New class of limited - memory variationally - derived variable metric methods 1

We present a new family of limited-memory variationally-derived variable metric (VM) line search methods with quadratic termination property for unconstrained minimization. Starting with x0 ∈ RN , VM line search methods (see [6], [3]) generate iterations xk+1 ∈ RN by the process xk+1 = xk + sk, sk = tkdk, where the direction vectors dk ∈ RN are descent, i.e. g k dk < 0, k ≥ 0, and the stepsizes tk > 0 satisfy f(xk+1)− f(xk) ≤ ε1tkg k dk, g k+1dk ≥ ε2g k dk, (1) k ≥ 0, with 0 < ε1 < 1/2 and ε1 < ε2 < 1, where f is an objective function, gk = ∇f(xk). We denote yk = gk+1 − gk, k ≥ 0 and by ‖.‖F the Frobenius matrix norm. We describe a new family in Section 1 and in Section 2 a correction formula, which uses the previous vectors sk−1, yk−1. Numerical results are presented in Section 3.