Nonlinear Programming by Mesh Adaptive Direct Searches

Abstract : This paper is intended not as a survey, but as an introduction to some ideas behind the class of mesh adaptive direct search (MADS) methods. Space limitations dictate a brief description of various key topics to be provided along with several references, which themselves provide further references. The convergence theory for the methods presented here make a case for closing the gap between nonlinear optimizers and nonsmooth analysts. However these methods are certainly not of purely theoretical interest; they are successful on difficult practical problems. To encourage further use, we give references to available implementations. MADS is implemented in the direct search portion of the MathWorks MATLAB Genetic Algorithm and Direct Search (GADS) Toolbox.

[1]  Virginia Torczon,et al.  On the Convergence of Pattern Search Algorithms , 1997, SIAM J. Optim..

[2]  Charles Audet,et al.  Catalytic Combustion Kinetics: Using a Direct Search Algorithm to Evaluate Kinetic Parameters from Light-Off Curves , 2002 .

[3]  Charles Audet,et al.  Pattern Search Algorithms for Mixed Variable Programming , 2000, SIAM J. Optim..

[4]  J. Dennis,et al.  Mixed Variable Optimization of the Number and Composition of Heat Intercepts in a Thermal Insulation System , 2001 .

[5]  Søren Nymand Lophaven,et al.  DACE - A Matlab Kriging Toolbox , 2002 .

[6]  Natalia Alexandrov,et al.  Multidisciplinary design optimization : state of the art , 1997 .

[7]  J. Hiriart-Urruty,et al.  Generalized Hessian matrix and second-order optimality conditions for problems withC1,1 data , 1984 .

[8]  Søren Nymand Lophaven,et al.  DACE - A Matlab Kriging Toolbox, Version 2.0 , 2002 .

[9]  C. J. Price,et al.  On the Convergence of Grid-Based Methods for Unconstrained Optimization , 2000, SIAM J. Optim..

[10]  V. Torczon,et al.  RANK ORDERING AND POSITIVE BASES IN PATTERN SEARCH ALGORITHMS , 1996 .

[11]  Charles Audet,et al.  Generalized pattern searches with derivative information , 2002, Math. Program..

[12]  A. J. Booker,et al.  A rigorous framework for optimization of expensive functions by surrogates , 1998 .

[13]  Mark A. Abramson,et al.  Second-Order Behavior of Pattern Search , 2005, SIAM J. Optim..

[14]  Tamara G. Kolda,et al.  Asynchronous Parallel Pattern Search , 2000 .

[15]  Charles Audet,et al.  A Pattern Search Filter Method for Nonlinear Programming without Derivatives , 2001, SIAM J. Optim..

[16]  Charles Audet,et al.  Analysis of Generalized Pattern Searches , 2000, SIAM J. Optim..

[17]  Nicholas I. M. Gould,et al.  GALAHAD, a library of thread-safe Fortran 90 packages for large-scale nonlinear optimization , 2003, TOMS.

[18]  Owen J. Eslinger,et al.  Optimization of Automotive Valve Train Components with Implicit Filtering , 2000 .

[19]  R. Rockafellar.  Generalized Directional Derivatives and Subgradients of Nonconvex Functions , 1980, Canadian Journal of Mathematics.

[20]  J. Jahn.  Introduction to the Theory of Nonlinear Optimization , 1994 .

[21]  Alison L. Marsden,et al.  Aerodynamic noise control by optimal shape design , 2004 .

[22]  J. Dennis,et al.  Pattern search algorithms for mixed variable general constrained optimization problems , 2003 .

[23]  Charles Audet,et al.  Second-Order Convergence of Mesh-Adaptive Direct Search , 2005 .

[24]  Thomas J. Santner,et al.  Design and analysis of computer experiments , 1998 .

[25]  C. J. Price,et al.  Positive Bases in Numerical Optimization , 2002, Comput. Optim. Appl..

[26]  Tamara G. Kolda,et al.  On the Convergence of Asynchronous Parallel Pattern Search , 2002, SIAM J. Optim..

[27]  I. Coope,et al.  Frame Based Methods for Unconstrained Optimization , 2000 .

[28]  Charles Audet,et al.  A surrogate-model-based method for constrained optimization , 2000 .

[29]  F. Clarke.  Optimization And Nonsmooth Analysis , 1983 .

[30]  Luís N. Vicente,et al.  Using Sampling and Simplex Derivatives in Pattern Search Methods , 2007, SIAM J. Optim..

[31]  C. J. Price,et al.  Frames and Grids in Unconstrained and Linearly Constrained Optimization: A Nonsmooth Approach , 2003, SIAM J. Optim..

[32]  Indraneel Das.  On characterizing the “knee” of the Pareto curve based on Normal-Boundary Intersection , 1999 .

[33]  Charles Audet,et al.  Convergence Results for Pattern Search Algorithms are Tight , 2002 .

[34]  Charles Audet,et al.  Finding Optimal Algorithmic Parameters Using the Mesh Adaptive Direct Search Algorithm , 2004 .

[35]  John E. Dennis,et al.  Numerical methods for unconstrained optimization and nonlinear equations , 1983, Prentice Hall series in computational mathematics.

[36]  I. Coope,et al.  Frame-Based Ray Search Algorithms in Unconstrained Optimization , 2003 .

[37]  M. Abramson.  Mixed Variable Optimization of a Load-Bearing Thermal Insulation System Using a Filter Pattern Search Algorithm , 2004 .

[38]  Sven Leyffer,et al.  Nonlinear programming without a penalty function , 2002, Math. Program..

[39]  Charles Audet,et al.  Mesh Adaptive Direct Search Algorithms for Constrained Optimization , 2006, SIAM J. Optim..