Optimization methods, Ill-posed problems, Numerical methods for extremum problems, ordinary differential and integral equations, Mathematical economics. Parameterization method for optimal control problems, which is contained in a finite parameterization of a seeking control function and calculation of first and second derivatives on the parameters of the problem"s functionals by using of conjugate variables, is created. The normal spline-collocation method for solution of integral and differential equations, particularly degenerated ones, is created. Regularization methods for nonlinear equations, inequalities and extremum problems, which are based on an explicit parameterization of input data and on using of multi-valued mappings, are created. The most effective method of extended minimization could be assumed as an expansion on nonlinear problems the last Tikhonov's regularization concept (1980–1985), according to which for obtaining a stable approximation of a solution of an ill-posed problem one have to find an element of minimal completeness in the join of solution set of the family of problems which are equivalent to the initial one with respect to input data accuracy. For inverse problem of the consumer demand theory solution methods in the class of differentiable utility functions are created. Numerical Analysis, Mathematical Economics
Main publications:
Regularization of degenerated equations and inequalities under explicit data parameterization // Journal of Inv. and Ill-Posed Problems, 2001, v. 9, no. 6. \begin{thebibliography}{9}
\Bibitem{1}
\by Gorbunov V.K.
\paper The parametrization method for optimal control problems
\paperinfo A method of numerical solution of optimal control problems is proposed, which consists in a finite parameterization of the required optimal control. The calculation of derivatives (the first and second order) of the problem functional and of constrains with respect to control parameters is fulfilled with the use of conjugated systems (first and second order).
\jour Computational Mathemathics and Mathemathical Physics
\yr 1979
\vol 19
\issue 2
\pages 18-30
\Bibitem{2}
\by Gorbunov V.K.
\paper The method of normal spline-collocation
\paperinfo A method of numerical solution of integro-differential equations (including initial and boundary value problems of ODEs and integral equations of the first kind) is proposed. The method consists in transfer from the equation to a collocation system with arbitrary nodes, and a statement of the problem of minimizing some Sobolev-Hilbert norm of solutions of the collocation system. The arisen point and integral (for the integral component of the equation) functionals are transformed to canonical form, that is to the scalar product. The method is applicable (converges) for equations with arbitrary degenerate principal part.
\jour Computational Mathemathics and Mathemathical Physics
\yr 1989
\vol 29
\issue 2
\pages 145-154
\Bibitem{3}
\by Gorbunov V.K., Sviridov V.Yu/
\paper A method of normal splines for linear DAEs on the number semi-axes
\paperinfo The method of normal spline-collocation, applicable to a wide class of optimal control problems, of ordinary linear singular differential and integral equations, is specified for the boundary value problems for DAEs of second order on the number semi-axes.
\jour Applied Numerical Mathematics
\yr 2009
\vol 59
\issue 3-4
\pages 655-670
\Bibitem{4}
\by Gorbunov V.K., I.V. Lutoshkin,
Yu.V. Martynenko
\paper A parametrization method for numerical solution of singular differential equations
\paperinfo The paper extends the numerical parametrization method, originally created for optimal control problems, for classical calculus of variational problems that arise in connection wuth singular implicite and DAEs in frame or their regularization.
\jour Applied Numerical Mathematics
\yr 2009
\vol 59
\issue 3-4
\pages 639-655