Abnormal problems with a nonclosed image

[No abstract available

Inverse function theorem and conditions of extremum for abnormal problems with non-closed range

The following two classical problems are considered: the existence and the estimate of a solution of an equation defined by a map F in the neighbourhood of a point x*; necessary conditions for an extremum at x* of a smooth function under equality-type constraints defined in terms of a non-linear map F. If the range of the first derivative of F at x* is not closed, then one cannot use classical methods of analysis based on inverse function theorems and Lagrange's principle. The results on these problems obtained in this paper are of interest in the case when the range of the first derivative of F at x* is non-closed; these are a further development of classical results extending them to abnormal problems with nonclosed range

Inverse function theorem and conditions of extremum for abnormal problems with non-closed range

The following two classical problems are considered: the existence and the estimate of a solution of an equation defined by a map F in the neighbourhood of a point x*; necessary conditions for an extremum at x* of a smooth function under equality-type constraints defined in terms of a non-linear map F. If the range of the first derivative of F at x* is not closed, then one cannot use classical methods of analysis based on inverse function theorems and Lagrange's principle. The results on these problems obtained in this paper are of interest in the case when the range of the first derivative of F at x* is non-closed; these are a further development of classical results extending them to abnormal problems with nonclosed range

Abnormal problems with a nonclosed image

[No abstract available

Necessary conditions for an extremum in a mathematical programming problem

For minimization problems with equality and inequality constraints, first-and second-order necessary conditions for a local extremum are presented. These conditions apply when the constraints do not satisfy the traditional regularity assumptions. The approach is based on the concept of 2-regularity; it unites and generalizes the authors' previous studies based on this concept. © Nauka/Interperiodica 2007

On second-order necessary optimality conditions in finite-dimensional abnormal optimization problems

[No abstract available

Directional metric regularity of mappings and stability theorems

A stability theorem, based on the concept of directional matric regularity of mappings is described. Robinson's stability theorem can be used to derive results on the quantitative stability of the feasible set which play a central role in sensitivity analysis for optimization problems. The Robinson regularity condition, if violated, the underlying smooth mapping is not metrically regular. A mapping is said to be regular at a point in the direction where cone denotes the conical hull of a set. In the context of optimization problems, a condition is known as Gollan's regularity condition and it is extended to the general case and in parametric optimization, the condition is known as the directional regularity condition. An analysis performs along such feasible arcs yields the most accurate known quantitative sensitivity results in the case when the solution is Lipschitz stable

Covering mappings and their applications to differential equations unsolved for the derivative

We continue to study the properties of covering mappings of metric spaces and present their applications to differential equations. To extend the applications of covering mappings, we introduce the notion of conditionally covering mapping. We prove that the solvability and the estimates for solutions of equations with conditionally covering mappings are preserved under small Lipschitz perturbations. These assertions are used in the solvability analysis of differential equations unsolved for the derivative. © 2009 Pleiades Publishing, Ltd

Directional regularity and metric regularity

For general constraint systems in Banach spaces, we present the directional stability theorem based on the appropriate generalization of the directional regularity condition, suggested earlier in [A. V. Arutyunov and A. F. Izmailov, Math. Oper. Res., 31 (2006), pp. 526-543]. This theorem contains Robinson's stability theorem but does not reduce to it. Furthermore, we develop the related concept of directional metric regularity which is stable subject to small Lipschitzian perturbations of the constraint mapping, and which is equivalent to directional regularity for sufficiently smooth mappings. Finally, we discuss some applications in sensitivity theory. © 2007 Society for Industrial and Applied Mathematics

An investigation of smooth maps in a neighbourhood of an abnormal point

We study the solubility of systems of non-linear equations in a neighbourhood of an abnormal point and prove inverse function theorems that guarantee the existence of solutions satisfying a linear-root estimate in the neighbourhood of the abnormal point. We also address the related issue of necessary conditions for an extremum at an abnormal point in a finite-dimensional constrained problem. We obtain second-order necessary optimality conditions that improve the known results. © 2014 Russian Academy of Sciences (DoM), London Mathematical Society, Turpion Ltd