New gradient methods with 3 dimensional quadratic termination

Abstract

A new stepsize for gradient method is proposed. Combining it with the exact line search stepsizes, the gradient method achieves the optimal solution in 5 steps for 3 dimensional quadratic function minimization problem. The new stepsize is plugged in the cyclic stepsize update strategy, and a new gradient method is proposed. By applying the quadratic interpolation for Cauchy approximation, the proposed gradient method is extended to solve general unconstrained problem. With the improved GLL line search, the global convergence of the proposed method is proved. Furthermore, its sublinear convergence rate for convex problems and R-linear convergence rate for problems with quadratic functional growth property are analyzed. Numerical results show that our proposed algorithm enjoys good performances in terms of computational cost, and line search requires very few trial stepsizes.