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.     

Convex relaxations of chance constrained optimization problems

In this paper we develop convex relaxations of chance constrained optimization problems in order to obtain lower bounds on the optimal value. Unlike existing statistical lower bounding techniques, our approach is designed to provide deterministic lower bounds. We show that a version of the proposed scheme leads to a tractable convex relaxation when the chance constraint function is affine with respect to the underlying random vector and the random vector has independent components. We also propose an iterative improvement scheme for refining the bounds.     

Confidence intervals in the solution of stochastic integer linear programming problems

A method is proposed to estimate confidence intervals for the solution of integer linear programming (ILP) problems where the technological coefficients matrix and the resource vector are made up of random variables whose distribution laws are unknown and only a sample of their values is available. This method, based on the theory of order statistics, only requires knowledge of the solution of the relaxed integer linear programming (RILP) problems which correspond to the sampled random parameters. The confidence intervals obtained in this way have proved to be more accurate than those estimated by the current methods which use the integer solutions of the sampled ILP problems.This research was partially supported by the Italian National Research Council contract no. 82.001 14.93 (P.F. Trasporti).     

Decision making under uncertainty: A semi-infinite programming approach

The subject of this paper is the formulation and discussion of a semi-infinite linear vector optimization problem which extends multiple objective linear programming problems to those with an infinite number of objective functions and constraints. Furthermore it generalizes in some way semi-infinite programming. Besides the statement of some immediately derived results which are related to known results in semi-infinite linear programming and vector optimization, the problem mentioned above is interpreted as a decision model, under risk or uncertainty containing continuous random variables. Thus we treat the case of an infinite number of occuring states of nature. These types of problems frequently occur within aspects of decision theory in management science.     

Applications of conditional comonotonicity to some optimization problems

In this article, we study two optimization problems. The first is finding the best L₁-approximant of a given random vector on some affine subspaces subject to a measurability condition. The second is finding the optimal allocation of policy limits such that the expected retained loss is minimized. Explicit solutions of both problems are constructed by utilizing the notion of conditional comonotonicity.     

Spectral analysis of the fourth moment matrix

The fourth moment of a random vector is a matrix whose elements are all moments of order four which can be obtained from the random vector itself. In this paper, we give a lower bound for its dominant eigenvalue and show that its eigenvectors corresponding to positive eigenvalues are vectorized symmetric matrices. Fourth moments of standardized and exchangeable random vectors are examined in more detail.     

Two-stage stochastic problems with correlated normal variables: computational experiences

Two-stage models are frequently used when making decisions under the influence of randomness. The case of normally distributed right hand side vector – with independent or correlated components – is treated here. The expected recourse function is computed by an enhanced Monte Carlo integration technique. Successive regression approximation technique is used for computing the optimal solution of the problem. Computational issues of the algorithm are discussed, improvements are proposed and numerical results are presented for random right hand side and a random matrix in the second stage problems.     

On an application of game theory to a problem of testing statistical hypothesis

Consider an-dimensional random vector with known covariance matrix. The expectation values of its single components may take arbitrary values subject to the restriction that their sum is a prescribed positive constant. Now choose a linear combination of these components, take its expectation value and divide this by the square root of its variance. This quotient, which is of importance in some problems of test theory serves as the pay-off function of a two-person zero-sum game. Player I wants to maximize the quotient by forming suitable linear combinations and player II wants to minimize it by choosing appropriate expectation values of the single components of the random vector subject to the restriction stated above. It is shown that the game possesses an essentially unique equilibrium point. In the more complicated case, when the strategies of the second player are confined to non-negative expectation values of the random vector's components, there is also an essentially unique equilibrium point of the game. It coincides with that one of the unconstrained case if and only if the row sums of the random vector's covariance matrix are all nonnegative.     

Student's t Vector Random Fields with Power-Law and Log-Law Decaying Direct and Cross Covariances

This article deals with the Student's t vector random field, which is formulated as a scale mixture of Gaussian vector random fields, and whose finite-dimensional distributions decay in power-law and have heavy tails. There are two classes of Student's t vector random fields, one with second-order moments, and the other without a second-order moment. A Cauchy vector random field is an example of Student's t vector random fields without a first-order moment, and is also an example of Stable vector random fields. A second-order Student's t vector random field allows for any given correlation structure, just as a Gaussian vector random field does. We propose four types of covariance matrix structures for second-order Student's t vector random fields, which decay in power-law or log-law.     

Weakly admissible vector equilibrium problems

We establish lower semi-continuity and strict convexity of the energy functionals for a large class of vector equilibrium problems in logarithmic potential theory. This, in particular, implies the existence and uniqueness of a minimizer for such vector equilibrium problems. Our work extends earlier results in that we allow unbounded supports without having strongly confining external fields. To deal with the possible noncompactness of supports, we map the vector equilibrium problem onto the Riemann sphere and our results follow from a study of vector equilibrium problems on compacts in higher dimensions. Our results cover a number of cases that were recently considered in random matrix theory and for which the existence of a minimizer was not clearly established yet.     

An alternative method for computing mean and covariance matrix of some multivariate distributions

Computing the mean and covariance matrix of some multivariate distributions, in particular, multivariate normal distribution and Wishart distribution are considered in this article. It involves a matrix transformation of the normal random vector into a random vector whose components are independent normal random variables, and then integrating univariate integrals for computing the mean and covariance matrix of a multivariate normal distribution. Moment generating function technique is used for computing the mean and covariances between the elements of a Wishart matrix. In this article, an alternative method that uses matrix differentiation and differentiation of the determinant of a matrix is presented. This method does not involve any integration.     

Convexity and optimization with copulæ structured probabilistic constraints

Probability constraints play a key role in optimization problems involving uncertainties. These constraints request that an inequality system depending on a random vector has to be satisfied with a high enough probability. In specific settings, copulæ can be used to model the probabilistic constraints with uncertainty on the left-hand side. In this paper, we provide eventual convexity results for the feasible set of decisions under local generalized concavity properties of the constraint mappings and involved copulæ. The results cover all Archimedean copulæ. We consider probabilistic constraints wherein the decision and random vector are separated, i.e. left/right-hand side uncertainty. In order to solve the underlying optimization problem, we propose and analyse convergence of a regularized supporting hyperplane method: a stabilized variant of generalized Benders decomposition. The algorithm is tested on a large set of instances involving several copulæ among which the Gaussian copula. A Numerical comparison with a (pure) supporting hyperplane algorithm and a general purpose solver for non-linear optimization is also presented.     

一类隐式集值向量均衡问题解的存在性和解集的稳定性

研究一类隐式集值向量均衡问题,它是隐式向量均衡问题、隐式变分不等式问题、隐式相补问题、向量均衡问题和向量变分不等式问题等的推广.利用截口定理,在Hausdorff拓扑线性空间的非紧子集上得出了一些隐式集值向量均衡问题解的存在性结果.并且还讨论了该隐式集值向量均衡问题解集的通有稳定性.这些结果推广和统一了近期的一些相关结果.     

Adaptive reliability analysis based on a support vector machine and its application to rock engineering

The response surface method (RSM), a simple and effective approximation technique, is widely used for reliability analysis in civil engineering. However, the traditional RSM needs a considerable number of samples and is computationally intensive and time-consuming for practical engineering problems with many variables. To overcome these problems, this study proposes a new approach that samples experimental points based on the difference between the last two trial design points. This new method constructs the response surface using a support vector machine (SVM); the SVM can build complex, nonlinear relations between random variables and approximate the performance function using fewer experimental points. This approach can reduce the number of experimental points and improve the efficiency and accuracy of reliability analysis. The advantages of the proposed method were verified using four examples involving random variables with different distributions and correlation structures. The results show that this approach can obtain the design point and reliability index with fewer experimental points and better accuracy. The proposed method was also employed to assess the reliability of a numerically modeled tunnel. The results indicate that this new method is applicable to practical, complex engineering problems such as rock engineering problems.     

On fuzzy random multiobjective quadratic programming

In this paper, a multiobjective quadratic programming problem having fuzzy random coefficients matrix in the objective and constraints and the decision vector are fuzzy pseudorandom variables is considered. First, we show that the efficient solutions of fuzzy quadratic multiobjective programming problems are resolved into series-optimal-solutions of relative scalar fuzzy quadratic programming. Some theorems are proved to find an optimal solution of the relative scalar quadratic multiobjective programming with fuzzy coefficients, having decision vectors as fuzzy variables. At the end, numerical examples are illustrated in the support of the obtained results.     

Postoptimality for mean-risk stochastic mixed-integer programs and its application

The mean-risk stochastic mixed-integer programs can better model complex decision problems under uncertainty than usual stochastic (integer) programming models. In order to derive theoretical results in a numerically tractable way, the contamination technique is adopted in this paper for the postoptimality analysis of the mean-risk models with respect to changes in the scenario set, here the risk is measured by the lower partial moment. We first study the continuity of the objective function and the differentiability, with respect to the parameter contained in the contaminated distribution, of the optimal value function of the mean-risk model when the recourse cost vector, the technology matrix and the right-hand side vector in the second stage problem are all random. The postoptimality conclusions of the model are then established. The obtained results are applied to two-stage stochastic mixed-integer programs with risk objectives where the objective function is nonlinear with respect to the probability distribution. The current postoptimality results for stochastic programs are improved.     

封闭型保单组未来损失现值向量的联合分布及渐近分布

本文针对封闭型保单组,利用历年死亡人数随机向量D,将保单组的未来给付现值随机变量和未来损失现值随机变量表达为某个满秩矩阵和D的乘积,根据D服从多项分布的性质,得到未来损失现值随机向量渐近服从多元正态分布的结果,为分析责任准备金提供了一个新的框架.     

SOME MULTIVARIATE DMRL AND NBUE DEFINITIONS BASED ON CONDITIONAL STOCHASTIC ORDER

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Vector optimization problems with nonconvex preferences

In this paper, some vector optimization problems are considered where pseudo-ordering relations are determined by nonconvex cones in Banach spaces. We give some characterizations of solution sets for vector complementarity problems and vector variational inequalities. When the nonconvex cone is the union of some convex cones, it is shown that the solution set of these problems is either an intersection or an union of the solution sets of all subproblems corresponding to each of these convex cones depending on whether these problems are defined by the nonconvex cone itself or its complement. Moreover, some relations of vector complementarity problems, vector variational inequalities, and minimal element problems are also given. While this paper was being revised in September 2006, Professor Alex Rubinov (the second author of the paper) left us due to the illness. This is a very sad news to us. We dedicate this paper to the memory of Professor Rubinov as a mathematician and truly friend.