Convergence of Minima of Integral Functionals, with Applications to Optimal Control and Stochastic Optimization

Epi-convergence of integral functionals is derived under new conditions that can be used in the infinite dimensional case. Applications include: the convergence of the solutions of approximating optimal control problems and of stochastic optimization problems

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

A simple randomised algorithm for convex optimisation: Application to two-stage stochastic programming

We consider maximising a concave function over a convex set by a simple randomised algorithm. The strength of the algorithm is that it requires only approximate function evaluations for the concave function and a weak membership oracle for the convex set. Under smoothness conditions on the function and the feasible set, we show that our algorithm computes a near-optimal point in a number of operations which is bounded by a polynomial function of all relevant input parameters and the reciprocal of the desired precision, with high probability. As an application to which the features of our algorithm are particularly useful we study two-stage stochastic programming problems. These problems have the property that evaluation of the objective function is #P-hard under appropriate assumptions on the models. Therefore, as a tool within our randomised algorithm, we devise a fully polynomial randomised approximation scheme for these function evaluations, under appropriate assumptions on the models. Moreover, we deal with smoothing the feasible set, which in two-stage stochastic programming is a polyhedron

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

On the continuity of the value of a linear program and of related polyhedral-valued multifunctions

Results about the continuity of the optimal value of a linear program and of related polyhedral-valued multifunctions (determined by the constraints) are reviewed. A framework is provided for studying their interconnections

Algorithmic Procedures for Stochastic Optimization

For purposes of preliminary discussion, it is convenient to identify stochastic optimization problems with: findxεRnthatminimizesz=E{f(x,ξ∼)} where ξ is a random N-vector with distribution function, P, f:Rn x RN → R U +∞ is a lower semicontinuous function, possibly convex, where dom f(.ξ) = x |f(x,ξ) is finite, corresponds to the set of acceptable choices for x when ξ is the observed value of the random vector ξ, and E{f(x,ξ∼)}=∫f(x,ξ)dp(ξ

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