Homotopy optimization methods for global optimization.

We define a new method for global optimization, the Homotopy Optimization Method (HOM). This method differs from previous homotopy and continuation methods in that its aim is to find a minimizer for each of a set of values of the homotopy parameter, rather than to follow a path of minimizers. We define a second method, called HOPE, by allowing HOM to follow an ensemble of points obtained by perturbation of previous ones. We relate this new method to standard methods such as simulated annealing and show under what circumstances it is superior. We present results of extensive numerical experiments demonstrating performance of HOM and HOPE.

[1]  R. Jennrich,et al.  Application of Stepwise Regression to Non-Linear Estimation , 1968 .

[2]  N. Metropolis,et al.  Equation of State Calculations by Fast Computing Machines , 1953, Resonance.

[3]  Dianne P. O'Leary,et al.  HOPE: A Homotopy Optimization Method for Protein Structure Prediction , 2005, J. Comput. Biol..

[4]  Lester Ingber,et al.  Simulated annealing: Practice versus theory , 1993 .

[5]  Masha Sosonkina,et al.  Algorithm 777: HOMPACK90: a suite of Fortran 90 codes for globally convergent homotopy algorithms , 1997, TOMS.

[6]  Layne T. Watson,et al.  Theory of Globally Convergent Probability-One Homotopies for Nonlinear Programming , 2000, SIAM J. Optim..

[7]  Daniel Michael Dunlavy.  Homotopy Optimization Methods and Protein Structure Prediction , 2005 .

[8]  Bernardetta Addis,et al.  Local optima smoothing for global optimization , 2005, Optim. Methods Softw..

[9]  Emile H. L. Aarts,et al.  Simulated Annealing: Theory and Applications , 1987, Mathematics and Its Applications.

[10]  Robert L. Smith,et al.  Improving Hit-and-Run for global optimization , 1993, J. Glob. Optim..

[11]  Layne T. Watson,et al.  Algorithm 652: HOMPACK: a suite of codes for globally convergent homotopy algorithms , 1987, TOMS.

[12]  E. Allgower,et al.  Introduction to Numerical Continuation Methods , 1987 .

[13]  Fabio Schoen,et al.  Stochastic techniques for global optimization: A survey of recent advances , 1991, J. Glob. Optim..

[14]  L. Watson,et al.  Modern homotopy methods in optimization , 1989 .

[15]  Elizabeth A. Burroughs,et al.  LOCA 1.0 Library of Continuation Algorithms: Theory and Implementation Manual , 2002 .

[16]  Werner Ebeling,et al.  EVOLUTIONARY STRATEGIES OF OPTIMIZATION , 1997 .

[17]  H. Scheraga,et al.  Global optimization of clusters, crystals, and biomolecules. , 1999, Science.

[18]  P. Pardalos,et al.  From local to global optimization , 2001 .

[19]  M. Locatelli.  Simulated Annealing Algorithms for Continuous Global Optimization , 2002 .

[20]  Emile H. L. Aarts,et al.  Global optimization and simulated annealing , 1991, Math. Program..

[21]  János D. Pintér,et al.  Global Optimization: Software, Test Problems, and Applications , 2002 .

[22]  Sven Leyffer,et al.  A Trust-Region Algorithm for Global Optimization , 2006, Comput. Optim. Appl..

[23]  Immo Diener,et al.  Trajectory Methods in Global Optimization , 1995 .

[24]  P. Pardalos,et al.  Handbook of global optimization , 1995 .

[25]  Editors , 1986, Brain Research Bulletin.

[26]  Perla B. Balbuena,et al.  Molecular dynamics : from classical to quantum methods , 1999 .

[27]  R. Horst,et al.  Global Optimization: Deterministic Approaches , 1992 .

[28]  Willy Govaerts,et al.  MATCONT: A MATLAB package for numerical bifurcation analysis of ODEs , 2003, TOMS.

[29]  M. C. Biggs.  Minimization Algorithms Making Use of Non-quadratic Properties of the Objective Function , 1971 .

[30]  Jorge J. Moré,et al.  Testing Unconstrained Optimization Software , 1981, TOMS.

[31]  Panos M. Pardalos,et al.  Recent Advances in Global Optimization , 1991 .

[32]  Amy Cha-Tien Sun,et al.  Homotopy-continuation algorithm for global optimization , 1992 .

[33]  Sigurd Schelstraete,et al.  Chapter 5 - Energy minimization by smoothing techniques: a survey , 1999 .

[34]  Thomas F. Fairgrieve,et al.  AUTO 2000 : CONTINUATION AND BIFURCATION SOFTWARE FOR ORDINARY DIFFERENTIAL EQUATIONS (with HomCont) , 1997 .

[35]  F. Schoen.  TWO-PHASE METHODS FOR GLOBAL OPTIMIZATION , 2002 .

[36]  C. D. Gelatt,et al.  Optimization by Simulated Annealing , 1983, Science.

[37]  A. Neumaier.  Complete search in continuous global optimization and constraint satisfaction , 2004, Acta Numerica.

[38]  E. J. Doedel,et al.  AUTO: a program for the automatic bifurcation analysis of autonomous systems , 1980 .

[39]  R. Meyer.  Theoretical and Computational Aspects of Nonlinear Regression , 1970 .

[40]  Ferdinand Freudenstein,et al.  Numerical Solution of Systems of Nonlinear Equations , 1963, JACM.