Introduction to linear and nonlinear programming

We study the asymptotic properties of estimators of the tail of a distribution based on the excesses over a threshold. A key idea is the use of Pickands' generalised Pareto distribution and its fitting, in most cases, by the method of maximum likelihood. The results cover all three limiting types of extreme value theory. We propose a new estimator for an index of regular variation and show that it often performs better than Hill's estimator. We give new results for estimating the endpoint of a distribution, extending earlier work by Hall and by Smith and Weissman. Finally, we give detailed results for the domain of attraction of $\exp(-e^{-x})$ and show that, in most cases, our proposed estimator is more efficient than two others, one based on the exponential distribution and the other due to Davis and Resnick. We also touch briefly on the problem of large deviations from a statistical point of view. The results make extensive use of existing work on rates of convergence.

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Estimating Tails of Probability Distributions

The Benders decomposition algorithm: A literature review

The Structure of Inference.

The traditional supply chain network planning problem is stated as a multi-period resource allocation model involving 0–1 discrete strategic decision variables. The MIP structure of this problem makes it fairly intractable for practical applications, which involve multiple products, factories, warehouses and distribution centres (DCs). The same problem formulated and studied under uncertainty makes it even more intractable. In this paper we consider two related modelling approaches and solution techniques addressing this issue. The first involves scenario analysis of solutions to “wait and see” models and the second involves a two-stage integer stochastic programming (ISP) representation and solution of the same problem. We show how the results from the former can be used in the solution of the latter model. We also give some computational results based on serial and parallel implementations of the algorithms.

Computational solution of capacity planning models under uncertainty

The purpose of this paper is to introduce a new test-problem collection for stochastic linear programming that the authors have recently begun to assemble. While there are existing stochastic programming test-problem collections, our new collection has three features that distinguish it from existing collections. First, our collection is web-based with free public access, and we intend to enrich it as new test problems become available. Indeed, we encourage submissions of new test problems. Second, along with the collection we provide documentation of the problems, so that researchers can quickly find information about each family without reading through the original source. Third, all of the data in our collection are provided in SMPS (Birge et al. 1987, Gassmann and Schweitzer 2001) format. In this paper, we provide an introduction to the stochastic linear program, give a brief description of each problem family currently in the test-problem collection, and describe the documentation that accompanies the collection.

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On a New Collection of Stochastic Linear Programming Test Problems

This paper is concerned with collinear scaling algorithms for unconstrained minimization where the underlying local approximants are forced to interpolate the objective function value and gradient at only the two most recent iterates. By suitably modifying the procedure of Sorensen (1980) for deriving such algorithms, we show that two members of the algorithm class derived related to the DFP and BFGS methods respectively are locally and q-superlinearly convergent. This local analysis as well as the results they yield exhibit the same sort of "duality" exhibited by those of Broyden, Dennis and More (1973) and Dennis and More (1974) for the DFP and BFGS methods. The results in this paper also imply the local and q-superlinear convergence of collinear scaling algorithms of Sorensen (1982, pp. 154---156) related to the DFP and BFGS methods.

Deriving collinear scaling algorithms as extension of quasi-Newton methods and the local convergence of DFP-and BFGS-related collinear scaling algorithms

Semidefinite programs are a class of optimization problems that have been studied extensively during the past 15 years. Semidefinite programs are naturally related to linear programs, and both are defined using deterministic data. Stochastic programs were introduced in the 1950s as a paradigm for dealing with uncertainty in data defining linear programs. In this paper, we introduce stochastic semidefinite programs as a paradigm for dealing with uncertainty in data defining semidefinite programs.

Stochastic semidefinite programming: a new paradigm for stochastic optimization

We describe a benchmark parallel version of the Van Slyke and Wets algorithm for two-stage stochastic programs and an implementation of that algorithm on the Sequent/Balance. We also report results of a numerical experiment using random test problems and our implementation. These performance results, to the best of our knowledge, are the first available for the Van Slyke and Wets algorithm on a parallel processor. They indicate that the benchmark implementation parallelizes well, and that even with the use of parallel processing, problems with random variables having large numbers of realizations can take prohibitively large amounts of computation for solution. Thus, they demonstrate the need for exploiting both parallelization and approximation for the solution of stochastic programs. 15 refs., 18 tabs.

K. A. Ariyawansa

Papers

On a New Collection of Stochastic Linear Programming Test Problems

Deriving collinear scaling algorithms as extension of quasi-Newton methods and the local convergence of DFP-and BFGS-related collinear scaling algorithms

Stochastic semidefinite programming: a new paradigm for stochastic optimization

Performance of a benchmark parallel implementation of the Van Slyke and wets algorithm for two-stage stochastic programs on the sequent/balance

A preliminary set of applications leading to stochastic semidefinite programs and chance-constrained semidefinite programs