By Alexey Abramov

The current publication is the end result of efforts to introduce topological connectedness as one of many simple instruments for the learn of priceless stipulations for an extremum. it appears this monograph is the 1st booklet within the thought of maxima and minima the place topological connectedness is used so largely for this goal. Its software allows us to acquire new ends up in this sphere and to contemplate the classical effects from a nonstandard viewpoint. concerning the form of the current ebook it's going to be remarked that it's relatively uncomplicated. the writer has made consistent efforts to make the booklet as self-contained as attainable. definitely, familiarity with the elemental proof of topology, useful research, and the idea of optimization is thought. The e-book is written for utilized mathematicians and graduate scholars attracted to the speculation of optimization and its purposes. We current the synthesis of the well-known Dybovitskii'-Milyutin ap proach for the examine of valuable stipulations for an extremum, in keeping with sensible research, and topological equipment. This synthesis permits us to teach that during a few circumstances now we have the subsequent vital consequence: if the Euler equation has no non trivial answer at some extent of an extremum, then a few inclusion is legitimate for the functionals belonging to the twin area. This common result's acquired for an optimization challenge thought of in a lin ear topological area. We additionally convey an software of our end result to a couple difficulties of nonlinear programming and optimum control.

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20 it follows that int = Thus, M(h,U,EO) C R; and we obtain the equality M(h,U,Eo) n Si = 0. , the vector h is not tangential to the set Si at the point x*. Thus, we have hE CKi , where CKi is the complement of the cone Ki with respect to E. Hence CHi ~ CKi and whence it follows that K z ~ Hi. So we obtain the inverse inclusion. Thus, K i = Hi. D Next we consider more complicated cases. Let SI. SI R;. n-l where sets Si, i = m + 1, ... 18. Denote by K the cone of tangential directions to the set Q at the point x*.

23. Corollary. 17, then for any neighborhood O(x*) of the point x* the set Si n O(x*) satisfies this assumption, too. Moreover, the cones of possible directions for sets Si and S, n O( x*) are the same at the point X*. 18 simultaneously. 24. Example. , M1 = {(xl, x 2 ) : x2 = Xl sin :1' o} . xl < 28 1. 17. 18, then the set S satisfies this modified assumption. p( xl) on the negative abscissa half-axis. Let O(x*) = R2 and It is easy to see that sets RI and R 2 are nonempty and separated. 25. Proposition.

P( xl) on the negative abscissa half-axis. Let O(x*) = R2 and It is easy to see that sets RI and R 2 are nonempty and separated. 25. Proposition. 6) at the point x* have the empty intersection: s1 ](1, ](; n](; = 0. 9) Proof. Suppose on the contrary that n =je 0. Let us choose some vector h E Kr n Kl. Then there is a neighborhood U of the vector h such §1. 8) we have 29 Sl n Sr. But in this case, for example, the point x* + ~ Eoh is internal for the set Si, because this point belongs to sets SI and Sl together with the neighborhood x* + ~ EoU.