to avoid or to minimize such situations we propose that the coupling of two oscillators could benefit from theory of arnold diffusion such phenomenon studies o 1 energy variations in hamiltonian systems and hence could be very useful in energy harvesting applications this article is a first step towards this goal we consider two piezoelectric beams submitted to a small forcing and coupled through an electric circuit by considering the coupling damping and forcing as perturbations we prove that the unperturbed system possesses a 4 dimensional normally hyperbolic invariant manifold with 5 and 4 dimensional stable and unstable manifolds respectively these are locally unique after the perturbation by means of the parameterization method we numerically compute parameterizations of the perturbed manifold its stable and unstable manifolds and study its inner dynamics we show evidence of homoclinic connections when the perturbation is switched on