About the Lipschitz property of the metric projection in the Hilbert space

Balashov, M. V.; Golubev, Maxim O.

doi:10.1016/j.jmaa.2012.05.024

Cited by 15 publications

(7 citation statements)

References 11 publications

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“…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].…”

Section: The Gradient Projection Methodsmentioning

confidence: 99%

“…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.…”

Section: Gradient Methods For Problems With Strongly Convex Feasible Setmentioning

confidence: 99%

“…[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.…”

Section: Introductionmentioning

confidence: 99%

“…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).…”

Section: Introductionmentioning

confidence: 99%

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

Veliov

Vuong

2018

Appl Math Optim

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The paper presents new results about convergence of the gradient projection and the conditional gradient methods for abstract minimization problems on strongly convex sets. In particular, linear convergence is proved, although the objective functional does not need to be convex. Such problems arise, in particular, when a recently developed discretization technique is applied to optimal control problems which are affine with respect to the control. This discretization technique has the advantage to provide higher accuracy of discretization (compared with the known discretization schemes) and involves strongly convex constraints and possibly non-convex objective functional. The applicability of the abstract results is proved in the case of linear-quadratic affine optimal control problems. A numerical example is given, confirming the theoretical findings. Keywords Optimal control • Mathematical programming • Numerical methods • Gradient methods • Affine control systems • Bang-bang control Mathematics Subject Classification 49M25 • 90C25 • 90C48 • 49M37

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Section: The Gradient Projection Methodsmentioning

confidence: 99%

Section: Gradient Methods For Problems With Strongly Convex Feasible Setmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Veliov

Vuong

2018

Appl Math Optim

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“…In a similar way, one can prove the gradient projection algorithm for the problem of minimization of a function with Lipschitz continuous gradient on a strongly convex set. For a convex function it has been done in [1].…”

Section: Theorem 32mentioning

confidence: 99%

Maximization of a Function with Lipschitz Continuous Gradient

Balashov

2015

J Math Sci

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In the present paper, we consider (nonconvex in the general case) functions that have Lipschitz continuous gradient. We prove that the level sets of such functions are proximally smooth and obtain an estimate for the constant of proximal smoothness. We prove that the problem of maximization of such function on a strongly convex set has a unique solution if the radius of strong convexity of the set is sufficiently small. The projection algorithm (similar to the gradient projection algorithm for minimization of a convex function on a convex set) for solving the problem of maximization of such a function is proposed. The algorithm converges with the rate of geometric progression.

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Local Strong Convexity in Hilbert Space

Golubev

2020

Communications in Computer and Information Science

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About the Lipschitz property of the metric projection in the Hilbert space

Cited by 15 publications

References 11 publications

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

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

Maximization of a Function with Lipschitz Continuous Gradient

Local Strong Convexity in Hilbert Space

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