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Optimization by Direct Search: New Perspectives on Some Classical and Modern Methods
 SIAM REVIEW VOL. 45, NO. 3, PP. 385–482
, 2003
"... Direct search methods are best known as unconstrained optimization techniques that do not explicitly use derivatives. Direct search methods were formally proposed and widely applied in the 1960s but fell out of favor with the mathematical optimization community by the early 1970s because they lacked ..."
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Cited by 198 (14 self)
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Direct search methods are best known as unconstrained optimization techniques that do not explicitly use derivatives. Direct search methods were formally proposed and widely applied in the 1960s but fell out of favor with the mathematical optimization community by the early 1970s because they lacked coherent mathematical analysis. Nonetheless, users remained loyal to these methods, most of which were easy to program, some of which were reliable. In the past fifteen years, these methods have seen a revival due, in part, to the appearance of mathematical analysis, as well as to interest in parallel and distributed computing. This review begins by briefly summarizing the history of direct search methods and considering the special properties of problems for which they are well suited. Our focus then turns to a broad class of methods for which we provide a unifying framework that lends itself to a variety of convergence results. The underlying principles allow generalization to handle bound constraints and linear constraints. We also discuss extensions to problems with nonlinear constraints.
COPASI  a COmplex PAthway SImulator
 BIOINFORMATICS
, 2006
"... Motivation: Simulation and modeling is becoming a standard approach to understand complex biochemical processes. Therefore, there is a big need for software tools that allow access to diverse simulation and modeling methods as well as support for the usage of these methods. Results: Here, we present ..."
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Cited by 159 (2 self)
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Motivation: Simulation and modeling is becoming a standard approach to understand complex biochemical processes. Therefore, there is a big need for software tools that allow access to diverse simulation and modeling methods as well as support for the usage of these methods. Results: Here, we present COPASI, a platformindependent and userfriendly biochemical simulator that offers several unique features. We discuss numerical issues with these features, in particular the criteria to switch between stochastic and deterministic simulation methods, hybrid deterministicstochastic methods, and the importance of random number generator numerical resolution in stochastic simulation. Availability: The complete software is available in binary (executable) for MS Windows, OS X, Linux (Intel), and Sun Solaris (SPARC), as well as the full source code under an open source license from
Optimization of Mutual Information for Multiresolution Image Registration
 IEEE Transactions on Image Processing
, 2000
"... We propose a new method for the intermodal registration of images using a criterion known as mutual information. Our main contribution is an optimizer that we specifically designed for this criterion. We show that this new optimizer is well adapted to a multiresolution approach because it typically ..."
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Cited by 122 (6 self)
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We propose a new method for the intermodal registration of images using a criterion known as mutual information. Our main contribution is an optimizer that we specifically designed for this criterion. We show that this new optimizer is well adapted to a multiresolution approach because it typically converges in fewer criterion evaluations than other optimizers. We have built a multiresolution image pyramid, along with an interpolation process, an optimizer, and the criterion itself, around the unifying concept of splineprocessing. This ensures coherence in the way we model data and yields good performance. We have tested our approach in a variety of experimental conditions and report excellent results. We claim an accuracy of about a hundredth of a pixel under ideal conditions. We are also robust since the accuracy is still about a tenth of a pixel under very noisy conditions. In addition, a blind evaluation of our results compares very favorably to the work of several other researchers.
Packing Lines, Planes, etc.: Packings in Grassmannian Spaces
, 1996
"... We address the question: How should N ndimensional subspaces of mdimensional Euclidean space be arranged so that they are as far apart as possible? The results of extensive computations for modest values of N; n; m are described, as well as a reformulation of the problem that was suggested by th ..."
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Cited by 98 (10 self)
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We address the question: How should N ndimensional subspaces of mdimensional Euclidean space be arranged so that they are as far apart as possible? The results of extensive computations for modest values of N; n; m are described, as well as a reformulation of the problem that was suggested by these computations. The reformulation gives a way to describe n dimensional subspaces of mspace as points on a sphere in dimension (m \Gamma 1)(m+2), which provides a (usually) lowerdimensional representation than the Pl ucker embedding, and leads to a proof that many of the new packings are optimal. The results have applications to the graphical display of multidimensional data via Asimov's grand tour method.
Direct search methods: once scorned, now respectable
 in Numerical Analysis 1995 (Proceedings of the 1995 Dundee Biennial Conference in Numerical Analysis
, 1996
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Direct search methods: then and now
, 2000
"... We discuss direct search methods for unconstrained optimization. We give a modern perspective on this classical family of derivativefree algorithms, focusing on the development of direct search methods during their golden age from 1960 to 1971. We discuss how direct search methods are characterized ..."
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Cited by 82 (3 self)
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We discuss direct search methods for unconstrained optimization. We give a modern perspective on this classical family of derivativefree algorithms, focusing on the development of direct search methods during their golden age from 1960 to 1971. We discuss how direct search methods are characterized by the absence of the construction of a model of the objective. We then consider a number of the classical direct search methods and discuss what research in the intervening years has uncovered about these algorithms. In particular, while the original direct search methods were consciously based on straightforward heuristics, more recent analysis has shown that in most — but not all — cases these heuristics actually
Direct Search Algorithms for Optimization Calculations
, 1998
"... : Many different procedures have been proposed for optimization calculations when first derivatives are not available. Further, several researchers have contributed to the subject, including some who wish to prove convergence theorems, and some who wish to make any reduction in the least calculated ..."
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Cited by 76 (2 self)
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: Many different procedures have been proposed for optimization calculations when first derivatives are not available. Further, several researchers have contributed to the subject, including some who wish to prove convergence theorems, and some who wish to make any reduction in the least calculated value of the objective function. There is not even a key idea that can be used as a foundation of a review, except for the problem itself, which is the adjustment of variables so that a function becomes least, where each value of the function is returned by a subroutine for each trial vector of variables. Therefore the paper is a collection of essays on particular strategies and algorithms, in order to consider the advantages, limitations and theory of several techniques. The subjects that are addressed are line search methods, the restriction of vectors of variables to discrete grids, the use of geometric simplices, conjugate direction procedures, trust region algorithms that form linear or...
Pattern Search Methods for Linearly Constrained Minimization
, 2000
"... We extend pattern search methods to linearly constrained minimization. We develop a general class of feasible point pattern search algorithms and prove global convergence to a Karush–Kuhn–Tucker point. As in the case of unconstrained minimization, pattern search methods for linearly constrained pr ..."
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Cited by 71 (7 self)
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We extend pattern search methods to linearly constrained minimization. We develop a general class of feasible point pattern search algorithms and prove global convergence to a Karush–Kuhn–Tucker point. As in the case of unconstrained minimization, pattern search methods for linearly constrained problems accomplish this without explicit recourse to the gradient or the directional derivative of the objective. Key to the analysis of the algorithms is the way in which the local search patterns conform to the geometry of the boundary of the feasible region.
A GLOBALLY CONVERGENT AUGMENTED LAGRANGIAN PATTERN SEARCH ALGORITHM FOR OPTIMIZATION WITH GENERAL CONSTRAINTS AND SIMPLE BOUNDS
, 2002
"... We give a pattern search methodfor nonlinearly constrained optimization that is an adaption of a bound constrained augmented Lagrangian method first proposed by Conn, Gould, andToint [SIAM J. Numer. Anal., 28 (1991), pp. 545–572]. In the pattern search adaptation, we solve the bound constrained subp ..."
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Cited by 61 (7 self)
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We give a pattern search methodfor nonlinearly constrained optimization that is an adaption of a bound constrained augmented Lagrangian method first proposed by Conn, Gould, andToint [SIAM J. Numer. Anal., 28 (1991), pp. 545–572]. In the pattern search adaptation, we solve the bound constrained subproblem approximately using a pattern search method. The stopping criterion proposedby Conn, Gould, andToint for the solution of the subproblem requires explicit knowledge of derivatives. Such information is presumed absent in pattern search methods; however, we show how we can replace this with a stopping criterion basedon the pattern size in a way that preserves the convergence properties of the original algorithm. In this way we proceedby successive, inexact, bound constrained minimization without knowing exactly how inexact the minimization is. As far as we know, this is the first provably convergent direct search method for general nonlinear programming.