valid inequalities and restrictions for stochastic programming problems with first order stochastic dominance constraints stochastic dominance relations are well studied in statistics decision theory and economics recently there has been significant interest in introducing dominance relations into stochastic optimization problems as constraints in the discrete case stochastic optimization models involving second order stochastic dominance constraints can be solved by linear programming however problems involving first order stochastic dominance constraints are potentially hard due to the non convexity of the associated feasible regions in this paper we consider a mixed 0 1 linear programming formulation of a discrete first order constrained optimization model and present a relaxation based on second order constraints we derive some valid inequalities and restrictions by employing the probabilistic structure of the problem we also generate cuts that are valid inequalities for the disjunctive relaxations arising from the underlying combinatorial structure of the problem by applying the lift and project procedure we describe three heuristic algorithms to construct feasible solutions based on conditional second order constraints variable fixing and conditional value at risk finally we present numerical results for several instances of a real world portfolio optimization problem

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 cutting surface method for uncertain linear programs with polyhedral stochastic dominance constraints in this paper we study linear optimization problems with a newly introduced concept of multidimensional polyhedral linear second order stochastic dominance constraints by using the polyhedral properties of this dominance condition we present a cutting surface algorithm and show its finite convergence the cut generation problem is a difference of convex functions dc optimization problem we exploit the polyhedral structure of this problem to present a novel branch and cut algorithm that incorporates concepts from concave minimization and binary integer programming a linear programming problem is formulated for generating concavity cuts in our case where the polyhedra are unbounded we also present duality results for this problem relating the dual multipliers to utility functions without the need to impose constraint qualifications which again is possible because of the polyhedral nature of the problem numerical examples are presented showing the nature of solutions of our model a corrected pdf has been appended to the original

lie ideals from pure algebra to c algebras how does the associative structure of the algebra a e ect the lie structure of a in particular is it possible to describe lie ideals of a through associative ideals of a this and related questions such as the link between lie ideals and conjugate invariant subspaces of unital algebras have been an active area of research for more than 50 years they have been studied in pure algebra see e g a bfm h1 h2 h3 jr lm mm1 mm2 mu and more or less independently also in functional analysis particularly in operator algebras see e g bm cy fm fr fms fn hms hp ma mamu ms mi to one of the goals of this paper is to glue these two areas that is we will apply purely algebraic results derived in the first part of the paper to the second part dealing with banach algebras especially with w algebras and c algebras

optimization with multivariate stochastic dominance constraints we consider stochastic optimization problems where risk aversion is expressed by a stochastic ordering constraint the constraint requires that a random vector depending on our decisions stochastically dominates a given benchmark random vector we identify a suitable multivariate stochastic order and describe its generator in terms of von neumann morgenstern utility functions we develop necessary and sufficient conditions of optimality and duality relations for optimization problems with this constraint assuming convexity we show that the lagrange multipliers corresponding to dominance constraints are elements of the generator of this order thus refining and generalizing earlier results for optimization under univariate stochastic dominance constraints furthermore we obtain necessary conditions of optimality for non convex problems under additional smoothness assumptions

optimization with reference based robust preference constraints we propose an optimization model using a novel robust preference relationship reference based almost stochastic dominance rsd the concept of rsd addresses the two problems in utility based dec

standard stochastic dominance abstract we propose a new stochastic dominance sd criterion based on standard risk aversion which assumes decreasing absolute risk aversion and decreasing absolute prudence to implement the proposed criterion we develop linear systems of optimality conditions for a given prospect relative to a discrete or polyhedral choice opportunity set in a general state space model an empirical application to historical stock market data shows that small loser stocks are more appealing to standard risk averters than the existing mean variance mv and higher order sd criteria suggest due to their upside potential depending on the assumed trading strategy and evaluation horizon accounting for standardness increases the estimated abnormal returns of these stocks by about 50 to 200 basis points per annum relative to mv and higher order sd criteria an analysis of the mv tangency portfolio shows that the opportunity cost of the mv approximation to direct utility maximization can be substantial

sample average approximation of stochastic dominance constrained programs in this paper we study optimization problems with second order stochastic dominance constraints this class of problems allows for the modeling of optimization problems where a risk averse decision maker wants to ensure that the solution produced by the model dominates certain benchmarks here we deal with the case of multi variate stochastic dominance under general distributions and nonlinear functions we introduce the concept of mathcal c dominance which generalizes some notions of multi variate dominance found in the literature we apply the sample average approximation saa method to this problem which results in a semi infinite program and study asymptotic convergence of optimal values and optimal solutions as well as the rate of convergence of the feasibility set of the resulting semi infinite program as the sample size goes to infinity we develop a finitely convergent method to find an epsilon optimal solution of the saa problem an important aspect of our contribution is the construction of practical statistical lower and upper bounds for the true optimal objective value we also show that the bounds are asymptotically tight as the sample size goes to infinity

higher degree stochastic dominance optimality and efficiency abstract we characterize a range of stochastic dominance sd relations by means of finite systems of convex inequalities for sd optimality of degree 1 to 4 and sd efficiency of degree 2 to 5 we obtain exact systems that can be implemented using linear programming or convex quadratic programming for sd optimality of degree five and higher and sd efficiency of degree six and higher we obtain necessary conditions we use separate model variables for the values of the derivatives of all relevant orders at all relevant outcome levels which allows for preference restrictions beyond the standard sign restrictions our systems of inequalities can be interpreted in terms of piecewise polynomial utility functions with a number of pieces that increases with the number of outcomes and the degree of sd an empirical study analyzes the relevance of higher order risk preferences for comparing a passive stock market index with actively managed stock portfolios in standard data sets from the empirical asset pricing literature

direct approach to the band completion problem we study the problem of completion of a matrix with a specified band in such a way that the inverse be a band matrix with the same width for the general block matrix case a new approach to solve this and related problems is proposed the relations with the class of quasiseparable matrices play an important role

stochastic dynamic optimization with discounted stochastic dominance constraints we introduce a stochastic dynamic optimization problem where risk aversion is expressed by a stochastic ordering constraint the constraint requires that a random reward sequence depending on our decisions dominates a given benchmark random sequence the dominance is defined by discounting both processes with a family of discount sequences and by applying a univariate order we describe the generator of this order we develop necessary and sufficient conditions of optimality for convex stochastic control problems with the new ordering constraint and we derive an equivalent control problem featuring implied utility functions furthermore we prove the existence of an optimal random discount sequence such that the solution of the risk averse problem is also a solution of an expected value problem with this discount finally we derive a version of the maximum principle for the problem with discounted dominance constraints

robust and stochastically weighted multiobjective optimization models and reformulations we introduce and study a family of models for multiexpert multiobjective criteria decision making these models use a concept of weight robustness to generate a risk averse decision in particular the multiexpert multicriteria robust weighted sum approach mcrow introduced in this paper identifies a robust pareto decision that minimizes the worst case weighted sum of objectives over a given weight region the corresponding objective value called the robust value of a decision is shown to be increasing and concave in the weight set we study compact reformulations of the mcrow model with polyhedral and conic descriptions of the weight regions the mcrow model is developed further for stochastic multiexpert multicriteria decision making by allowing ambiguity or randomness in the weight region as well as the objective functions the properties of the proposed approach are illustrated with a few textbook examples the usefulness of the stochastic mcrow model is demonstrated using a disaster planning example and an agriculture revenue management example

optimization with stochastic dominance constraints we introduce stochastic optimization problems involving stochastic dominance constraints we develop necessary and sufficient conditions of optimality and duality theory for these models and show that the lagrange multipliers corresponding to dominance constraints are concave nondecreasing utility functions the models and results are illustrated on a portfolio optimization problem

tractable almost stochastic dominance abstract ll almost stochastic dominance ll asd is a relaxation of the stochastic dominance sd concept proposed by leshno and levy that explains more of realistic preferences observed in practice than sd alone does unfortunately numerical applications of this concept such as identifying if a given portfolio is efficient or determining a marketed portfolio that dominates a given benchmark are computationally prohibitive due to the structure of ll asd we propose a new almost stochastic dominance asd concept that is computationally tractable for instance a marketed dominating portfolio can be identified by solving a simple linear programming problem moreover the new concept performs well on all the intuitive examples from the literature and in some cases leads to more realistic predictions than the earlier concept we develop some properties of asd formulate efficient optimization models and apply the concept to analyzing investors preferences between bonds and stocks for the long run

nilpotency and generalized lie ideals the motivation for this paper arises from two desires the first is to generalize in a natural way the notion of a lie inner ideal of an associative ring originally studied by benkart cl we recall that a lie inner idea of a ring r is an additive subgroup v such that v v r c v the second is to find a lie type notion that will not only include lie ideals but will also include in an appropriate sense nilpotent associative subrings since these sometimes arise in a lie setting e g strictly triangular matrices the concept we are looking for appears to be that of a generalized lie ideal which we proceed to define after making a few remarks concerning notation let v and w be subsets of a ring r we define i w inductively as follows v w v w vcm l w v vcm w the notation i will mean v mil v v will have its usual associative meaning

on time stochastic dominance induced by mixed integer linear recourse in multistage stochastic programs abstract we propose in this work a new multistage risk averse strategy based on time stochastic dominance tsd along a given horizon it can be considered as a mixture of the two risk averse measures based on first and second order stochastic dominance constraints induced by mixed integer linear recourse respectively given the dimensions of medium sized problems augmented by the new variables and constraints required by this new risk measure it is unrealistic to solve the problem up to optimality by plain use of mip solvers in a reasonable computing time at least instead of it decomposition algorithms of some type should be used we present an extension of our branch and fix coordination algorithm so named bfc tsd where a special treatment is given to cross scenario group constraints that link variables from different scenario groups a broad computational experience is presented by comparing the risk neutral approach and the tested risk averse strategies the performance of the new version of the bfc algorithm versus the plain use of a state of the art mip solver is also reported

the qr algorithm for unitary hessenberg matrices abstract let h be an n n unitary right hessenberg matrix with positive subdiagonal elements using what we call the schur parameterization of h we show how one step of the shifted qr algorithm for h can be carried out in o n arithmetic operations coupled with the shift strategy of eberlein and huang 3 this will permit computation of the spectrum of h to machine precision in o n 2 operations one potential application is the computation of gauss szego quadrature formulas 12 given the associated schur parameters 7 the weights can also be computed by direct analogy with 6

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

robust decision making using a general utility set abstract we address the problem of ambiguity and inconsistency in a decision maker s dm assessed utility function by using a maxmin framework in this framework dm s utility function belongs to a set of functions the member functions of the set are nondecreasing and satisfy additional boundary and auxiliary conditions the maxmin framework provides robustness in decision making alternatively it allows us to perform sensitivity or parametric analysis for the optimal decision when a reference utility function is known for this purpose we use a cost of ambiguity concept and show that this cost is increasing and concave when extent of ambiguity is parametrically increased next we develop a lagrangian based solution approach for the decision problem we show that under suitable conditions a sample average approximation saa of the lagrangian model can be solved using a mixed integer linear program milp we also show that the set of optimal solutions of the saa converges to that of its true counterpart and the optimum objective value of the saa converges to the true objective value at an exponential rate we use this convergence property to develop a heuristic for identifying a solution of the saa milp with increasing sample size we illustrate the properties of the maxmin model using two examples of portfolio investment and streaming bandwidth we provide a discussion on the performance of a commercial solver used to solve saa milps and the quality of solutions generated by the proposed heuristic

the unitary completion and qr iterations for a class of structured matrices we consider the problem of completion of a matrix with a specified lower triangular part to a unitary matrix in this paper we obtain the necessary and sufficient conditions of existence of a unitary completion without any additional constraints and give a general formula for this completion the paper is mainly focused on matrices with the specified lower triangular part of a special form for such a specified part the unitary completion is a structured matrix and we derive in this paper the formulas for its structure next we apply the unitary completion method to the solution of the eigenvalue problem for a class of structured matrices via structured qr iterations

generalized lie ideals in prime rings abstract let r be a prime ring with skew elements k extended centroid c and central closure rc for u w r we define u n w inductively u 1 w u w u n 1 w u u n w an additive subgroup v of k is called a generalized lie ideal gli of k of index n if v n k v the notion of a gli includes that of a lie ideal of k a lie inner ideal of k and an additive subgroup of k which generates a nilpotent subring of r theorem if char r 0 and v k such that v n k 0 then v c b where b is a nilpotent subring of rc theorem suppose is an involution of the first kind v is a gli of k of index n and t t k v t k v if char r 2 and t t 0 then i k k v for some nonzero ideal i of r if char r 0 and t t 0 then v 6 n 7 0 a similar result is obtained when is of the second kind

new formulations for optimization under stochastic dominance constraints stochastic dominance constraints allow a decision maker to manage risk in an optimization setting by requiring his or her decision to yield a random outcome which stochastically dominates a reference random outcome we present new integer and linear programming formulations for optimization under first and second order stochastic dominance constraints respectively these formulations are more compact than existing formulations and relaxing integrality in the first order formulation yields a second order formulation demonstrating the tightness of this formulation we also present a specialized branching strategy and heuristics which can be used with the new first order formulation computational tests illustrate the potential benefits of the new formulations

the restarted qr algorithm for eigenvalue computation of structured matrices the qr algorithm is a popular numerical method for the computation of eigenvalues of matrices all eigenvalues of a general n n upper hessenberg matrix typically can be computed in o n3 arithmetic floating point operations using o n2 storage locations when the upper hessenberg matrix is hermitian or unitary then it can be represented by o n parameters and there are variants of the qr algorithm that reduce the operation count for computing all eigenvalues to o n2 arithmetic floating point operations and the storage requirement to o n locations however for many structured matrices that can be represented with o n storage locations available implementations of the qr algorithm require o n3 arithmetic floating point operations and o n2 storage locations to determine all eigenvalues this note shows that for some of the latter matrices the operation count can be reduced to o n2 arithmetic floating point operations and the memory requirement to o n storage locations by periodically restarting the qr algorithm