Sublinear upper bounds for stochastic programs with recourse

Separable sublinear functions are used to provide upper bounds on the recourse function of a stochastic program. The resulting problem's objective involves the inf-convolution of convex functions. A dual of this problem is formulated to obtain an implementable procedure to calculate the bound. Function evaluations for the resulting convex program only require a small number of single integrations in contrast with previous upper bounds that require a number of function evaluations that grows exponentially in the number of random variables. The sublinear bound can often be used when other suggested upper bounds are intractable. Computational results indicate that the sublinear approximation provides good, efficient bounds on the stochastic program objective value.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/47918/1/10107_2005_Article_BF01582286.pd

Ordmet: A general algorithm for constructing all numerical solutions to ordered metric structures

The algorithm is applicable to structures such as are obtained from additive conjoint measurement designs, unfolding theory, general Fechnerian scaling, some special types of multidimensional scaling, and ordinal multiple regression. A description is obtained of the space containing all possible numerical representations which can satisfy the structure, the size and shape of which is informative. The Abelson-Tukey maximin r 2 solution is provided.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/45736/1/11336_2005_Article_BF02291758.pd

Models and model value in stochastic programming

Finding optimal decisions often involves the consideration of certain random or unknown parameters. A standard approach is to replace the random parameters by the expectations and to solve a deterministic mathematical program. A second approach is to consider possible future scenarios and the decision that would be best under each of these scenarios. The question then becomes how to choose among these alternatives. Both approaches may produce solutions that are far from optimal in the stochastic programming model that explicitly includes the random parameters. In this paper, we illustrate this advantage of a stochastic program model through two examples that are representative of the range of problems considered in stochastic programming. The paper focuses on the relative value of the stochastic program solution over a deterministic problem solution.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/44253/1/10479_2005_Article_BF02031741.pd

Uncertainty Quantification using Exponential Epi-Splines

Proceedings of the International Conference on Structural Safety and Reliability, New York, NY.We quantify uncertainty in complex systems by a flexible, nonparametric framework for estimating probability density functions of output quantities of interest. The framework systematically incorporates soft information about the system from engineering judgement and experience to improve the estimates and ensure that they are consistent with prior knowledge. The framework is based on a maximum likelihood criterion, with epi-splines facilitating rapid solution of the resulting optimization problems. In four numerical examples with few realizations of the system output, we identify the main features of output densities even for nonsmooth and discontinuous system function and high-dimensional inputs

Deriving the continuity of maximum-entropy basis functions via variational analysis

In this paper, we prove the continuity of maximum-entropy basis functions using Variational Analysis techniques. The use of information-theoretic variational principles to derive basis functions is a recent development. In this setting, data approximation is viewed as an inductive inference problem, with the basis functions being synonymous with a discrete probability distribution and the polynomial reproducing conditions acting as the linear constraints. For a set of distinct nodes {x i} n i=1 in IRd, the convex approximation of a function u(x) is: u h (x) = � n i=1 pi(x)ui, where {pi} n i=1 are non-negative basis functions, and uh (x) must reproduce affine functions: � n i=1 pi(x) = 1, � n i=1 pi(x)x i = x. Given these constraints, we compute pi(x) by minimizing the relative entropy functional (Kullback-Leibler distance), D(p�m) = � n i=1 pi(x) ln � pi(x)/mi(x) � , where mi(x) is a known prior weight function distribution. To prove the continuity of the basis functions, we appeal to the theory of epi-convergence

A sublinear approximation method for Stochastic programming

http://deepblue.lib.umich.edu/bitstream/2027.42/3669/5/bal9393.0001.001.pdfhttp://deepblue.lib.umich.edu/bitstream/2027.42/3669/4/bal9393.0001.001.tx