On the convergence properties of the projected gradient method for convex optimization

Abstract

Abstract. When applied to an unconstrained minimization problem with a convex objective, the steepest descent method has stronger convergence properties than in the noncovex case: the whole sequence converges to an optimal solution under the only hypothesis of existence of minimizers (i.e. without assuming e.g. boundedness of the level sets). In this paper we look at the projected gradient method for constrained convex minimization. Convergence of the whole sequence to a minimizer assuming only existence of solutions has also been already established for the variant in which the stepsizes are exogenously given and square summable. In this paper, we prove the result for the more standard (and also more efficient) variant, namely the one in which the stepsizes are determined through an Armijo search. 90C25, 90C30. Mathematical subject classification:

Finite Termination of Inexact Proximal Point Algorithms in Hilbert Spaces

“…The projected gradient method (cf. [39][40][41] and the references therein) for solving (52) is defined as follows:…”

An explicit algorithm for monotone variational inequalities

“…Next we discuss the extent to which vectors y k and x kþ1 satisfying (12)- (14) can be determined through an explicit procedure.…”

“…P C (x) ¼ argmin y2C kx À yk. See [2,3,14] for convergence properties of this method for the case in which f is convex, which are related to results in this article. An immediate extension of the method (1)- (2) to VIP(T, C ) for the case in which T is point-to-set is the iterative procedure given by…”

A Strongly Convergent Method for Nonsmooth Convex Minimization in Hilbert Spaces

“…For differentiable functions the projected subgradient method coincides with the projected gradient method and different variants rules are used to choose the stepsizes in order to ensure functional decrease at each iteration; see [13]. Its extension to the generalized convex case is studied in [7,14,21].…”