National Institute for Research and Development in Informatics – ICI Bucharest
8-10 Mareșal Alexandru Averescu Av., 011455, Bucharest, România
Abstract: A new quasi-Newton method with a diagonal updating matrix is suggested, where the diagonal elements are determined by minimizing the measure function of Byrd and Nocedal subject to the weak secant equation of Dennis and Wolkowicz. The Lagrange multiplier of this minimization problem is computed by using an adaptive procedure based on the conjugacy condition. The convergence of the algorithm is proved for twice differentiable, convex and bounded below functions using only the trace and the determinant. Using a set of 80 unconstrained optimization test problems and some applications from the MINPACK-2 collection we have the computational evidence that the algorithm is more efficient and more robust than the steepest descent, than the Barzilai and Borwein algorithm, than the Cauchy algorithm with Oren and Luenberger scaling and than the classical BFGS algorithms with the Wolfe line search conditions.
Keywords: Unconstrained optimization. Weak secant. Diagonal quasi-Newton update. Measure function of Byrd and Nocedal. Numerical comparisons.