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P-order necessary and sufficient conditions for optimality in singular calculus of variations

100%

Prusińska A. , Tret'yakov A.

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

2010

tom 30

nr 2

269-279

This paper is devoted to singular calculus of variations problems with constraint functional not regular at the solution point in the sense that the first derivative is not surjective. In the first part of the paper we pursue an approach based on the constructions of the p-regularity theory. For p-regular calculus of variations problem we formulate and prove necessary and sufficient conditions for optimality in singular case and illustrate our results by classical example of calculus of variations problem.

On existence of solutions to degenerate nonlinear optimization problems

88%

Prusińska A. , Tret'yakov A.

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

2007

tom 27

nr 1

151-164

We investigate the existence of the solution to the following problem min φ(x) subject to G(x)=0, where φ: X → ℝ, G: X → Y and X,Y are Banach spaces. The question of existence is considered in a neighborhood of such point x₀ that the Hessian of the Lagrange function is degenerate. There was obtained an approximation for the distance of solution x* to the initial point x₀.

Generalization of p-regularity notion and tangent cone description in the singular case

75%

Grzegorczyk W. , Medak B. , Tret’yakov A. A.

Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica

2012

tom 66

nr 2

The theory of p-regularity has approximately twenty-five years’ history and many results have been obtained up to now. The main result of this theory is description of tangent cone to zero set in singular case. However there are numerous nonlinear objects for which the p-regularity condition fails, especially for p > 2. In this paper we generalize the p-regularity notion as a starting point for more detailed consideration based on different p-factor operators constructions.

The p-Factor Method for Nonlinear Optimization

75%

Szczepanik E. , Prusińska A. , Tret’yakov A.

Schedae Informaticae

2012

tom Vol. 21

141--157

We present the main concept and results of the p-regularity theory (also known as p-factor analysis of nonlinear mappings) applied to nonlinear optimization problems. The approach is based on the construction of p-factor operator. The main result of this theory gives a detailed description of the structure of the zero set of irregular nonlinear mappings. Applications include a new numerical method for solving nonlinear optimization problems and p- order necessary and sufficient optimality conditions. We substantiate the rate of convergence of p-factor method.