A Derivative-free Trust-region Method for Optimization on the Ellipsoid

Abstract

Optimization methods play a crucial role in various fields and applications. In some optimization problems, the derivative information of the objective function is unavailable. Such black-box optimization problems need to be solved by derivative-free optimization methods. At the same time, optimization problems with ellipsoidal constraints are important and have widespread applications in various fields as well. Following the development of the late professor M. J. D. Powell’s efficient derivative-free trust-region optimization methods, this paper considers solving derivative-free optimization problems on the ellipsoid. Our new optimization solver EC-NEWUOA for problems on the ellipsoid in ℜ n is designed based on Powell’s derivative-free software NEWUOA for unconstrained optimization problems. The proposed techniques for our new method mainly include using the Courant penalty function, the augmented Lagrangian method, and the projection technique. Details about the method and theoretical analysis are included in this paper. We also compare our new method with other algorithms by solving test problems and then show the numerical advantages of our new method.