Approximation Methods in Optimization of Nonlinear Systems

Kogut, Peter I.; Kupenko, Olha P.

doi:10.1515/9783110668520

Cited by 12 publications

(10 citation statements)

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“…However, if the problem ( 16) admits a unique solution (u 0 i , p 0 i ) ∈ i , then (38) implies that this function is considered as a weak solution. On the other hand, if u * i ∈ i is some solution to (16), then setting in (34)…”

Section: Remarkmentioning

confidence: 99%

On a Variational Problem with a Nonstandard Growth Functional and Its Applications to Image Processing

et al. 2022

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We propose a new variational model in Sobolev–Orlicz spaces with non-standard growth conditions of the objective functional and discuss its applications to image processing. The characteristic feature of the proposed model is that the variable exponent, which is associated with non-standard growth, is unknown a priori and it depends on a particular function that belongs to the domain of objective functional. So, we deal with a constrained minimization problem that lives in variable Sobolev–Orlicz spaces. In view of this, we discuss the consistency of the proposed model, give the scheme for its regularization, derive the corresponding optimality system, and propose an iterative algorithm for practical implementations.

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Section: Remarkmentioning

confidence: 99%

On a Variational Problem with a Nonstandard Growth Functional and Its Applications to Image Processing

et al. 2022

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“…In particular, to specify the term ½y, y f , we have the following result (we refer to [20], Lemma 2.1) where this result was proven for a particular nonlinearity f ðyÞ = e y (see also [27,28,32] for the more general cases).…”

Section: Preliminariesmentioning

confidence: 99%

On Regularization of an Optimal Control Problem for Ill-Posed Nonlinear Elliptic Equations

Kogut

Kupenko

Manzo

2020

Abstract and Applied Analysis

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We discuss the existence issue to an optimal control problem for one class of nonlinear elliptic equations with an exponential type of nonlinearity. We deal with the control object when we cannot expect to have a solution of the corresponding boundary value problem in the standard functional space for all admissible controls. To overcome this difficulty, we make use of a variant of the classical Tikhonov regularization scheme. In particular, we eliminate the PDE constraints between control and state and allow such pairs run freely by introducing an additional variable which plays the role of “compensator” that appears in the original state equation. We show that this fictitious variable can be determined in a unique way. In order to provide an approximation of the original optimal control problem, we define a special family of regularized optimization problems. We show that each of these problems is consistent, well-posed, and their solutions allow to attain an optimal solution of the original problem as the parameter of regularization tends to zero. As a consequence, we prove the existence of optimal solutions to the original problem and propose a way for their approximation.

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“…Many interesting and well-suited approaches for non-convex optimization problems were proposed including augmented Lagrangian methods, penalty methods, alternative direction minimization, and others. In this section, by analogy with the recent results developed in [6] (see also [29][30][31][32]34]), we make use of the relaxation approach passing from the non-convex constrained set |C(x)| = 1 to the convex unit ball |C(x)| 1 in R M with further penalization of this constraint in the corresponding minimization cost functional. With that in mind, for any real number ε > 0 and any given B 0 ∈ BV (Ω\D) and u SAR ∈ L ∞ (Ω), we consider the convex functional…”

Section: On Relaxation Of the Chromaticity Recovery Problemmentioning

confidence: 99%

Variational Approach for the Reconstruction of Damaged Optical Satellite Images Through Their Co-Registration with Synthetic Aperture Radar

Kogut

Uvarov²

2020

JODEA

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Approximation Methods in Optimization of Nonlinear Systems

Cited by 12 publications

References 0 publications

On a Variational Problem with a Nonstandard Growth Functional and Its Applications to Image Processing

On a Variational Problem with a Nonstandard Growth Functional and Its Applications to Image Processing

On Regularization of an Optimal Control Problem for Ill-Posed Nonlinear Elliptic Equations

Variational Approach for the Reconstruction of Damaged Optical Satellite Images Through Their Co-Registration with Synthetic Aperture Radar

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