Accuracy of Enriched Multipoint Cubic Approximations for Large-Scale Optimization
Ronald Roberts, Robert A. Canfield · 2010
Previous work demonstrated an optimization method using an Enriched Multipoint Cubic Approximation in place of nonlinear objective and constraint functions. The approximation technique creates a cubic function based on a reduced design space of previous design points. The reduced space is enriched using available gradient information at previous design points. The Enriched Multipoint Cubic Approximation is shown to accurately reproduce the function and gradient values at each previous design point used to construct the cubic function. By optimizing the cubic approximation between each function evaluation, this method may reduce the number of exact objective and constraint function calculations required to achieve an optimum solution. In some cases, the approximation requires more iterations to achieve an optimum solution. The accuracy of the cubic approximation is examined for a variety of nonlinear functions and a large number of design variables. The approximate functions are compared to actual function values within the design space. One interesting feature of the Enriched Multipoint Cubic Approximation is that it exhibits a very high accuracy along a line connecting the expansion point and any design point used to construct the approximation. It is proposed that a more thorough understanding of the approximation and its’properties will enable the development of a more e¢ cient optimization method.