Split Sampling: Expectations, Normalisation and Rare Events

In this paper we develop a methodology that we call split sampling methods to estimate high dimensional expectations and rare event probabilities. Split sampling uses an auxiliary variable MCMC simulation and expresses the expectation of interest as an integrated set of rare event probabilities. We derive our estimator from a Rao-Blackwellised estimate of a marginal auxiliary variable distribution. We illustrate our method with two applications. First, we compute a shortest network path rare event probability and compare our method to estimation to a cross entropy approach. Then, we compute a normalisation constant of a high dimensional mixture of Gaussians and compare our estimate to one based on nested sampling. We discuss the relationship between our method and other alternatives such as the product of conditional probability estimator and importance sampling. The methods developed here are available in the R package: SplitSampling

Prior Reduced Fill-In in Solving Equations in Interior Point Algorithms

The efficiency of interior-point algorithms for linear programming is related to the effort required to factorize the matrix used to solve for the search direction at each iteration. When the linear program is in symmetric form (i.e., the constraints are Ax b, x > 0 ), then there are two mathematically equivalent forms of the search direction, involving different matrices. One form necessitates factoring a matrix whose sparsity pattern has the same form as that of (A AT). The other form necessitates factoring a matrix whose sparsity pattern has the same form as that of (ATA). Depending on the structure of the matrix A, one of these two forms may produce significantly less fill-in than the other. Furthermore, by analyzing the fill-in of both forms prior to starting the iterative phase of the algorithm, the form with the least fill-in can be computed and used throughout the algorithm. Finally, this methodology can be applied to linear programs that are not in symmetric form, that contain both equality and inequality constraints

On the Convergence of L-shaped Algorithms for Two-Stage Stochastic Programming

In this paper, we design, analyze, and implement a variant of the two-loop L-shaped algorithms for solving two-stage stochastic programming problems that arise from important application areas including revenue management and power systems. We consider the setting in which it is intractable to compute exact objective function and (sub)gradient information, and instead, only estimates of objective function and (sub)gradient values are available. Under common assumptions including fixed recourse and bounded (sub)gradients, the algorithm generates a sequence of iterates that converge to a neighborhood of optimality, where the radius of the convergence neighborhood depends on the level of the inexactness of objective function estimates. The number of outer and inner iterations needed to find an approximate optimal iterate is provided. Finally, we show a sample complexity result for the algorithm with a Polyak-type step-size policy that can be extended to analyze other situations. We also present a numerical study that verifies our theoretical results and demonstrates the superior empirical performance of our proposed algorithms over classic solvers.Comment: 39 pages, 2 figure

Redistricting to maximize the preservation of political boundaries

Redefining legislative districts is a task undertaken by the states after each census in order to ensure equitable representation. Many criteria have been proposed as objectives in forming districts but specific definitions of an optimal plan have not been enforced. In attempting to eliminate political concerns from the effort, the Michigan Supreme Court defined criteria based on the preservation of county and municipality borders. A quadratic programming formulation is given for this problem, and a heuristic solution procedure is proposed.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/25116/1/0000549.pd

The value of the stochastic solution in stochastic linear programs with fixed recourse

Stochastic linear programs have been rarely used in practical situations largely because of their complexity. In evaluating these problems without finding the exact solution, a common method has been to find bounds on the expected value of perfect information. In this paper, we consider a different method. We present bounds on the value of the stochastic solution, that is, the potential benefit from solving the stochastic program over solving a deterministic program in which expected values have replaced random parameters. These bounds are calculated by solving smaller programs related to the stochastic recourse problem.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/47912/1/10107_2005_Article_BF01585113.pd

Methods for a network design problem in solar power systems

We consider the problem of minimizing cable connections between a central computer and a field of heliostats in the design of solar power systems. This practical task can be modeled as a p-median problem with additional constraints in a weighted graph. We compare an exact branch-and-bound method with two approximate algorithms. For the latter two methods, estimations of time complexity and accuracy are presented. Computational results are shown which should be useful in the design of such large-scale power systems.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/25844/1/0000407.pd

Continuous approximation schemes for stochastic programs

One of the main methods for solving stochastic programs is approximation by discretizing the probability distribution. However, discretization may lose differentiability of expectational functionals. The complexity of discrete approximation schemes also increases exponentially as the dimension of the random vector increases. On the other hand, stochastic methods can solve stochastic programs with larger dimensions but their convergence is in the sense of probability one. In this paper, we study the differentiability property of stochastic two-stage programs and discuss continuous approximation methods for stochastic programs. We present several ways to calculate and estimate this derivative. We then design several continuous approximation schemes and study their convergence behavior and implementation. The methods include several types of truncation approximation, lower dimensional approximation and limited basis approximation.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/44240/1/10479_2005_Article_BF02031698.pd

Setting Single-Period Optimal Capacity Levels and Prices for Substitutable Products

In this paper, we consider how a company that has the flexibility to produce two substitutable products would determine optimal capacity levels and prices for these products in a single-period problem. We first consider the case where the firm is a price taker but can determine optimal capacity levels for both products. We then consider the case where the firm can set the price for one product and the optimal capacity level for the other. Finally, we consider the case where capacity is fixed for both products, but the firm can set prices. For each case, we examine the sensitivity of optimal prices and capacities to the problem parameters. Finally, we consider the case where each product is managed by a product manager trying to maximize individual product profits rather than overall firm profits and analyze how optimal price and capacity decisions are affected.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/45464/1/10696_2004_Article_184045.pd