Convergence of the Gradient Projection Method for Generalized Convex Minimization

This paper develops convergence theory of the gradient projection method by Calamai and Moré (Math. Programming, vol. 39, 93–116, 1987) which, for minimizing a continuously differentiable optimization problem min{f(x) : x ε Ω} where Ω is a nonempty closed convex set, generates a sequence xk+1 = P(xk − αk ∇ f(xk)) where the stepsize αk > 0 is chosen suitably. It is shown that, when f(x) is a pseudo-convex (quasi-convex) function, this method has strong convergence results: either xk → x* and x* is a minimizer (stationary point); or ‖xk‖ → ∞ arg min{f(x) : x ε Ω} = ∅, and f(xk) ↓ inf{f(x) : x ε Ω}.