Gradient Projection Algorithm for Strongly Convex Set

Abstract

В работе рассматривается стандартный метод проекции градиента в случае, когда множество является R-сильно выпуклым, а функция выпукла, дифференцируема и имеет липшицев градиент. Доказано, что при некоторых естественных дополнительных условиях метод сходится со скоростью геометрической прогрессии.Ключевые слова: гильбертово пространство, метод проекции градиента, метрическая проекция, R-сильно выпуклое множество.

Gradient Methods on Strongly Convex Feasible Sets and Optimal Control of Affine Systems

“…If f is only convex (but not necessarily strongly convex), the sequence {w k } converges weakly [20]. When K is strongly convex, the linear convergence of {w k } is obtained under additional conditions (too strong for our main application) in [5,6,15].…”

“…In contrast, in our results below the function f does not even need to be convex, while the set K is assumed strongly convex. Some convergence results for smooth convex functions f and strongly convex sets K are obtained in [6,15], but under suppositions that (apart from the convexity of f ) are not satisfied in our main motivation as described in the introduction (see Remark 2.3 below). The convergence results presented in this section are substantially stronger.…”

“…Using the contractivity of the projection onto strongly convex sets, Balashov and Golubev [6] and Golubev [15] obtained the linear convergence of the GPM for smooth, convex optimization problem with the following additional conditions:…”

“…[18,19] and the references therein), but results about linear convergence of the generated sequence of iterates seem to be available only for problems with strongly convex objective functions. Exceptions are the papers [6,15], where strong convexity is assumed for the feasible set instead of the objective function. However, as clarified in the end of Sect.…”

“…2.1 we prove linear convergence of the sequence of approximate solutions generated by the GPM, provided that the step sizes are appropriately chosen. Apart from the applicability for non-convex objective functionals, this result does not require the additional conditions in [6,15]. As usual, the "appropriate" choice of the step sizes is expressed by some constants related to the data of the problem, which are often not available (or very roughly estimated).…”