The armijo step rule adapted to gradient path algorithms

Abstract

In a previous work of the author, the convergence of the algorithm

$$x_{k + 1} = x^k (t_k ) = {\text{x}}_{\text{k}} - s_k (t_k )\nabla f({\text{x}}_{\text{k}} )k = 0,1,2...$$

for the minimization of f: IRⁿ → IR, was analyzed. Due to the nonlinearity of the matrix S_k the cost of evaluation of x^k(t) for several values of t can be expensive, hence the choice of the first trial α_k is critical.

In this work a procedure for the choice α_k is proposed. This choice is based on the form of the curve x^k(t), t≥0 and under the convergence hypothesis of the algorithm.