Regularized parametric Kuhn-Tucker theorem in a Hilbert space

Sumin, M. I.

doi:10.1134/s0965542511090156

Cited by 22 publications

(21 citation statements)

References 3 publications

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“…So-called regularized Kuhn-Tucker theorem in nondifferential sequential form was proved for Problem (P) with strongly convex objective functional and with parameters in constraints in [7]. This theorem is an assertion in terms of minimizing sequences (more precisely, in terms of minimizing approximate solutions in the sense of J. Warga) about possibility of approximation of the solution of the problem by minimizers of its regular Lagrangian without any regularity assumptions.…”

Section: Introductionmentioning

confidence: 95%

“…For convex programming problems with infinite-dimensional constraints, the nonvalidity of this classical theorem can be regarded as their characteristic property. In this case a simple meaningful example can be found in [7] (see Example 2 in [7]) also.…”

Section: Introductionmentioning

confidence: 99%

“…At the same time, it is well-known that optimization problems and their duals are typically unsolvable. The mentioned above examples from [7] show that the unsolvability of dual problems fully reveals itself even in very simple (at first glance) convex programming problems. On the one hand, optimality conditions for such problems cannot be written in terms of optimal elements.…”

Section: Introductionmentioning

confidence: 99%

“…If the regularity of the problem is not assumed, then, in general, the classical theorem does not hold even for the simplest finite-dimensional convex programming problems. In particular, the corresponding one-dimensional example can be found in [7] (see Example 1 in [7]). For convex programming problems with infinite-dimensional constraints, the nonvalidity of this classical theorem can be regarded as their characteristic property.…”

Section: Introductionmentioning

confidence: 99%

“…It contains its classical analogue and its proof is based on the dual regularization method (see, e.g. [4][5][6][7]). The specified above minimizers are constructively generated by means of this dual regularization method.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

On the Stable Sequential Kuhn-Tucker Theorem and Its Applications

Sumin¹

2012

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The Kuhn-Tucker theorem in nondifferential form is a well-known classical optimality criterion for a convex programming problems which is true for a convex problem in the case when a Kuhn-Tucker vector exists. It is natural to extract two features connected with the classical theorem. The first of them consists in its possible "impracticability" (the Kuhn-Tucker vector does not exist). The second feature is connected with possible "instability" of the classical theorem with respect to the errors in the initial data. The article deals with the so-called regularized Kuhn-Tucker theorem in nondifferential sequential form which contains its classical analogue. A proof of the regularized theorem is based on the dual regularization method. This theorem is an assertion without regularity assumptions in terms of minimizing sequences about possibility of approximation of the solution of the convex programming problem by minimizers of its regular Lagrangian, that are constructively generated by means of the dual regularization method. The major distinctive property of the regularized Kuhn-Tucker theorem consists that it is free from two lacks of its classical analogue specified above. The last circumstance opens possibilities of its application for solving various ill-posed problems of optimization, optimal control, inverse problems.

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Section: Introductionmentioning

confidence: 95%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

On the Stable Sequential Kuhn-Tucker Theorem and Its Applications

Sumin¹

2012

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Stable Sequential Pontryagin Maximum Principle as a Tool for Solving Unstable Optimal Control and Inverse Problems for Distributed Systems

Sumin

2016

IFIP Advances in Information and Communication Technology

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This article is devoted to studying dual regularization method as applied to parametric convex optimal control problem of controlled third boundary-value problem for parabolic equation with boundary control and with equality and inequality pointwise state constraints. These constraints are understood as ones in the Hilbert space L2. A major advantage of the constraints of the original problem which are understood as ones in L2 is that the resulting dual regularization algorithm is stable with respect to errors in the input data and leads to the construction of a minimizing approximate solution in the sense of J. Warga. Simultaneously, this dual algorithm yields the corresponding necessary and sufficient conditions for minimizing sequences, namely, the stable, with respect to perturbation of input data, sequential or, in other words, regularized Lagrange principle in nondifferential form and Pontryagin maximum principle for the original problem. Regardless of the fact that the stability or instability of the original optimal control problem, they stably generate a minimizing approximate solutions for it. For this reason, we can interpret these regularized Lagrange principle and Pontryagin maximum principle as tools for direct solving unstable optimal control problems and reducing to them unstable inverse problems.

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Regularized Lagrange Principle and Pontryagin Maximum Principle in Optimal Control and Inverse Problems

Sumin

2018

IFAC-PapersOnLine

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Regularized parametric Kuhn-Tucker theorem in a Hilbert space

Cited by 22 publications

References 3 publications

On the Stable Sequential Kuhn-Tucker Theorem and Its Applications

On the Stable Sequential Kuhn-Tucker Theorem and Its Applications

Stable Sequential Pontryagin Maximum Principle as a Tool for Solving Unstable Optimal Control and Inverse Problems for Distributed Systems

Regularized Lagrange Principle and Pontryagin Maximum Principle in Optimal Control and Inverse Problems

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