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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 222 (15 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.
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 70 (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.
Filter Pattern Search Algorithms for Mixed Variable Constrained Optimization Problems
 SIAM Journal on Optimization
, 2004
"... A new class of algorithms for solving nonlinearly constrained mixed variable optimization problems is presented. This class combines and extends the AudetDennis Generalized Pattern Search (GPS) algorithms for bound constrained mixed variable optimization, and their GPSfilter algorithms for gene ..."
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Cited by 55 (8 self)
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A new class of algorithms for solving nonlinearly constrained mixed variable optimization problems is presented. This class combines and extends the AudetDennis Generalized Pattern Search (GPS) algorithms for bound constrained mixed variable optimization, and their GPSfilter algorithms for general nonlinear constraints. In generalizing existing algorithms, new theoretical convergence results are presented that reduce seamlessly to existing results for more specific classes of problems. While no local continuity or smoothness assumptions are required to apply the algorithm, a hierarchy of theoretical convergence results based on the Clarke calculus is given, in which local smoothness dictate what can be proved about certain limit points generated by the algorithm. To demonstrate the usefulness of the algorithm, the algorithm is applied to the design of a loadbearing thermal insulation system. We believe this is the first algorithm with provable convergence results to directly target this class of problems.
Dynamical Modeling and MultiExperiment Fitting with PottersWheel – Supplement
, 2008
"... This supplement provides detailed information about the functionalities of the PottersWheel toolbox as described in the main text. For further information please use the ..."
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Cited by 52 (7 self)
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This supplement provides detailed information about the functionalities of the PottersWheel toolbox as described in the main text. For further information please use the
Theory and implementation of numerical methods based on RungeKutta integration for solving optimal control problems
, 1996
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Optimization With VariableFidelity Models Applied To Wing Design
, 2000
"... This work discusses an approach, the Approximation Management Framework (AMF), for solving optimization problems that involve computationally expensive simulations. AMF aims to maximize the use of lowerfidelity, cheaper models in iterative procedures with occasional, but systematic, recourse to high ..."
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Cited by 34 (2 self)
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This work discusses an approach, the Approximation Management Framework (AMF), for solving optimization problems that involve computationally expensive simulations. AMF aims to maximize the use of lowerfidelity, cheaper models in iterative procedures with occasional, but systematic, recourse to higherfidelity, more expensive models for monitoring the progress of the algorithm. The method is globally convergent to a solution of the original, highfidelity problem. Three versions of AMF, based on three nonlinear programming algorithms, are demonstrated on a 3D aerodynamic wing optimization problem and a 2D airfoil optimization problem. In both cases Euler analysis solved on meshes of various refinement provides a suite of variablefidelity models. Preliminary results indicate threefold savings in terms of highfidelity analyses in case of the 3D problem and twofold savings for the 2D problem. Key Words: Approximation concepts, approximation management, model management, surrogate optimi...
UncertaintyAware Circuit Optimization
 IN DAC
, 2002
"... Almost by definition, welltuned digital circuits have a large number of equally critical paths, which form a socalled "wall" in the slack histogram. However, by the time the design has been through manufacturing, many uncertainties cause these carefully aligned delays to spread out. Inac ..."
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Cited by 28 (1 self)
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Almost by definition, welltuned digital circuits have a large number of equally critical paths, which form a socalled "wall" in the slack histogram. However, by the time the design has been through manufacturing, many uncertainties cause these carefully aligned delays to spread out. Inaccuracies in parasitic predictions, clock slew, modeltohardware correlation, static timing assumptions and manufacturing variations all cause the performance to vary from prediction. Simple statistical principles tell us that the variation of the limiting slack is larger when the height of the wall is greater. Although the wall may be the optimum solution if the static timing predictions were perfect, in the presence of uncertainty in timing and manufacturing, it may no longer be the best choice. The application of formal mathematical optimization in transistor sizing increases the height of the wall, thus exacerbating the problem. There is also a practical matter that schematic restructuring downstream in the design methodology is easier to conceive when there are fewer equally critical paths. This paper describes a method that gives formal mathematical optimizers the incentive to avoid the wall of equally critical paths, while giving up as little as possible in nominal performance. Surprisingly, such a formulation reduces the degeneracy of the optimization problem and can render the optimizer more effective. This "uncertaintyaware" mode has been implemented and applied to several highperformance microprocessor macros. Numerical results are included.
A globally convergent linearly constrained Lagrangian method for nonlinear optimization
 SIAM J. Optim
, 2002
"... Abstract. For optimization problems with nonlinear constraints, linearly constrained Lagrangian (LCL) methods solve a sequence of subproblems of the form “minimize an augmented Lagrangian function subject to linearized constraints. ” Such methods converge rapidly near a solution but may not be relia ..."
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Cited by 25 (4 self)
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Abstract. For optimization problems with nonlinear constraints, linearly constrained Lagrangian (LCL) methods solve a sequence of subproblems of the form “minimize an augmented Lagrangian function subject to linearized constraints. ” Such methods converge rapidly near a solution but may not be reliable from arbitrary starting points. Nevertheless, the wellknown software package MINOS has proved effective on many large problems. Its success motivates us to derive a related LCL algorithm that possesses three important properties: it is globally convergent, the subproblem constraints are always feasible, and the subproblems may be solved inexactly. The new algorithm has been implemented in Matlab, with an option to use either MINOS or SNOPT (Fortran codes) to solve the linearly constrained subproblems. Only first derivatives are required. We present numerical results on a subset of the COPS, HS, and CUTE test problems, which include many large examples. The results demonstrate the robustness and efficiency of the stabilized LCL procedure.
Analysis and implementation of a dual algorithm for constrained optimization
 Journal of Optimization Theory and Applications
, 1993
"... Abstract. This paper analyzes a constrained optimization algorithm that combines an unconstrained minimization scheme like the conjugate gradient method, an augmented Lagrangian, and multiplier updates to obtain global quadratic convergence. Some of the issues that we focus on are the treatment of r ..."
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Cited by 19 (3 self)
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Abstract. This paper analyzes a constrained optimization algorithm that combines an unconstrained minimization scheme like the conjugate gradient method, an augmented Lagrangian, and multiplier updates to obtain global quadratic convergence. Some of the issues that we focus on are the treatment of rigid constraints that must be satisfied during the iterations and techniques for balancing the error associated with constraint violation with the error associated with optimality. A preconditioner is constructed with the property that the rigid constraints are satisfied while illconditioning due to penalty terms is alleviated. Various numerical linear algebra techniques required for the efficient implementation of the algorithm are presented, and convergence behavior is illustrated in a series of numerical experiments.