Minimization of Functions Having Lipschitz Continuous First Partial Derivatives
Abstract¶
A general convergence theorem for the gradient method is proved under hypotheses which are given below. It is then shown that the usual steepest descent and modified steepest descent algorithms converge under the some hypotheses. The modified steepest descent algorithm allows for the possibility of variable stepsize.
Summary¶
Proves convergence of steepest descent for continuously differentiable functions with Lipschitz continuous gradients under level-set conditions, and introduces a modified steepest descent procedure with an adaptive variable step size. The sufficient-decrease step selection in the modified method became known as the Armijo rule, a foundational ingredient in backtracking line search and modern smooth optimization algorithms.
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