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

beam search heuristic to solve stochastic integer problems under probabilistic constraints abstract this paper proposes a beam search heuristic strategy to solve stochastic integer programming problems under probabilistic constraints beam search is an adaptation of the classical branch and bound method in which at any level of the search tree only the most promising nodes are kept for further exploration whereas the remaining are pruned out permanently the proposed algorithm has been compared with the branch and bound method the numerical results collected on the probabilistic set covering problem show that the beam search technique is very efficient and appears to be a promising tool to solve difficult stochastic integer problems under probabilistic constraints

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

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

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

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

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

minimal concave cost rebalance of a portfolio to the efficient frontier one usually constructs a portfolio on the efficient frontier but it may not be efficient after say three months since the efficient frontier will shift as the elapse of time we then have to rebalance the portfolio if the deviation is no longer acceptable the method to be proposed in this paper is to find a portfolio on the new efficient frontier such that the total transaction cost required for this rebalancing is minimal this problem results in a nonconvex minimization problem if we use mean variance model in this paper we will formulate this problem by using absolute deviation as the measure of risk and solve the resulting linearly constrained concave minimization problem by a branch and bound algorithm successfully applied to portfolio optimization problem under concave transaction costs it will be demonstrated that this method is efficient and that it leads to a significant reduction of transaction costs

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

insdecm an interactive procedure for stochastic multicriteria decision problems abstract a new interactive technique for a discrete stochastic multiattribute decision making problem is proposed in this paper it is assumed that performance probability distribution for each action on each attribute is known two concepts are combined in the procedure stochastic dominance and interactive approach the first one is employed for generating efficient actions and constructing rankings of actions with respect to attributes the second concept is used when the communication between the dm and the model is conducted it is assumed that decision maker s restrictions are defined by specifying minimal or maximal values of scalar criteria measuring either expected outcome or variability of outcomes as such restrictions are in general not consistent with stochastic dominance rules we suggest verifying this consistency and asking the decision maker to redefine inconsistent restrictions

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

frontiers of stochastically nondominated portfolios we consider the problem of constructing a portfolio of finitely many assets whose returns are described by a discrete joint distribution we propose mean risk models that are solvable by linear programming and generate portfolios whose returns are nondomi nated in the sense of second order stochastic dominance next we develop a specialized parametric method for recovering the entire mean risk efficient frontiers of these models and we illustrate its operation on a large data set involving thousands of assets and real izations the problem of optimizing a portfolio of finitely many assets is a classical problem in theoretical and computational finance since the seminal work of markowitz 1952 it is generally agreed that portfolio performance should be measured in two distinct dimen sions the mean describing the expected return and the risk which measures the uncer tainty of the return in the mean risk approach we select from the universe of all possible portfolios those that are efficient for a given value of the mean they minimize the risk or equivalently for a given value of risk they maximize the mean such an approach has many advantages it allows one to formulate the problem as a parametric optimization problem and it facilitates the trade off analysis between mean and risk markowitz used the variance of the return as the measure of the risk it is easy to compute and it reduces the portfolio selection problem to a parametric quadratic pro gramming problem one can however construct simple counterexamples that show the imperfection of the variance as the risk measure it treats over performance equally as under performance and more importantly its use may suggest a portfolio that is always outperformed by another portfolio the use of the semivariance rather than the variance was already recommended by markowitz 1959 but even in this case significant deficien cies remain as we shall explain another theoretical approach to the portfolio selection problem is that of stochastic dominance see whitmore and findlay 1978 levy 1992 the usual first order defi nition of stochastic dominance gives a partial order in the space of real random variables more important from the portfolio point of view is the notion of second order dominance which is also defined as a partial order but which is equivalent to this statement a random variable y dominates the random variable z if e u y e u z for all nondecreasing concave functions u for which these expected values are finite thus no risk averse decision marker will prefer a portfolio with return z over a portfolio with return y while

design of entropy constrained multiple description scalar quantizers the problem of entropy constrained multiple description scalar quantizer design is posed as an optimization problem necessary conditions for optimality are derived and an iterative design algorithm is presented performance results are presented for a gaussian source along with comparisons to the multiple description rate distortion bound and a reference system

multiple description wavelet based image coding we consider the problem of coding images for transmission over error prone channels the impairments we target are transient channel shutdowns as would occur in a packet network when a packet is lost or in a wireless system during a deep fade when data is delivered it is assumed to be error free but some of the data may never reach the receiver the proposed algorithms are based on a combination of multiple description scalar quantizers with techniques successfully applied to the construction of some of the most efficient subband coders a given image is encoded into multiple independent packets of roughly equal length when packets are lost the quality of the approximation computed at the receiver depends only on the number of packets received but does not depend on exactly which packets are actually received when compared with previously reported results on the performance of robust image coders based on multiple descriptions on standard test images our coders attain similar psnr values using typically about 50 60 of the bit rate required by these other state of the art coders while at the same time providing significantly more freedom in the mechanism for allocation of redundancy among descriptions

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

hybrid digital analog hda joint source channel codes for broadcasting and robust communications we consider the problem of transmitting a band limited gaussian source on an additive band limited gaussian noise channel the well known threshold effect dictates that the more powerful a code is the more sensitive it is to the exact knowledge of the channel noise a code is said to be robust if it is asymptotically optimal for a wide range of channel noise thus robust codes have a graceful degradation characteristic and are free of the threshold effect it is demonstrated that robust codes exist whenever the source and channel bandwidths are equal in the unequal bandwidth case a collection of nearly robust joint source channel codes is constructed using a hybrid digital analog hda coding technique for designing nearly robust codes a matched tandem code whose channel encoder s output is partially fully matched to its input is proposed and the existence of an asymptotically optimal matched tandem code is shown the nearly robust codes achieve the shannon limit theoretically optimum distortion and have a less severe threshold effect finally for the case of two different noise conditions the distortion regions of these codes are determined

multiple description analog joint source channel coding to exploit the diversity in parallel channels the complexity and delay introduced by efficient digital coding strategies may be a barrier in some real time communications in this sense these last years joint source channel coding schemes based on analog mappings have gained prominence precisely for their simplicity and their implicit low delay in this work analog mappings originally designed for point to point communications are adapted to the case of parallel channels by following the multiple description strategy traditionally used in source coding in principle the coding scheme is designed to transmit over parallel awgn on off channels which are characterized by the possibility of having failures we also show that our scheme performs satisfactorily over slow rayleigh fading parallel channels

mean absolute deviation portfolio optimization model and its applications to tokyo stock market the purpose of this paper is to demonstrate that a portfolio optimization model using the l1 risk mean absolute deviation risk function can remove most of the difficulties associated with the classical markowitz s model while maintaining its advantages over equilibrium models in particular the l1 risk model leads to a linear program instead of a quadratic program so that a large scale optimization problem consisting of more than 1 000 stocks may be solved on a real time basis numerical experiments using the historical data of nikkei 225 stocks show that the l1 risk model generates a portfolio quite similar to that of the markowitz s model within a fraction of time required to solve the latter

solving non linear portfolio optimization problems with the primal dual interior point method stochastic programming is recognized as a powerful tool to help decision making under uncertainty in financial planning the deterministic equivalent formulations of these stochastic programs have huge dimensions even for moderate numbers of assets time stages and scenarios per time stage so far models treated by mathematical programming approaches have been limited to simple linear or quadratic models due to the inability of currently available solvers to solve nlp problems of typical sizes however stochastic programming problems are highly structured the key to the efficient solution of such problems is therefore the ability to exploit their structure interior point methods are well suited to the solution of very large non linear optimization problems in this paper we exploit this feature and show how portfolio optimization problems with sizes measured in millions of constraints and decision variables featuring constraints on semi variance skewness or non linear utility functions in the objective can be solved with the state of the art solver

random variables monotone relations and convex analysis random variables can be described by their cumulative distribution functions a class of nondecreasing functions on the real line those functions can in turn be identified after the possible vertical gaps in their graphs are filled in with maximal monotone relations such relations are known to be the subdifferentials of convex functions analysis of these connections yields new insights the generalized inversion operation between distribution functions and quantile functions corresponds to graphical inversion of monotone relations in subdifferential terms it corresponds to passing to conjugate convex functions under the legendre fenchel transform among other things this shows that convergence in distribution for sequences of random variables is equivalent to graphical convergence of the monotone relations and epigraphical convergence of the associated convex functions measures of risk that employ quantiles var and superquantiles cvar either individually or in mixtures are illuminated in this way formulas for their calculation are seen from a perspective that reveals how they were discovered the approach leads further to developments in which the superquantiles for a given distribution are interpreted as the quantiles for an overlying superdistribution in this way a generalization of koenker basset error is derived which lays a foundation for superquantile regression as a higher order extension of quantile regression

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 unified approach to uncertain optimization abstract in this paper we consider uncertain scalar optimization problems with infinite scenario sets we apply methods from vector optimization in general spaces set valued optimization and scalarization techniques to develop a unified characterization of different concepts of robust optimization and stochastic programming these methods provide new insights on the interrelation between different concepts for handling uncertainties and naturally lead to new concepts of robustness

second order stochastic dominance constrained portfolio optimization theory and computational tests abstract due to the definition of second order stochastic dominance ssd in terms of utility theory portfolio optimization with ssd constraints is of major practical interest we contribute to the field in two ways first we present a self contained theory with some new results and new proofs of known results second we perform a set of tests for computational efficiency we provide new and simple arguments for the formulation of ssd constraints in a mathematical programming framework for many individuals an ssd constraint may seem too severe wherefore various relaxations assd have been proposed we introduce yet another relaxation directional ssd where a candidate portfolio is admissible if a step from the benchmark in the direction of the candidate yields a dominating portfolio optimal step size depends on individual preferences reflected by the objective function we compare computational efficiency of seven approaches for sd constrained portfolio problems including ssd and assd constrained cases

process based risk measures and risk averse control of discrete time systems for controlled discrete time stochastic processes we introduce a new class of dynamic risk measures which we call process based their main feature is that they measure risk of processes that are functions of the history of a base process we introduce a new concept of conditional stochastic time consistency and we derive the structure of process based risk measures enjoying this property we show that they can be equivalently represented by a collection of static law invariant risk measures on the space of functions of the state of the base process we apply this result to controlled markov processes and we derive dynamic programming equations we also derive dynamic programming equations for multistage stochastic programming with decision dependent distributions

stochastically weighted stochastic dominance concepts with an application in capital budgeting abstract the problem of comparing random vectors arises in many applications we propose three new concepts of stochastically weighted dominance for comparing random vectors x and y the main idea is to use a random vector v to scalarize x and y as v t x and v t y and subsequently use available concepts from stochastic dominance and stochastic optimization for comparison for the case where the distributions of x y and v have finite support we give mixed integer linear inequalities that can be used for random vector comparison as well as for modeling of optimization problems where one of the random vectors depends on decisions to be optimized some advantages of the proposed new concepts are illustrated with the help of a capital budgeting example

on the distortion snr exponent of hybrid digital analog space time coding we consider the transmission of a real independent and identically distributed i i d analog source over a quasi static m input n output multiple input multiple output mimo block fading channel the relevant performance criterion is end to end average quadratic distortion d versus channel signal to noise ratio snr for given spectral efficiency eta defined as the ratio of the source bandwidth over the channel bandwidth in the limit of high snr we define the distortion snr exponent a eta as the largest a such that d esdot snr a over all possible source channel coding schemes of spectral efficiency eta we find a simple upper bound on a eta an achievable lower bound asep eta achievable by separated tandem source channel coding and a tighter lower bound ahybrid eta achievable by new hybrid digital analog space time coding schemes as a corollary we have that a eta is completely determined for the scalar case m n 1 and for the bandwidth compression case eta ges 2 min m n expiicit and simple construction of hybrid space time codes achieving ahybrid eta are also given

computational error bounds for multiple or nearly multiple eigenvalues abstract in this paper bounds for clusters of eigenvalues of non selfadjoint matrices are investigated we describe a method for the computation of rigorous error bounds for multiple or nearly multiple eigenvalues and for a basis of the corresponding invariant subspaces the input matrix may be real or complex dense or sparse the method is based on a quadratically convergent newton like method it includes the case of defective eigenvalues uncertain input matrices and the generalized eigenvalue problem computational results show that verified bounds are still computed even if other eigenvalues or clusters are nearby the eigenvalues under consideration

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

portfolio construction based on stochastic dominance and target return distributions mean risk models have been widely used in portfolio optimization however such models may produce portfolios that are dominated with respect to second order stochastic dominance and therefore not optimal for rational and risk averse investors this paper considers the problem of constructing a portfolio which is non dominated with respect to second order stochastic dominance and whose return distribution has specified desirable properties the problem is multi objective and is transformed into a single objective problem by using the reference point method in which target levels known as aspiration points are specified for the objective functions a model is proposed in which the aspiration points relate to ordered outcomes for the portfolio return this concept is extended by additionally specifying reservation points which act pre emptively in the optimization model the theoretical properties of the models are studied the performance of the models on real data drawn from the hang seng index is also investigated

optimality and duality theory for stochastic optimization problems with nonlinear dominance constraints we consider a new class of optimization problems involving stochastic dominance constraints of second order we develop a new splitting approach to these models optimality conditions and duality theory these results are used to construct special decomposition methods

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

design of multiple description scalar quantizers the design of scalar quantizers for communication systems that use diversity to overcome channel impairments is considered the design problem is posed as an optimization problem and necessary conditions for optimality are derived a design algorithm a generalization of s p lloyd s 1962 algorithm for quantizer design is developed unlike a single channel scalar quantizer the performance of a multiple description scalar quantizer is dependent on the index assignment the problem of index assignment is addressed good index assignments performance results and sample quantizer designs are presented for a memoryless gaussian source comparisons are made with rate distortion bounds for the multiple description problem