Regularization of degenerated equations and inequalities under explicit data parameterization

Gorbunov, V. K.

doi:10.1515/jiip.2001.9.6.575

Cited by 6 publications

(7 citation statements)

References 8 publications

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“…The probabilistic approach is approved mainly for financial markets (Lee et al, 2019), but statistics of consumer markets generally do not provide probabilistic characteristics for errors 28 . In this case, the methodology for ill-posed (irregular) problems, developed in natural science and engineering (Tikhonov and Arsenin, 1977;Gorbunov: 1991Gorbunov: , 2001a, can be used.…”

Section: Regularization Of Afriat's Inequalitiesmentioning

confidence: 99%

“…The regularization of the solution problem for an irregular inequality system (Gorbunov: 1999(Gorbunov: , 2001a) means, firstly, its approximation by a regular inequality system, and secondly, introducing a choice rule for the desired solution, which ensures its uniqueness and continuous dependence on the initial data. The method for regularization and solution of Afriat's systems presented further is the developing the relaxation-penalty method for degenerate inequalities and extremal problems (Gorbunov: 2001b(Gorbunov: , 2004(Gorbunov: , 2015.…”

Section: A Multi-valued Mapping A( )mentioning

confidence: 99%

“…The key problem here is solving the Afriat's inequalities, which determine utility values and Lagrange multipliers for statistical commodity quantities and expenditures. These inequalities may be unstable, and methods for solving them are based on special regularization methods (Gorbunov: 1991(Gorbunov: , 1999(Gorbunov: , 2001a(Gorbunov: , 2001b. Recent papers (Gorbunov and Lvov, 2019;Gorbunov et al, 2020) present and approve variative verification methods for choosing solutions to regularized Afriat's inequalities with different desired properties.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Market demand: a holistic theory and its verification

Gorbunov

2021

SSRN Journal

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Section: Regularization Of Afriat's Inequalitiesmentioning

confidence: 99%

Section: A Multi-valued Mapping A( )mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Market demand: a holistic theory and its verification

Gorbunov

2021

SSRN Journal

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“…Difficulties of numerical solution to the problem (1)-(3) appear when the Jacobi matrix of the finite subsystem (2) ∂h(x(t), u(t))/∂u (4) degenerates on the solution {x(t), u(t)}. There are some regular numerical methods for the initial value problem for the DAE (1) which have a finite index up to 3.…”

Section: Problem Statementmentioning

confidence: 99%

“…The PM handles this problem naturally and simply. In difficult unstable problems one can use some regularization of the initial problem [4].…”

Section: Introductionmentioning

confidence: 99%

The Parameterization Method in Singular Differential-Algebraic Equations

Gorbunov

Lutoshkin

2003

Lecture Notes in Computer Science

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The paper is devoted to the circumstantiation of the parameterization method for classical calculus of variation problems corresponding to the non-linear ODEs. The method is based on a finite parameterization of "control" functions (finitely entering in initial system) and on derivation of the problem functional with respect to control parameters. The first and the second derivatives are calculated with the help of adjoint vector and matrix impulses. The problems of arising degeneration of gradients and optimality conditions of the first order are overcome by using the Newton method. Results of the solution to degenerate DAEs, particularly with non-unique solutions, are presented.

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Variational splines for the numerical solution of singular differential equations

Gorbunov

Lutoshkin

Martynenko

2007

Proc Appl Math and Mech

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The numerical parametrization method (PM), originally created for optimal control problems, is specificated for classical calculus of variation problems that arise in connection with singular implicit (IDEs) and differential-algebraic equations (DAEs). The PM for IDEs is based on representation of the required solution as a spline with moving knots and on minimization of the discrepancy functional with respect to the spline parameters. Such splines are named variational splines. For DAEs only finite entering functions can be represented by splines, and the functional under minimization is the discrepancy of the algebraic subsystem. The first and the second derivatives of the functionals are calculated in two ways -for DAEs with the help of adjoint variables, and for IDE directly. The PM does not use the notion of differentiation index, and it is applicable to any singular equation having a solution.The last three decades mathematical modeling involves problems that are modeled as systems of implicit differential equationswhereSuch systems arise in designing electric circuits (microchips for computers), mechanics (cinematics equations), nuclear energy, and others [8]. Irresolvability of the system (1) with respect to the derivativesẋ (singularity) can be a consequence of the complexity of the corresponding algebraic and transcendental transformations, as well as a consequence of the degeneration of the Jacobi matrix ∂F (ẋ, x, t)/∂ẋ on the solution {x(t)}. The statement of initial or boundary value problems for such equations should be in accordance with the system (1). The IDEs (1) are also called 'unstructured differential-algebraic equations' [1].An important particular kind of IDEs is the so-called 'structured DAEs', or simply DAEṡwhereHere initial or boundary value problems should be in accordance with the finite subsystem, and difficulties of theory and numerical solution to the initial/boundary value problems appear when the Jacobi matrix of the finite subsystem of ∂g(x, u, t)/∂u degenerates on the solution {x(t)} or {x(t), u(t)}.It is known that singular differential equations has a set of pathologies with respect to regular ODEs. So, their solutions (if they exist) can depend non-continuously on input data (initial values and parameters of equations), and the set of solutions can be infinite-dimensional (see, for example, the recent investigation [2]). Correspondingly, in the common singular case, theory and numerical methods should be created in frame of the theory of ill-posed problems [4].Often the system IDE or DAE can be transformed to the regular normal (Cauchy) by a finite number of differentiations and by algebraic transformations. The minimal number of such differentiations is called the differentiation index (DI) [8]. In applications, DAEs with indices five and higher do appear, and there are DAEs which have variable degeneracy, and they are non-reducible to the normal form.For singular differential equations with DI up to three effective specialized step-type methods exist [8]. In view of the p...

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Regularization of degenerated equations and inequalities under explicit data parameterization

Cited by 6 publications

References 8 publications

Market demand: a holistic theory and its verification

Market demand: a holistic theory and its verification

The Parameterization Method in Singular Differential-Algebraic Equations

Variational splines for the numerical solution of singular differential equations

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