Browsing by Author "Trosset, Michael W."
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Item A New Formulation of the Nonmetric STRAIN Problem in Multidimensional Scaling(1993-08) Trosset, Michael W.A new formulation of nonmetric multidimensional scaling using the STRAIN criterion is proposed. Innovative features of the new formulation include: the parameterization of the p-dimensional distance matrices by the positive semidefinite matrices of rank >=p; optimization of the disparity variables, rather than the configuration coordinate variables; and a new nondegeneracy constraint, which restricts the set of disparities rather than the set of distances. It is demonstrated that this formulation allows the identification of globally optimal configurations. The method is applied to two famous data sets.Item A Study of the Stationary Configurations of the SStress Criterion for Metric Multidimensional Scaling(2000-01) Malone, Samuel W.; Trosset, Michael W.It is widely believed that both the stress and the sstress criteria for metric multidimensional scaling are plagued by the existence of nonglobal minimizers. At present, there is little theory that enlightens this belief. Trosset and Mathar (1997) established that nonglobal minimizers of the stress criterion can exist, while Glunt, Hayden, and Liu (1991) demonstrated that the distance matrices of all configurations for which the gradient of the sstress criterion vanishes lie on a certain sphere. This report extends existing theory in several directions. Emphasis is placed on the more tractable case of the sstress criterion. Because the stress and stress criteria must be minimized by numerical optimization, one result that is of immediate practical value is a simple device for improving the quality of the initial configurations from which optimization commences.Item Correcting an Inconsistent System of Linear Inequalities by Nonlinear Programming(2000-07) Amaral, Paula; Trosset, Michael W.; Barahona, PedroWe consider the problem of correcting an inconsistent system of linear inequalities, Ax <= b, subject to nonnegativity constraints, x >= 0. We formulate this problem as a nonlinear program and derive the corresponding Karush-Kuhn-Tucker conditions. These conditions reveal several interesting properties that solutions must satisfy and allow us to derive several equivalent problems that involve fewer decision variables and are more amenable to solution. We propose using a gradient projection method to minimize an objective function Ø(x) subject only to x >= 0. We also propose a hybrid approach that exploits an interesting relation between the correction problem and the method of total least squares.Item Designing and Analyzing Computational Experiments for Global Optimization(2000-07) Trosset, Michael W.; Padula, Anthony D.We consider a variety of issues that arise when designing and analyzing computational experiments for global optimization. We describe a probability model for objective functions and a method for generating pseudorandom objective functions. We argue in favor of evaluating the performance of global optimization algorithms by measuring the depth of the objective function achieved with a fixed number of function evaluations. We emphasize the importance of replication in computational experiments and describe some useful statistical techniques for assimilating results. We illustrate our methods by performing a small study that compares two multistart strategies for global optimization.Item Distance Matrix Completion by Numerical Optimization(1995-10) Trosset, Michael W.Consider the problem of determining whether or not a partial dissimilarity matrix can be completed to a Euclidean distance matrix. The dimension of the distance matrix may be restricted and the known dissimilarities may be permitted to vary subject to bound constraints. This problem, which naturally arises in the study of molecular conformation, can be formulated as an optimization problem. Completion is possible if and only if the global minimum of the optimization problem is zero; furthermore, using ideas from nonmetric multidimensional scaling, it is possible to construct a sequence of objective function values that is guaranteed to converge to the global minimum. Thus, this approach provides a constructive technique for obtaining approximate solutions to a very general class of distance matrix completion problems.Item I Know It When I See It: Toward a Definition of Direct Search Methods(1996-06) Trosset, Michael W.We discuss some ambiguities associated with the phrase "direct search method" and propose a formal definition.Item On the Use of Direct Search Methods for Stochastic Optimization(2000-06) Trosset, Michael W.We examine the conventional wisdom that commends the use of directe search methods in the presence of random noise. To do so, we introduce new formulations of stochastic optimization and direct search. These formulations suggest a natural strategy for constructing globally convergent direct search algorithms for stochastic optimization by controlling the error rates of the ordering decisions on which direct search depends. This strategy is successfully applied to the class of generalized pattern search methods. However, a great deal of sampling is required to guarantee convergence with probability one.Item Optimal Shapes for Kernel Density Estimations: An Historical Footnote(1991-09) Trosset, Michael W.In the early years of kernel density estimation, Watson and Leadbetter (1963) attempted to optimize kernel shape for fixed sample sizes by minimizing the expected L² distance between the kernel density estimate and the true density. Perhaps out of technical necessity, they did not impose the constraint that the kernel be a probability density function. The present paper uses recent developments in the theory of infinite programming to successfully impose that constraint. Necessary and sufficient conditions for solution of the constrained problem are derived. These conditions are not trivial; however, they can be exploited to demonstrate that symmetric densities with sufficiently light tails have optimal kernels with compact support.Item Taguchi and Robust Optimization(1996-10) Trosset, Michael W.This report is intended to facilitate dialogue between engineers and optimizers about the efficiency of Taguchi methods for robust design, especially in the context of design by computer simulation. Three approaches to robust design are described: 1)Robust optimization, i.e., specifying an objective function f and then minimizing a smoothed (robust) version of f by the methods of numerical optimization. 2) Taguchi's method of specifying the objective function as a certain signal-to-noise ratio, to be optimized by designing, performing, and analyzing a single massive element. 3) Specifying an expected loss function f and then minimizing a cheap-to-compute surrogate objective function f, to be obtained by designing and performing a single massive element.Item The Formulation and Solution of Multidimensional Scaling Problems(1993-11) Trosset, Michael W.Numerous experiments in a variety of applied disciplines involve measuring distances between pairs of objects. The statistical problem posed by such experiments is that of fitting the observed data with a model defined to be the Euclidean distances between an abstract configuration of points. Techniques for solving this problem are collectively known as multidimensional scaling. These techniques have a long history in psychometrics and multivariate statistics, a much shorter one in the application of distance geometry to problems of molecular conformation. This review attempts to integrate these two traditions, which presently exist almost unaware of each other. Emphasis is placed on the rigorous formulation of the defining optimization problems, and on the computational practices that have been developed for solving these problems. Recent developments suggest that multidimensional scaling has entered a new and exciting era, as researchers begin to apply the tools of modern computational mathematics.Item What is Simulated Annealing?(2000-02) Trosset, Michael W.Beginning in 1983, simulated annealing was marketed as a global optimization methodology that mimics the physical annealing process by which molten substances cool to crystalline lattices of minimal energy. This marketing strategy had a polarizing effect, attracting those who delighted in metaphor and alienating others who found metaphor insufficient at best and facile at worst. In fact, the emotional outbursts that accompany many discussions of simulated annealing are an unfortunate distraction. Whatever its pros and cons, simulated annealing can be grounded in rigorous mathematics. Here we provide an elementary, self-contained introduction to simulated annealing in terms of Markov chains.