Numerical analysis of two Derivative Free Optimization Algorithms

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

Córdoba

Congreso; I Congreso de Matemática Aplicada, Computacional e Industrial (MACI'2007) - XVI Congreso sobre Métodos Numéricos y sus Aplicaciones (ENIEF'2007); 2007

Sección Argentina de SIAM - Asociación Argentina de Mecánica Computacional (AMCA)

Derivative free optimization methods  are designed for solving nonlinear optimization  problem where  the derivatives  of the objective function (and   the    constraints   in    the constrainedcase)    are   not available. Formally, we consider the problem of minimize a real valued function f subject to x  in the n-dimensional space,  where f is  a smooth nonlinear  objective function  bounded below. We  assume that the gradient of  f(x) and the Hessian matrix of f(x) can not  be computed for any x.Our purpose is compare numerically two methods for solving this problem: DFO and NEWUOA, both of  them are based   on    trust-region   strategy   and multivariate   polynomial interpolation. One of the main difference between them is DFO use Newton polynomials instead of Lagrange polynomials for the interpolation.  This is an  very important aspect  because the geometry of  the interpolation set and  the linear  algebra involved depends  on such  polynomials.  We consider that having  a deep knowledge of the  implementation details of thecodes will  enable us to be much more exacting  when proposing a new code  in  the  near  future.  Specifically, we  compare  the performance (eficiency  and robustness)  of the  two codes  by using a set  of test problems,  taking into  account the  many differentoptions of  the two codes.