Otimização sem derivadas

Abstract

Trust-region methods are a class of iterative algorithms widely applied to nonlinear unconstrained optimization problems for which derivatives of the objective function are unavailable or inaccurate. One of the classical approaches involves the optimization of a polynomial model for the objective function, built at each iteration and based on a sample set.In a recent work, Scheinberg and Toint [SIAM Journal on Optimization, 20 (6) (2010), pp. 3512-3532 ] proved that, despite being essential for convergence results, the improvement of the geometry (poisedness) of the sample set might occur only in the Ąnal stage of the algorithm. Based on these ideas and incorporating them into a theoretical algorithm framework, the authors investigate analytically an interesting self-correcting geometry mechanism of the interpolating set, which becomes evident at unsuccessful iterations. Global convergence for the new algorithm is then proved as a consequence of this self-correcting property.In this work we study the positioning of the sample points within interpolation-based methods that rely on quadratic models and investigate the computational performance of the theoretical algorithm proposed by Scheinberg and Toint, whose parameters are based upon either choices of previous works or numerical experiments.