On the Identification Property of a Projected Gradient Method
Pasquale Luigi De Angelis, Gerardo Toraldo · SIAM Journal on Numerical Analysis · 1993
The authors study the convergence properties of a projected gradient algorithm for the general problem \[\min \{ f(x):x \in \Omega \} ,\] where $f:R^n \to R$ is a mapping continuously differentiable on a closed convex set $\Omega \subseteq R^n $. The algorithm, which requires only one projection per iteration, is a special version of the method of projection of the gradient by Demyanov and Rubinov [Approximate Methods in Optimization Problems, Elsevier, New York, 1970] where the step choice is made according to a scheme similar to the one used by Calamai and Moré [Math. Programming, 39 (1987), pp. 93–116]. The authors are mainly interested in analysing the identification property of the algorithm for the case where the set $\Omega $ is a polyhedron, that is, the ability to identify in a finite number of steps the face in which the final solution lies. The convergence results that are shown are very similar to those shown in [6] for the standard projected gradient method.