a convex analytic approach to risk aware markov decision processes in classical markov decision process mdp theory we search for a policy that say minimizes the expected infinite horizon discounted cost expectation is of course a risk neutral measure which does not suffice in many applications particularly in finance we replace the expectation with a general risk functional and call such models risk aware mdp models we consider minimization of such risk functionals in two cases the expected utility framework and conditional value at risk a popular coherent risk measure later we consider risk aware mdps wherein the risk is expressed in the constraints this includes stochastic dominance constraints and the classical chance constrained optimization problems in each case we develop a convex analytic approach to solve such risk aware mdps in most cases we show that the problem can be formulated as an infinite dimensional linear program lp in occupation measures when we augment the state space we provide a discretization method and finite approximati

minimax and risk averse multistage stochastic programming in this paper we study relations between the minimax risk averse and nested formulations of multistage stochastic programming problems in particular we discuss conditions for time consistency of such formulations of stochastic problems we also describe a connection between law invariant coherent risk measures and the corresponding sets of probability measures in their dual representation finally we discuss a minimax approach with moment constraints to the classical inventory model

computational methods for risk averse undiscounted transient markov models the total cost problem for discrete time controlled transient markov models is considered the objective functional is a markov dynamic risk measure of the total cost two solution methods value and policy iteration are proposed and their convergence is analyzed in the policy iteration method we propose two algorithms for policy evaluation the nonsmooth newton method and convex programming and we prove their convergence the results are illustrated on a credit limit control problem

sampling based decomposition methods for multistage stochastic programs based on extended polyhedral risk measures we define a risk averse nonanticipative feasible policy for multistage stochastic programs and propose a methodology to implement it the approach is based on dynamic programming equations written for a risk averse formulation of the problem this formulation relies on a new class of multiperiod risk functionals called extended polyhedral risk measures dual representations of such risk functionals are given and used to derive conditions of coherence in the one period case conditions for convexity and consistency with second order stochastic dominance are also provided the risk averse dynamic programming equations are specialized considering convex combinations of one period extended polyhedral risk measures such as spectral risk measures to implement the proposed policy the approximation of the risk averse recourse functions for stochastic linear programs is discussed in this context we detail a stochastic dual dynamic programming algorithm which converges to the optimal value of the risk averse p

optimal stopping under uncertainty in drift and jump intensity this paper studies the optimal stopping problem in the presence of model uncertainty ambiguity we develop a numerically implementable method to solve this problem in a general setting allowing

on consistency of stochastic dominance and mean semideviation models we analyze relations between two methods frequently used for modeling the choice among uncertain outcomes stochastic dominance and mean risk approaches new necessary conditions for stochastic dominance are developed these conditions compare values of a certain functional which contains two components the expected value of a random outcome and a risk term represented by the central semideviation of the corresponding degree if the weight of the semideviation in the composite objective does not exceed the weight of the expected value maximization of such a functional yields solutions which are efficient in terms of stochastic dominance the results are illustrated graphically

conditionally evenly convex sets and evenly quasi convex maps abstract evenly convex sets in a topological vector space are defined as the intersection of a family of open half spaces we introduce a generalization of this concept in the conditional framework and provide a generalized version of the bipolar theorem this notion is then applied to obtain the dual representation of conditionally evenly quasi convex maps which turns out to be a fundamental ingredient in the study of quasi convex dynamic risk measures

selecting optimal alternatives and risk reduction strategies in decision trees in this paper we conduct a quantitative analysis for a strategic risk management problem that involves allocating certain available failure mitigating and consequence alleviating resources to reduce the failure probabilities of system safety components and subsequent losses respectively together with selecting optimal strategic decision alternatives to minimize the risk or expected loss in the event of a hazardous occurrence using a novel decision tree optimization approach to represent the cascading sequences of probabilistic events as controlled by key decisions and investment alternatives the problem is modeled as a nonconvex mixed integer 0 1 factorable program we develop a specialized branch and bound algorithm in which lower bounds are computed via tight linear relaxations of the original problem that are constructed by utilizing a polyhedral outer approximation mechanism in concert with two alternative linearization schemes having different levels of tightness and complexity we also suggest three alternative branching schemes each of which is proven to guarantee convergence to a global optimum for the underlying problem extensive computational results and sensitivity analyses are presented to provide insights and to demonstrate the efficacy of the proposed algorithm

measures of residual risk with connections to regression risk tracking surrogate models and ambiguity measures of residual risk are developed as an extension of measures of risk they view a random variable of interest in concert with an auxiliary random vector that helps to manage predict and mitigate the risk in the original variable residual risk can be exemplified as a quantification of the improved situation faced by a hedging investor compared to that of a single asset investor but the notion reaches further with deep connections emerging with forecasting and generalized regression we establish the fundamental properties in this framework and show that measures of residual risk along with generalized regression can play central roles in the development of risk tuned approximations of random variables in tracking of statistics and in estimation of the risk of conditional random variables the paper ends with dual expressions for measures of residual risk which lead to further insights and a new class of distributionally robust optimization models

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

semidefinite relaxations of ordering problems ordering problems assign weights to each ordering and ask to find an ordering of maximum weight we consider problems where the cost function is either linear or quadratic in the first case there is a given profit if the element u is before v in the ordering in the second case the profit depends on whether u is before v and r is before s the linear ordering problem is well studied with exact solution methods based on polyhedral relaxations the quadratic ordering problem does not seem to have attracted similar attention we present a systematic investigation of semidefinite optimization based relaxations for the quadratic ordering problem extending and improving existing approaches we show the efficiency of our relaxations by providing computational experience on a variety of problem classes

stochastic programming approach to optimization under uncertainty in this paper we discuss computational complexity and risk averse approaches to two and multistage stochastic programming problems we argue that two stage say linear stochastic programming problems can be solved with a reasonable accuracy by monte carlo sampling techniques while there are indications that complexity of multistage programs grows fast with increase of the number of stages we discuss an extension of coherent risk measures to a multistage setting and in particular dynamic programming equations for such problems

median linear orders heuristics and a branch and bound algorithm abstract after recalling some facts about the median linear orders problem two heuristics are studied to solve it then a branch and bound method is designed which computes all the solutions

robust risk management estimating the probabilities by which different events might occur is usually a delicate task subject to many sources of inaccuracies moreover these probabilities can change over time leading to a very difficult evaluation of the risk induced by any particular decision given a set of probability measures and a set of nominal risk measures we define in this paper the concept of robust risk measure as the worst possible of our risks when each of our probability measures is likely to occur we study how some properties of this new object can be related with those of our nominal risk measures such as convexity or coherence we introduce a robust version of the conditional value at risk cvar and of entropy based risk measures we show how to compute and optimize the robust cvar using convex duality methods and illustrate its behavior using data from the new york stock exchange and from the nasdaq between 2005 and 2010

capital allocation a la aumann shapley for non differentiable risk measures abstract we study capital allocation rules satisfying suitable properties for convex and quasi convex risk measures by focusing in particular on a family of capital allocation rules based on the dual representation for risk measures and inspired by the aumann shapley allocation principle these rules extend some well known methods of capital allocation for coherent and convex risk measures to the case of non gateaux differentiable risk measures we also analyze the properties of the allocation principles here introduced and discuss their suitability in the quasi convex context

entropy coherent and entropy convex measures of risk we introduce two subclasses of convex measures of risk referred to as entropy coherent and entropy convex measures of risk entropy coherent and entropy convex measures of risk are special cases of coherent and convex measures of risk contrary to the classical use of coherent and convex measures of risk which for a given probabilistic model entails evaluating a financial position by considering its expected loss coherent and convex measures of risk evaluate a financial position under a given probabilistic model by considering its normalized expected loss we prove that i entropy coherent and entropy convex measures of risk are obtained by requiring coherent and convex measures of risk to be translation invariant ii convex entropy convex and entropy coherent measures of risk emerge as certainty equivalents under variational homothetic and multiple priors preferences upon requiring the certainty equivalents to be translation invariant and iii convex measures of risk are certainty equivalents under variational and homothetic preferences if and only if they are convex and entropy convex measures of risk in addition we study the properties of entropy coherent and entropy convex measures of risk derive their dual conjugate function and characterize entropy coherent and entropy convex measures of risk in terms of properties of the corresponding acceptance sets

risk trading and endogenous probabilities in investment equilibria a risky design equilibrium problem is an equilibrium system that involves n designers who invest in risky assets such as production plants evaluate these using convex or coherent risk measures and also trade financial securities in order to manage their risk our main finding is that in a complete risk market when all uncertainties can be replicated by financial products a risky design equilibrium problem collapses to what we call a risky design game i e a stochastic nash game in which the original design agents act as risk neutral and there emerges an additional system risk agent the system risk agent simultaneously prices risk and determines the probability density used by the other agents for their risk neutral evaluations this situation is stochastic endogenous the probability density used by agents to value uncertain investments is endogenous to the risky design equilibrium problem this result is most striking when design agents use coherent risk measures in which case the intersectio

optimization of convex risk functions we consider optimization problems involving convex risk functions by employing techniques of convex analysis and optimization theory in vector spaces of measurable functions we develop new representation theorems for risk models and optimality and duality theory for problems with convex risk functions

portfolio optimization with quasiconvex risk measures in this paper we focus on the portfolio optimization problem associated with a quasiconvex risk measure satisfying some additional assumptions for coherent convex risk measures the portfolio optimization problem has been already studied in the literature following the approach of ruszczynski and shapiro ruszczynski a shapiro a 2006 optimization of convex risk functions math oper res 31 3 433 452 but by means of quasiconvex analysis and notions of subdifferentiability we characterize optimal solutions of the portfolio problem associated with quasiconvex risk measures the shape of the efficient frontier in the mean risk space and some particular cases are also investigated

bounds on risk averse mixed integer multi stage stochastic programming problems with mean cvar abstract risk averse mixed integer multi stage stochastic programming forms a class of extremely challenging problems since the problem size grows exponentially with the number of stages the problem is non convex due to integrality restrictions and the objective function is nonlinear in general we propose a scenario tree decomposition approach namely group subproblem approach to obtain bounds for such problems with an objective of dynamic mean conditional value at risk mean cvar our approach does not require any special problem structure such as convexity and linearity therefore it can be applied to a wide range of problems we obtain lower bounds by using different convolution of mean cvar risk measures and different scenario partition strategies the upper bounds are obtained through the use of optimal solutions of group subproblems using these lower and upper bounds we propose a solution algorithm for risk averse mixed integer multi stage stochastic problems with mean cvar risk measures we test the performance of the proposed algorithm on a multi stage stochastic lot sizing problem and compare different choices of lower bounds and partition strategies comparison of the proposed algorithm to a commercial solver revealed that on the average the proposed algorithm yields 1 13 stronger bounds the commercial solver requires additional running time more than a factor of five on the average to reach the same optimality gap obtained by the proposed algorithm

risk optimization with p order conic constraints a linear programming approach the paper considers solving of linear programming problems with p order conic constraints that are related to a certain class of stochastic optimization models with risk objective or constraints the proposed approach is based on construction of polyhedral approximations for p order cones and then invoking a benders decomposition scheme that allows for efficient solving of the approximating problems the conducted case study of portfolio optimization with p order conic constraints demonstrates that the developed computational techniques compare favorably against a number of benchmark methods including second order conic programming methods

analysis of stochastic dual dynamic programming method abstract in this paper we discuss statistical properties and convergence of the stochastic dual dynamic programming sddp method applied to multistage linear stochastic programming problems we assume that the underline data process is stagewise independent and consider the framework where at first a random sample from the original true distribution is generated and consequently the sddp algorithm is applied to the constructed sample average approximation saa problem then we proceed to analysis of the sddp solutions of the saa problem and their relations to solutions of the true problem finally we discuss an extension of the sddp method to a risk averse formulation of multistage stochastic programs we argue that the computational complexity of the corresponding sddp algorithm is almost the same as in the risk neutral case

tight approximations of dynamic risk measures this paper compares two frameworks for measuring risk in a multiperiod setting the first corresponds to applying a single coherent risk measure to the cumulative future costs and the second involves applying a composition of one step coherent risk mappings we characterize several necessary and sufficient conditions under which one measurement always dominates the other and introduce a metric to quantify how close the two measures are using this notion we address the question of how tightly a given coherent measure can be approximated by lower or upper bounding compositional measures we exhibit an interesting asymmetry between the two cases the tightest upper bound can be exactly characterized and corresponds to a popular construction in the literature whereas the tightest lower bound is not readily available we show that testing domination and computing the approximation factors are generally np hard even when the risk measures are comonotonic and law invariant however we characterize conditio

risk preferences on the space of quantile functions we propose a novel approach to quantification of risk preferences on the space of nondecreasing functions when applied to law invariant risk preferences among random variables it compares their quantile functions the axioms on quantile functions impose relations among comonotonic random variables we infer the existence of a numerical representation of the preference relation in the form of a quantile based measure of risk using conjugate duality theory by pairing the banach space of bounded functions with the space of finitely additive measures on a suitable algebra varsigma we develop a variational representation of the quantile based measures of risk furthermore we introduce a notion of risk aversion based on quantile functions which enables us to derive an analogue of kusuoka representation of coherent law invariant measures of risk

robust stochastic dominance and its application to risk averse optimization we introduce a new preference relation in the space of random variables which we call robust stochastic dominance we consider stochastic optimization problems where risk aversion is expressed by a robust stochastic dominance constraint these are composite semi infinite optimization problems with constraints on compositions of measures of risk and utility functions we develop necessary and sufficient conditions of optimality for such optimization problems in the convex case in the nonconvex case we derive necessary conditions of optimality under additional smoothness assumptions of some mappings involved in the problem

the linear ordering problem revisited abstract the linear ordering problem is a popular combinatorial optimisation problem which has been extensively addressed in the literature however in spite of its popularity little is known about the characteristics of this problem this paper studies a procedure to extract static information from an instance of the problem and proposes a method to incorporate the obtained knowledge in order to improve the performance of local search based algorithms the procedure introduced identifies the positions where the indexes cannot generate local optima for the insert neighbourhood and thus global optima solutions this information is then used to propose a restricted insert neighbourhood that discards the insert operations which move indexes to positions where optimal solutions are not generated in order to measure the efficiency of the proposed restricted insert neighbourhood system two state of the art algorithms for the lop that include local search procedures have been modified conducted experiments confirm that the restricted versions of the algorithms outperform the classical designs systematically when a maximum number of function evaluations is considered as the stopping criterion the statistical test included in the experimentation reports significant differences in all the cases which validates the efficiency of our proposal moreover additional experiments comparing the execution times reveal that the restricted approaches are faster than their counterparts for most of the instances

risk averse control of undiscounted transient markov models we use markov risk measures to formulate a risk averse version of the undiscounted total cost problem for a transient controlled markov process using the new concept of a multikernel we derive conditions for a system to be risk transient that is to have finite risk over an infinite time horizon we derive risk averse dynamic programming equations satisfied by the optimal policy and we describe methods for solving these equations we illustrate the results on an optimal stopping problem and an organ transplantation problem

rectangular sets of probability measures in this paper we consider the notion of rectangularity of a set of probability measures from a somewhat different point of view we define rectangularity as a property of dynamic decomposition of a distributionally robust stochastic optimization problem and show how it relates to the modern theory of coherent risk measures consequently we discuss robust formulations of multistage stochastic optimization problems in frameworks of stochastic programming stochastic optimal control and markov decision processes

robust portfolio choice and indifference valuation we solve theoretically and numerically the problems of optimal portfolio choice and indifference valuation in a general continuous time setting the setting features i ambiguity and time consistent ambiguity averse preferences ii discontinuities in the asset price processes with a general and possibly infinite activity jump part next to a continuous diffusion part and iii general and possibly nonconvex trading constraints we characterize our solutions as solutions to backward stochastic differential equations bsdes generalizing kobylanski s result for quadratic bsdes to an infinite activity jump setting we prove existence and uniqueness of the solution to a general class of bsdes encompassing the solutions to our portfolio choice and valuation problems as special cases we provide an explicit decomposition of the excess return on an asset into a risk premium and an ambiguity premium and a further decomposition into a piece stemming from the diffusion part and a piece stemming from the ju

risk preferences and their robust representation to address the plurality of interpretations of the subjective notion of risk we describe it by means of a risk order and concentrate on the context invariant features of diversification and monotonicity our main results are uniquely characterized robust representations of lower semicontinuous risk orders on vector spaces and convex sets this representation covers most instruments related to risk and allows for a differentiated interpretation depending on the underlying context that is illustrated in different settings for random variables risk perception can be interpreted as model risk and we compute among others the robust representation of the economic index of riskiness for lotteries risk perception can be viewed as distributional risk and we study the value at risk for consumption patterns which excerpt an intertemporality dimension in risk perception we provide an interpretation in terms of discounting risk and discuss some examples

conditional risk mappings we introduce an axiomatic definition of a conditional convex risk mapping and we derive its properties in particular we prove a representation theorem for conditional risk mappings in terms of conditional expectations we also develop dynamic programming relations for multistage optimization problems involving conditional risk mappings

optimality conditions in portfolio analysis with general deviation measures optimality conditions are derived for problems of minimizing a general measure of deviation of a random variable with special attention to situations where the random variable could be the rate of return from a portfolio of financial instruments general measures of deviation go beyond standard deviation in satisfying axioms that do not demand symmetry between ups and downs the optimality conditions are applied to characterize the generalized master funds which elsewhere have been developed in extending classical portfolio theory beyond the case of standard deviation the consequences are worked out for deviation based on conditional value at risk and its variants in particular

the domain of definition of the complex monge ampere operator we give a precise characterization of those plurisubharmonic functions for which one can well define the monge ampere operator as a regular borel measure

optimization under probabilistic envelope constraints chance constraints are an important modeling tool in stochastic optimization providing probabilistic guarantees that a solution succeeds in satisfying a given constraint although they control the probability of success they provide no control whatsoever in the event of a failure that is they do not distinguish between a slight overshoot or undershoot of the bounds and more catastrophic violation in short they do not capture the magnitude of violation of the bounds this paper addresses precisely this topic focusing on linear constraints and ellipsoidal gaussian like uncertainties we show that the problem of requiring different probabilistic guarantees at each level of constraint violation can be reformulated as a semi infinite optimization problem we provide conditions that guarantee polynomial time solvability of the resulting semi infinite formulation we show further that this resulting problem is what has been called a comprehensive robust optimization problem in the literature as a byproduct we provide tight probabilistic bounds for comprehensive robust optimization thus analogously to the connection between chance constraints and robust optimization we provide a broader connection between probabilistic envelope constraints and comprehensive robust optimization

risk averse shape optimization risk averse optimization has attracted much attention in finite dimensional stochastic programming in this paper we propose a risk averse approach in the infinite dimensional context of shape optimization we consider elastic materials under stochastic loading as measures of risk awareness we investigate the expected excess and the excess probability the developed numerical algorithm is based on a regularized gradient flow acting on an implicit description of the shapes based on level sets we incorporate topological derivatives to allow for topological changes in the shape optimization procedure numerical results in two dimensions demonstrate the impact of the risk averse modeling on the optimal shapes and on the cost distribution over the set of scenarios

a lagrangean based branch and bound algorithm for single machine sequencing with precedence constraints to minimize total weighted completion time the single machine sequencing problem is considered in which there are precedence constraints on the jobs the objective is to minimize the sum of weighted completion times a lower bound is obtained by successively performing a lagrangean relaxation of appropriate constraints each lagrange multiplier is chosen to provide the maximum increment to the lower bound subject to retaining the nonnegativity of the coefficients of the variables when no further suitable constraints can be introduced into the lagrangean function the variables having zero cost coefficient are used to obtain a feasible sequence which provides an upper bound the gap between the lower and upper bound is reduced by removing some constraints from the lagrangean function and replacing them with others this lower bounding procedure is used in a branch and bound algorithm computational results indicate that the algorithm can satisfactorily solve problems with up to 100 jobs

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

two stage non cooperative games with risk averse players this paper formally introduces and studies a non cooperative multi agent game under uncertainty the well known nash equilibrium is employed as the solution concept of the game while there are several formulations of a stochastic nash equilibrium problem we focus mainly on a two stage setting of the game wherein each agent is risk averse and solves a rival parameterized stochastic program with quadratic recourse in such a game each agent takes deterministic actions in the first stage and recourse decisions in the second stage after the uncertainty is realized each agent s overall objective consists of a deterministic first stage component plus a second stage mean risk component defined by a coherent risk measure describing the agent s risk aversion we direct our analysis towards a broad class of quantile based risk measures and linear quadratic recourse functions for this class of non cooperative games under uncertainty the agents objective functions can be shown to be convex in their own decision variables provided that the deterministic component of these functions have the same convexity property nevertheless due to the non differentiability of the recourse functions the agents objective functions are at best directionally differentiable such non differentiability creates multiple challenges for the analysis and solution of the game two principal ones being 1 a stochastic multi valued variational inequality is needed to characterize a nash equilibrium provided that the players optimization problems are convex 2 one needs to be careful in the design of algorithms that require differentiability of the objectives moreover the resulting multi valued variational formulation cannot be expected to be of the monotone type in general the main contributions of this paper are as follows a prior to addressing the main problem of the paper we summarize several approaches that have existed in the literature to deal with uncertainty in a non cooperative game b we introduce a unified formulation of the two stage snep with risk averse players and convex quadratic recourse functions and highlight the technical challenges in dealing with this game c to handle the lack of smoothness we propose smoothing schemes and regularization that lead to differentiable approximations d to deal with non monotonicity we impose a generalized diagonal dominance condition on the players smoothed objective functions that facilitates the application and ensures the convergence of an iterative best response scheme e to handle the expectation operator we rely on known methods in stochastic programming that include sampling and approximation f we provide convergence results for various versions of the best response scheme particularly for the case of private recourse functions overall this paper lays the foundation for future research into the class of sneps that provides a constructive paradigm for modeling and solving competitive decision making problems with risk averse players facing uncertainty this paradigm is very much at an infancy stage of research and requires extensive treatment in order to meet its broad applications in many engineering and economics domains

from stochastic dominance to mean risk models semideviations as risk measures abstract two methods are frequently used for modeling the choice among uncertain outcomes stochastic dominance and mean risk approaches the former is based on an axiomatic model of risk averse preferences but does not provide a convenient computational recipe the latter quantifies the problem in a lucid form of two criteria with possible trade off analysis but cannot model all risk averse preferences in particular if variance is used as a measure of risk the resulting mean variance markowitz model is in general not consistent with stochastic dominance rules this paper shows that the standard semideviation square root of the semivariance as the risk measure makes the mean risk model consistent with the second degree stochastic dominance provided that the trade off coefficient is bounded by a certain constant similar results are obtained for the absolute semideviation and for the absolute and standard deviations in the case of symmetric or bounded distributions in the analysis we use a new tool the outcome risk o r diagram which appears to be particularly useful for comparing uncertain outcomes

risk forms representation disintegration and application to partially observable two stage systems we introduce the concept of a risk form which is a real functional of two arguments a measurable function on a polish space and a measure on that space we generalize the duality theory and the kusuoka representation to this setting for a risk form acting on a product of polish spaces we define marginal and conditional forms and we prove a disintegration formula which represents a risk form as a composition of its marginal and conditional forms we apply the proposed approach to two stage stochastic programming problems with partial information and decision dependent observation distribution

on kusuoka representation of law invariant risk measures in this paper we discuss representations of law invariant coherent risk measures in a form of integrals of the average value at risk measures we show that such an integral representation exists iff the dual set of the considered risk measure is generated by one of its elements and this representation is uniquely defined on the other hand representation of risk measures as a maximum of such integral forms is not unique the suggested approach gives a constructive way for writing such representations

sensitivity analysis in applications with deviation risk regret and error measures the envelope formula is obtained for optimization problems with positively homogeneous convex functionals defined on a space of random variables those problems include linear regression with general error measures and optimal portfolio selection with the objective function being either a general deviation measure or a coherent risk measure subject to a constraint on the expected rate of return the obtained results are believed to be novel even for markowitz s mean variance portfolio selection but are far more general and include explicit envelope relationships for the rates of return of portfolios that minimize lower semivariance mean absolute deviation deviation measures of cal l p type and semi cal l p type and conditional value at risk in each case the envelope theorem yields explicit estimates for the absolute value of the difference between deviation risk of optimal portfolios with the unperturbed and perturbed asset probability distributions in terms of a norm of the perturbation

optimal stopping under probability distortions in this paper we study optimal stopping problems with respect to distorted expectations with concave distortion functions our starting point is a seminal work of xu and zhou in 2013 who gave an explicit solution of such a stopping problem under a rather large class of distortion functionals in this paper we continue this line of research and prove a novel representation which relates the solution of an optimal stopping problem under distorted expectation to the sequence of standard optimal stopping problems and hence makes the application of the standard dynamic programming based approaches possible furthermore by means of the well known kusuoka representation we extend our results to optimal stopping under general law invariant coherent risk measures finally based on our representations we develop several monte carlo approximation algorithms and illustrate their power for optimal stopping under absolute semideviation risk measures

convergence analysis of sampling based decomposition methods for risk averse multistage stochastic convex programs we consider a class of sampling based decomposition methods to solve risk averse multistage stochastic convex programs we prove a formula for the computation of the cuts necessary to build the outer linearizations of the recourse functions this formula can be used to obtain an efficient implementation of stochastic dual dynamic programming applied to convex nonlinear problems we prove the almost sure convergence of these decomposition methods when the relatively complete recourse assumption holds we also prove the almost sure convergence of these algorithms when applied to risk averse multistage stochastic linear programs that do not satisfy the relatively complete recourse assumption the analysis is first done assuming the underlying stochastic process is interstage independent and discrete with a finite set of possible realizations at each stage we then indicate two ways of extending the methods and convergence analysis to the case when the process is interstage dependent

risk averse stochastic structural topology optimization abstract a novel approach for risk averse structural topology optimization under uncertainties is presented which takes into account stochastic data of the state equation specifically random material properties and random forces for the distribution of material a phase field approach is employed which allows for arbitrary topological changes during the iterative optimization the state equation is assumed to be a high dimensional pde parametrized in a truncated finite set of random variables the examined case employs linearized elasticity with a parametric elasticity tensor for practical purposes instead of an optimization with respect to the expectation of the involved random fields the designed structures should in particular be robust with respect to rather unlikely and possibly critical events for this as a common risk measure the conditional value at risk cvar is introduced to the cost functional of the minimization procedure the proposed method is illustrated with numerical examples based on monte carlo sampling for different risk values and compared with the result of the deterministic formulation it is observed that the resulting shapes dependent on the risk parameter of the functional and can deviate significantly from the deterministic case

a branch and bound algorithm for the acyclic subgraph problem abstract a branch and bound algorithm for the acyclic subgraph problem feedback are set problem is described the branching scheme lexicographically enumerates all permutations skipping initial segments known by some easy tests not to have any optimal completion a lower bound for the number of feedback arcs is given by the size of any collection of disjoint cycles we propose a heuristic algorithm to find a large collection the size of the problems our branch and bound algorithm can solve varies from 25 to 34 nodes depending on the nature of the problem

risk averse two stage stochastic linear programming modeling and decomposition we formulate a risk averse two stage stochastic linear programming problem in which unresolved uncertainty remains after the second stage the objective function is formulated as a composition of conditional risk measures we analyze properties of the problem and derive necessary and sufficient optimality conditions next we construct a new decomposition method for solving the problem that exploits the composite structure of the objective function we illustrate its performance on a portfolio optimization problem

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

risk averse stochastic path detection abstract we introduce the stochastic path detection problem and propose a risk averse solution approach the problem comprises an invader and a protector that both operate on a network with a number of possible source destination paths the protector aims at allocating security resources on the network such that the invader s path is detected with high probability the invader s choice of path is known to the protector in terms of a probability distribution reflecting the protector s beliefs errors in these beliefs induce the risk of a low detection probability we derive a linear programming approximation and leverage the theory of coherent risk measures to consider risk aversion with respect to errors in the protector s beliefs the performance of the resulting risk averse detection policy is numerically compared with the performance of the risk neutral policy we show that the risk averse policy significantly mitigates the risk of facing a low detection probability in the presence of large errors in the protector s beliefs

simulating sensitivities of conditional value at risk conditional value at risk cvar is both a coherent risk measure and a natural risk statistic it is often used to measure the risk associated with large losses in this paper we study how to estimate the sensitivities of cvar using monte carlo simulation we first prove that the cvar sensitivity can be written as a conditional expectation for general loss distributions we then propose an estimator of the cvar sensitivity and analyze its asymptotic properties the numerical results show that the estimator works well furthermore we demonstrate how to use the estimator to solve optimization problems with cvar objective and or constraints and compare it to a popular linear programming based algorithm

evaluations of risk measures for different probability measures stochastic optimization problems which arise in different areas such as simple asset allocation problems or problems in insurance often involve coherent risk measures in these real world problems the considered risk measure is frequently built on empirical distributions it is therefore of interest to understand the potential deviation which occurs when evaluating a risk measure for a perturbed or slightly changed probability measure this paper addresses the potential deviation it turns out that the wasserstein distance a well known distance for probability measures provides a valuable notion of distance in the present context many risk measures allow a precise quantification in terms of the wasserstein distance and important risk measures are continuous with respect to this distance for specific random variables which often occur in concrete real world problems it is moreover demonstrated that the derived constants describing the continuity properties cannot be improved the associated

risk tuning with generalized linear regression a framework is set up in which linear regression as a way of approximating a random variable by other random variables can be carried out in a variety of ways which moreover can be tuned to the needs of a particular model in finance or operations research more broadly although the idea of adapting the form of regression to the circumstances at hand has already found advocates in promoting quantile regression as an alternative to classical least squares approaches it is carried here much farther than that axiomatic concepts of error measure deviation measure and risk measure are coordinated with certain statistics that likewise say something about a random variable problems of regression utilizing these concepts are analyzed and the character of their solutions is explored in a range of examples special attention is paid to parametric forms of regression which arise in connection with factor models it is argued that when different aspects of risk enter an optimization problem different forms of regression ought to be invoked for each of those aspects

var as the cvar sensitivity var minimization is a complex problem playing a critical role in many actuarial and financial applications of mathematical programming the usual methods of convex programming do not apply due to the lack of sub additivity the usual methods of differentiable programming do not apply either due to the lack of continuity taking into account that the cvar may be given as an integral of var one has that var becomes a first order mathematical derivative of cvar this property will enable us to give accurate approximations in var optimization since the optimization var and cvar will become quite closely related topics applications in both finance and insurance will be given

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

equilibrium routing under uncertainty we provide a brief introduction to the basic models used to describe traffic on congested networks both in urban transport and telecommunications we discuss traffic equilibrium models covering atomic and non atomic routing games with emphasis on situations where the travel times are subject to random fluctuations we use convex optimization to present the models in a unified framework that stresses the common underlying structures as a prototypical example of traffic equilibrium with elastic demands we discuss some models for routing and congestion control in telecommunications we also describe a class of stochastic dynamics that model the adaptive behavior of agents and which provides a plausible micro foundation for the equilibrium finally we present some recent ideas on how risk averse behavior might be incorporated in the equilibrium models

additive consistency of risk measures and its application to risk averse routing in networks we investigate the use of risk measures and theories of choice to model risk averse routing and traffic equilibrium on networks with random travel times we interpret the postulates of these theories in the context of routing identifying additive consistency as a plausible condition that allows to reduce risk averse route choice to a standard shortest path problem within the classical theories of choice we show that the only preferences that satisfy this condition are the ones induced by the entropic risk measures

variational theory for optimization under stochastic ambiguity stochastic ambiguity provides a rich class of uncertainty models that includes those in stochastic robust risk based and semi infinite optimization and that accounts for uncertainty about parameter values as well as incompleteness of the description of uncertainty we provide a novel unifying perspective on optimization under stochastic ambiguity that rests on two pillars first ambiguity is formulated in terms of the cumulative probability distribution associated with the random elements more specifically ambiguity is expressed by letting this distribution belong to a subfamily of distributions that might or might not depend on the decision variable we derive a series of estimates by introducing a metric for the space of distribution functions based on the hypo distance between upper semicontinuous functions in the process we show that this metric is consistent with convergence in distribution weak star convergence of the associated probability measures second we rely on the theory

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

risk adjusted probability measures in portfolio optimization with coherent measures of risk abstract we consider the problem of optimizing a portfolio of n assets whose returns are described by a joint discrete distribution we formulate the mean risk model using as risk functionals the semideviation deviation from quantile and spectral risk measures using the modern theory of measures of risk we derive an equivalent representation of the portfolio problem as a zero sum matrix game and we provide ways to solve it by convex optimization techniques in this way we reconstruct new probability measures which constitute part of the saddle point of the game these risk adjusted measures always exist irrespective of the completeness of the market we provide an illustrative example in which we derive these measures in a universe of 200 assets and we use them to evaluate the market portfolio and optimal risk averse portfolios

on the acyclic subgraph polytope the acyclic subgraph problem can be formulated as follows given a digraph with arc weights find a set of arcs containing no directed cycle and having maximum total weight we investigate this problem from a polyhedral point of view and determine several classes of facets for the associated acyclic subgraph polytope we also show that the separation problem for the facet defining dicycle inequalities can be solved in polynomial time this implies that the acyclic subgraph problem can be solved in polynomial time for weakly acyclic digraphs this generalizes a result of lucchesi for planar digraphs

minimizing measures of risk by saddle point conditions the minimization of risk functions is becoming a very important topic due to its interesting applications in mathematical finance and actuarial mathematics this paper addresses this issue in a general framework many types of risk function may be involved a general representation theorem of risk functions is used in order to transform the initial optimization problem into an equivalent one that overcomes several mathematical caveats of risk functions this new problem involves banach spaces but a mean value theorem for risk measures is stated and this simplifies the dual problem then optimality is characterized by saddle point properties of a bilinear expression involving the primal and the dual variable this characterization is significantly different if one compares it with previous literature furthermore the saddle point condition very easily applies in practice four applications in finance and insurance are presented

time consistent decisions and temporal decomposition of coherent risk functionals in management and planning it is commonplace for additional information to become available gradually over time it is well known that most risk measures risk functionals are time inconsistent in the following sense it may happen that at a given time period some loss distribution appears to be less risky than another one but looking at the conditional distribution at a later time the opposite relation holds almost surely the extended conditional risk functionals introduced in this paper enable a temporal decomposition of the initial risk functional that can be used to ensure consistency between past and future preferences the central result is a decomposition theorem which allows recomposing the initial coherent risk functional by compounding the conditional risk functionals without losing information or preferences it follows from our results that the revelation of partial information in time must change the decision maker s preferences for consistency reasons among the remaining courses of act

goal driven optimization we develop a goal driven stochastic optimization model that considers a random objective function in achieving an aspiration level target or goal our model maximizes the shortfall aware aspiration level criterion which encompasses the probability of success in achieving the aspiration level and an expected level of underperformance or shortfall the key advantage of the proposed model is its tractability we can obtain its solution by solving a small collection of stochastic linear optimization problems with objectives evaluated under the popular conditional value at risk cvar measure using techniques in robust optimization we propose a decision rule based deterministic approximation of the goal driven optimization problem by solving subproblems whose number is a polynomial with respect to the accuracy with each subproblem being a second order cone optimization problem socp we compare the numerical performance of the deterministic approximation with sampling based approximation and report the computational insights on a multiproduct newsvendor problem

risk averse dynamic programming for markov decision processes we introduce the concept of a markov risk measure and we use it to formulate risk averse control problems for two markov decision models a finite horizon model and a discounted infinite horizon model for both models we derive risk averse dynamic programming equations and a value iteration method for the infinite horizon problem we develop a risk averse policy iteration method and we prove its convergence we also propose a version of the newton method to solve a nonsmooth equation arising in the policy iteration method and we prove its global convergence finally we discuss relations to min max markov decision models

a cutting plane algorithm for the linear ordering problem the linear ordering problem is an np hard combinatorial optimization problem with a large number of applications including triangulation of input output matrices archaeological senation minimizing total weighted completion time in one machine scheduling and aggregation of individual preferences in a former paper we have investigated the facet structure of the 0 1 polytope associated with the linear ordering problem here we report on a new algorithm that is based on these theoretical results the main part of the algorithm is a cutting plane procedure using facet defining inequalities this procedure is combined with various heuristics and branch and bound techniques our computational results compare favorably with the results of existing codes in particular we could triangulate all input output matrices of size up to 60 60 available to us within acceptable time bounds

subdifferential representations of risk measures measures of risk appear in two categories risk capital measures serve to determine the necessary amount of risk capital in order to avoid ruin if the outcomes of an economic activity are uncertain and their negative values may be interpreted as acceptability measures safety measures pure risk measures risk deviation measures are natural generalizations of the standard deviation while pure risk measures are typically convex acceptability measures are typically concave in both cases the convexity concavity implies under mild conditions the existence of subgradients supergradients the present paper investigates the relation between the subgradient supergradient representation and the properties of the corresponding risk measures in particular we show how monotonicity properties are reflected by the subgradient representation once the subgradient supergradient representation has been established it is extremely easy to derive these monotonicity properties we give a list of examples

controlled markov decision processes with avar criteria for unbounded costs in this paper we consider the control problem with the average value at risk avar criteria of the possibly unbounded l1 costs in infinite horizon on a markov decision process mdp with a suitable state aggregation and by choosing a priori a global variable s heuristically we show that there exist optimal policies for the infinite horizon problem for possibly unbounded costs

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