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.     

A note on taxation,incentives and risk sharing

The necessary and sufficient conditions for a tax system to guarantee the consistency of private and public evaluation of risky projects are derived. A stochastic dominance approach is used.     

Relaxations of linear programming problems with first order stochastic dominance constraints

Linear stochastic programming problems with first order stochastic dominance (FSD) constraints are non-convex. For their mixed 0-1 linear programming formulation we present two convex relaxations based on second order stochastic dominance (SSD). We develop necessary and sufficient conditions for FSD, used to obtain a disjunctive programming formulation and to strengthen one of the SSD-based relaxations.     

The Singularity/Nonsingularity Problem for Matrices Satisfying Diagonal Dominance Conditions in Terms of Directed Graphs

The paper considers the singularity/nonsingularity problem for matrices satisfying certain conditions of diagonal dominance. The conditions considered extend the classical diagonal dominance conditions and involve the directed graph of the matrix in question. Furthermore, in the case of the so-called mixed diagonal dominance, the corresponding conditions are allowed to involve both row and column sums for an arbitrary finite set of matrices diagonally conjugated to the original matrix. Conditions sufficient for the nonsingularity of quasi-irreducible matrices strictly diagonally dominant in certain senses are established, as well as necessary and sufficient conditions of singularity/nonsingularity for weakly diagonally dominant matrices in the irreducible case. The results obtained are used to describe inclusion regions for eigenvalues of arbitrary matrices. In particular, a direct extension of the Gerschgorin (r = 1) and Ostrowski-Brauer (r = 2) theorems to r ≥ 3 is presented. Bibliography: 18 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 309, 2004, pp. 40–83.     

服务系统中位相型分布的随机比较

本文运用应用概率中的随机占优研究位相型(PH)分布的随机比较问题,具体给出在一阶、二阶随机占优下比较两个离散PH分布或两个连续PH分布的充分条件及充分必要条件。研究表明,比较两个离散PH分布可变性的条件与比较两个连续PH分布可变性的条件不同,在二阶随机占优意义下比较两个连续PH分布的条件与均值无关,而比较两个离散PH分布的条件与均值有关。本文的结果可用于研究PH分布的最小变异系数问题和可变性问题,也可用于研究带有PH到达间隔或PH服务的排队系统中到达过程或服务时间可变性对系统队长或等待时间的影响。     

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. Received: September 15, 1998 / Accepted: October 1, 2000?Published online December 15, 2000     

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.     

On circle-valued cocycles of an ergodic measure-preserving transformation

Analytic necessary and sufficient conditions are given for a circle-valued functionf to generate a cocycle which is a multiple of a coboundary. These conditions are then used to derive some other new criteria for cocycles to be coboundaries. This research was supported in part by an NSF grant DMS8600753.     

Optimal endpoints

A complete set of necessary and sufficient conditions for selecting optimal endpoints for extremals obtained from the variational Bolza problem in control notation has been developed. The method used to obtain these conditions is based on a seldom used concept of performing a dichotomy on the general optimization problem. With this concept, the problem of Bolza is decomposed into two problems, the first of which involves the selection of optimal paths with the endpoints considered fixed. The second problem involves the selection of optimal endpoints with the paths between the endpoints taken to be stationary curves. The convenience of the dichotomy in deriving the necessary and sufficient conditions for endpoints lies in its simplicity and elementary character; well-known necessary and sufficient conditions from the theory of ordinary maxima and minima are used.An endpoint necessary condition is first obtained which is simply the well-known transversality condition. An additional condition is then developed which, together with the transversality condition, leads to a set of necessary and sufficient conditions for a given extremal to be locally optimal with respect to endpoint variations. While the second condition presented is akin to the classical focal-point condition, the result is new in form and is directly applicable to the optimal control problem. In addition, it is relatively simple to apply and is easy to implement numerically when an analytical solution is not possible. It should be useful in situations where the transversality conditions yield more than one choice for an optimal endpoint.An analytic solution for a simple geodetics problem is presented to illustrate the theory. A discussion of numerical implementation of the sufficiency conditions and its application to an orbit transfer example is also included.This work was supported in part by the National Aeronautics and Space Administration, Grant No. NGR-03-002-001.     

Conjugate-point conditions for variational problems with delayed argument

A variational problem with delayed argument is investigated. The existing necessary conditions are first reviewed. New results, two conjugate-point conditions, are then derived for this problem. The method of proof is similar to that used by Bliss for the classical problem. An example shows that the two conditions are not equivalent, and that the first-order necessary conditions, the strengthened Legendre conditions, and the conjugate-point conditions do not in general constitute a set of sufficient conditions for the delay problem. It is shown that a special case, referred to as the separated-integrand problem, leads to considerable simplification of the results for the general problem.     

A new efficient approach to the characterization of D‐stable matrices

The concept of D‐stability is relevant for stable square matrices of any order, especially when they appear in ordinary differential systems modeling physical problems. Indeed, D‐stability was treated from different points of view in the last 50 years, but the problem of characterization of a general D‐stable matrix was solved for low‐order matrices only (ie, up to order 4). Here, a new approach is proposed within the context of numerical linear algebra. Starting from a known necessary and sufficient condition, other simpler equivalent necessary and sufficient conditions for D‐stability are proved. Such conditions turn out to be computationally more appealing for symbolic software, as discussed in the reported examples. Therefore, a new symbolic method is proposed to characterize matrices of order greater than 4, and then it is used in some numerical examples, given in details.     

一类由概率分布生成的新度量

主要得到了一类由概率分布生成的新度量.以信息理论中的重要概念-相对熵为基础,对前人文章中的重要结论进行推广,通过利用改进的初等方法在离散的可测空间中得到了这类新度量,并且证明了得到的新度量成立的充要条件.由此再将这类新度量推广到连续的可测空间中,得到了同样的结果.最后讨论了新度量的最值问题.     

Normal form backward induction for decision trees with coherent lower previsions

We examine normal form solutions of decision trees under typical choice functions induced by lower previsions. For large trees, finding such solutions is hard as very many strategies must be considered. In an earlier paper, we extended backward induction to arbitrary choice functions, yielding far more efficient solutions, and we identified simple necessary and sufficient conditions for this to work. In this paper, we show that backward induction works for maximality and E-admissibility, but not for interval dominance and Γ-maximin. We also show that, in some situations, a computationally cheap approximation of a choice function can be used, even if the approximation violates the conditions for backward induction; for instance, interval dominance with backward induction will yield at least all maximal normal form solutions.     

A note on the two-stage hybrid flow shop problem with dedicated machines

In this paper we introduce a new dominance rule for the two-stage hybrid flow shop problem with dedicated machines. The rule is then used to construct a dominating set. The efficiency of the proposed rule is shown through an analysis of the dominating set cardinality.     

A hybrid genetic algorithm with dominance properties for single machine scheduling with dependent penalties

In this paper, a hybrid genetic algorithm is developed to solve the single machine scheduling problem with the objective to minimize the weighted sum of earliness and tardiness costs. First, dominance properties of (the conditions on) the optimal schedule are developed based on the switching of two adjacent jobs i and j. These dominance properties are only necessary conditions and not sufficient conditions for any given schedule to be optimal. Therefore, these dominance properties are further embedded in the genetic algorithm and we call it genetic algorithm with dominance properties (GADP). This GADP is a hybrid genetic algorithm. The initial populations of schedules in the genetic algorithm are generated using these dominance properties. GA can further improve the performance of these initial solutions after the evolving procedures. The performances of hybrid genetic algorithm (GADP) have been compared with simple genetic algorithm (SGA) using benchmark instances. It is shown that this hybrid genetic algorithm (GADP) performs very well when compared with DP or SGA alone.     

A new decomposition approach for the single machine total tardiness scheduling problem

This paper deals with the single machine total tardiness problem. From Emmons’ basic dominance conditions a new partition theorem is derived which generalises Lawler’s decomposition rule and leads to a new double decomposition procedure. This procedure is embedded into a branch and bound method which applies a new lower bound based on due dates reassignment. The branch and bound method is tested on problems with size up to 150 jobs.     

基于熵权与优势关系的教学效果评价方案

为了提高教学效果评价的质量,利用熵权方法来确定评价指标的权向量,进一步结合优势关系建立一种综合排序方法,并应用文献(中国卫生统计,2009,26(3)314-316.)的一个实例表明方法是有效和可行的.     

On determinantal diagonal dominance conditions

The paper suggests sufficient nonsingularity conditions for matrices in terms of certain determinantal relations of diagonal dominance type, which improve and generalize some known results. These conditions are used to describe new eigenvalue inclusion sets and to derive new two-sided bounds on the determinants of matrices satisfying them. Bibliography: 8 titles.     

New existence results for some nonconvex optimization problems

We present new existence results for a family of nonconvex variational boundary-value problems. Our method is based on an argument that has already been used in such problems, although only in a very partial fashion. In order to be able to use the full strength of this argument, one is led to introduce technical hypotheses whose generality is not self-evident. Another component of this study is then to show that these hypotheses are actually generic, at least under sufficient smoothness of the data. There follows a generally calculable sufficient criterion for existence of solutions. What it really takes to go through the necessary calculations is illustrated by several examples. Closely related sufficient conditions to obtain uniqueness as well are also given.     

Multivariate robust second-order stochastic dominance and resulting risk-averse optimization

ABSTRACT

By utilizing a min-biaffine scalarization function, we define the multivariate robust second-order stochastic dominance relationship to flexibly compare two random vectors. We discuss the basic properties of the multivariate robust second-order stochastic dominance and relate it to the nonpositiveness of a functional which is continuous and subdifferentiable everywhere. We study a stochastic optimization problem with multivariate robust second-order stochastic dominance constraints and develop the necessary and sufficient conditions of optimality in the convex case. After specifying an ambiguity set based on moments information, we approximate the ambiguity set by a series of sets consisting of discrete distributions. Furthermore, we design a convex approximation to the proposed stochastic optimization problem with multivariate robust second-order stochastic dominance constraints and establish its qualitative stability under Kantorovich metric and pseudo metric, respectively. All these results lay a theoretical foundation for the modelling and solution of complex stochastic decision-making problems with multivariate robust second-order stochastic dominance constraints.