A Direct Search Algorithm for Optimization with Noisy Function Evaluations

We consider the unconstrained optimization of a function when each function evaluation is subject to a random noise. We assume that there is some control over the variance of the noise term, in the sense that additional computational effort will reduce the amount of noise. This situation may occur when function evaluations involve simulation or the approximate solution of a numerical problem. It also occurs in an experimental setting when averaging repeated observations at the same point can lead to a better estimate of the underlying function value. We describe a new direct search algorithm for this type of problem. We prove convergence of the new algorithm when the noise is controlled so that the standard deviation of the noise approaches zero faster than the step size. We also report some numerical results on the performance of the new algorithm.

[1]  J. Dennis,et al.  Direct Search Methods on Parallel Machines , 1991 .

[2]  K. I. M. McKinnon,et al.  Convergence of the Nelder-Mead Simplex Method to a Nonstationary Point , 1998, SIAM J. Optim..

[3]  Y. Wardi.  Stochastic algorithms with armijo stepsizes for minimization of functions , 1990 .

[4]  Manel Poch,et al.  Comparison of the Powell and simplex methods in the optimization of flow-injection systems. Simulation on modelled experimental surfaces and experimental optimizations , 1990 .

[5]  Virginia Torczon,et al.  On the Convergence of the Multidirectional Search Algorithm , 1991, SIAM J. Optim..

[6]  V. Fabian.  Stochastic Approximation Methods for Constrained and Unconstrained Systems (Harold L. Kushner and Dean S. Clark) , 1980 .

[7]  A. Gustavsson,et al.  Design and evaluation of modified simplex methods , 1982 .

[8]  Peter Hedlund,et al.  Design and evaluation of modified simplex methods having enhanced convergence ability , 1992 .

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

[10]  Paul Tseng,et al.  Fortified-Descent Simplicial Search Method: A General Approach , 1999, SIAM J. Optim..

[11]  A. Goldstein,et al.  Optimization of functions whose values are subject to small errors , 1977 .

[12]  Harold J. Kushner,et al.  wchastic. approximation methods for constrained and unconstrained systems , 1978 .

[13]  M. Deaton,et al.  Response Surfaces: Designs and Analyses , 1989 .

[14]  G. R. Hext,et al.  Sequential Application of Simplex Designs in Optimisation and Evolutionary Operation , 1962 .

[15]  M. Powell.  A Direct Search Optimization Method That Models the Objective and Constraint Functions by Linear Interpolation , 1994 .

[16]  P. Toint,et al.  An Algorithm using Quadratic Interpolation for Unconstrained Derivative Free Optimization , 1996 .

[17]  Francesco Zirilli,et al.  Algorithm 667: Sigma—a stochastic-integration global minimization algorithm , 1988, TOMS.

[18]  Robert Hooke,et al.  `` Direct Search'' Solution of Numerical and Statistical Problems , 1961, JACM.

[19]  O. Nelles,et al.  An Introduction to Optimization , 1996, IEEE Antennas and Propagation Magazine.

[20]  Feller William,et al.  An Introduction To Probability Theory And Its Applications , 1950 .

[21]  J. S. Ivey,et al.  Nelder-Mead simplex modifications for simulation optimization , 1996 .

[22]  Jeffrey C. Lagarias,et al.  Convergence Properties of the Nelder-Mead Simplex Method in Low Dimensions , 1998, SIAM J. Optim..

[23]  Y. Wardi.  A stochastic steepest-descent algorithm , 1988 .

[24]  M. J. D. Powell,et al.  An efficient method for finding the minimum of a function of several variables without calculating derivatives , 1964, Comput. J..

[25]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1967 .

[26]  Susana Gomez,et al.  Advances in optimization and numerical analysis : proceedings of the sixth Workshop on Optimization and Numerical Analysis, Oaxaca, Mexico , 1994 .

[27]  John A. Nelder,et al.  A Simplex Method for Function Minimization , 1965, Comput. J..

[28]  A. Neumaier,et al.  A grid algorithm for bound constrained optimization of noisy functions , 1995 .

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

[30]  P. Gill,et al.  Computing Forward-Difference Intervals for Numerical Optimization , 1983 .

[31]  R. Brent.  Table errata: Algorithms for minimization without derivatives (Prentice-Hall, Englewood Cliffs, N. J., 1973) , 1975 .

[32]  L. R. Parker,et al.  Comparison of simplex algorithms , 1985 .

[33]  P. Dupuis,et al.  On sampling controlled stochastic approximation , 1991 .