Active-set strategy in Powell's method for optimization without derivatives

MARÍA B. AROUXÉT; NÉLIDA ECHEBEST; ELVIO A. PILOTTA

MATEMATICA APLICADA E COMPUTACIONAL

SOC BRASILEIRA MATEMATICA APLICADA & COMPUTACIONAL

Año: 2011 vol. 30 p. 171 - 196

0101-8205

In this article we present an algorithm  for solving bound constrained optimization problems without derivatives based on Powell´s method for derivative-free optimization. First we consider the unconstrained optimization problem. At each iteration  a quadratic interpolation model of the objective function  is constructed around the current iterate and this model is minimized to obtain a new trial point. The whole process is embedded within a trust-region framework. Our algorithm uses infinity norm instead of the Euclidean norm and we solve a box constrained quadratic subproblem using an active-set strategy to explore faces of the box. Therefore, a bound constrained optimization algorithm is easily extended. We compare our implementation with NEWUOA and BOBYQA, Powell´s algorithms for unconstrained and bound constrained derivative free optimization respectively. Numerical experiments show that, in general, ouralgorithm require less functional evaluations than Powell´s algorithms.