New Direct Numerical Methods for Some Multidimensional Problems of the Calculus of Variations

Долгополик, М. В.

doi:10.1080/01630563.2017.1378231

Cited by 2 publications

(8 citation statements)

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“…By the Fubini theorem the function Av is correctly defined and A is a continuous linear operator mapping L 2 (Ω) to L 2 (Ω). The following result holds true (see [12,Thm. 3]).…”

Section: P and Sobolev Spacesmentioning

confidence: 70%

“…We will also use a Sobolev-like function space introduced by the author in [12]. Let us recall the definition of this space.…”

Section: P and Sobolev Spacesmentioning

confidence: 99%

“…Suppose by contradiction that the vectors ( 14) are linearly dependent. Then bearing in mind (12) one gets that there exist real numbers α j , j ∈ {1, . .…”

Section: Arguing In a Similar Way One Can Readily Verify Thatmentioning

confidence: 99%

“…In order to compute the gradients of the functionals I i , we will utilise an extension of the "transition into the space of derivatives" technique to the multidimensional case, developed by the author in [12]. To this end, suppose that all KKT points (u * , λ * ) of problem (15) one has u * ∈ u + M W n,2 0 (Ω; R d ) (see Subsection 1.2).…”

Section: Isoperimetric Problemsmentioning

confidence: 99%

“…Applying (17) and integrating by parts (see [12] for more details) one obtains that the functionals I i , i ∈ {0, 1}, are continuously Fréchet differentiable and their Fréchet derivatives have the form…”

Section: Isoperimetric Problemsmentioning

confidence: 99%

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Exact augmented Lagrangians for constrained optimization problems in Hilbert spaces I: theory

Долгополик

2022

Optimization

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This two-part study is devoted to the analysis of the so-called exact augmented Lagrangians, introduced by Di Pillo and Grippo for finite dimensional optimization problems, in the case of optimization problems in Hilbert spaces. In the second part of our study we present applications of the general theory of exact augmented Lagrangians to several constrained variational problems and optimal control problems, including variational problems with additional constraints at the boundary, isoperimetric problems, problems with nonholonomic equality constraints (PDE constraints), and optimal control problems for linear evolution equations. We provide sufficient conditions for augmented Lagrangians for these problems to be globally/completely exact, that is, conditions under which a constrained variational problem/optimal control problem becomes equivalent to the problem of unconstrained minimization of the corresponding exact augmented Lagrangian in primal and dual variables simultaneously. PreliminariesLet us introduce notation and recall some auxiliary definitions from the first part of our study [14] and the theory of Sobolev spaces that will be utilised throughout this article.

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“…By the Fubini theorem the function Av is correctly defined and A is a continuous linear operator mapping L 2 (Ω) to L 2 (Ω). The following result holds true (see [12,Thm. 3]).…”

Section: P and Sobolev Spacesmentioning

confidence: 70%

“…We will also use a Sobolev-like function space introduced by the author in [12]. Let us recall the definition of this space.…”

Section: P and Sobolev Spacesmentioning

confidence: 99%

“…Suppose by contradiction that the vectors ( 14) are linearly dependent. Then bearing in mind (12) one gets that there exist real numbers α j , j ∈ {1, . .…”

Section: Arguing In a Similar Way One Can Readily Verify Thatmentioning

confidence: 99%

Section: Isoperimetric Problemsmentioning

confidence: 99%

Section: Isoperimetric Problemsmentioning

confidence: 99%

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