Generalized Gradients and Applications

نویسندگان

  • FRANK H. CLARKE
  • F. H. CLARKE
چکیده

A theory of generalized gradients for a general class of functions is developed, as well as a corresponding theory of normals to arbitrary closed sets. It is shown how these concepts subsume the usual gradients and normals of smooth functions and manifolds, and the subdifferentials and normals of convex analysis. A theorem is proved concerning the differentiability properties of a function of the form max{g(x, u):u e if}. This result unifies and extends some theorems of Danskin and others. The results are then applied to obtain a characterization of flow-invariant sets which yields theorems of Bony and Brezis as corollaries. Introduction. Some of the most important recent advances in optimization have come about as a result of the systematic replacement of smoothness assumptions by convexity. This is exemplified by the work of Rockafellar [12], [13], which has extended the boundaries of treatable problems and has in addition led to new techniques for dealing with problems of a familiar nature. It is natural to ask whether analogous results can be proven without either smoothness or convexity. A general theory of necessary conditions for such problems has been obtained [3] and the results described in [4]. The conditions are expressed, in part, by means of generalized "gradients". They subsume the results of the smooth and convex cases and they yield, among other things, significant extensions of the Pontryagin maximum principle of optimal control theory. We describe in this article the generalized theory of gradients and some of its consequences. As mentioned, the principal application of this theory has been to variational problems, but the two main applications given here concern the differential properties of max functions (§2) and flow-invariant sets (§4). 1. Locally Lipschitz functions. Throughout this section, we shall be dealing with a function f:R" —► R which will be assumed locally Lipschitz. That is, for each bounded subset B of R" there exists a constant K such that \f(xx)-f(x2)\<K\xx-x2\ Received by the editors January 29, 1974. AMS (MOS) subject classifications (1970). Primary 26A27, 26A51 ; Secondary 34A99.

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

منابع مشابه

Maximal elements of $mathscr{F}_{C,theta}$-majorized mappings and applications to‎ ‎generalized games

In the paper‎, ‎some new existence theorems of maximal elements for‎ ‎$mathscr{F}_{C,theta}$-mappings and $mathscr{F}_{C,theta}$-majorized mappings are established‎. ‎As applications, ‎some new existence theorems of equilibrium points for‎ ‎one-person games‎, ‎qualitative games and generalized games are obtained‎. ‎Our results unify and generalize most known results‎ ‎in recent literature‎.

متن کامل

Generalized Regular Fuzzy Irresolute Mappings and Their Applications

In this paper, the notion of generalized regular fuzzy irresolute, generalized regular fuzzy irresolute open  and generalized regular fuzzy irresolute closed maps in fuzzy  topological spaces are introduced and studied. Moreover, some separation axioms and $r$-GRF-separated sets are established. Also, the relations between generalized regular fuzzy continuous maps and generalized regular fuzzy ...

متن کامل

On generalized Hermite-Hadamard inequality for generalized convex function

In this paper, a new inequality for generalized convex functions which is related to the left side of generalized Hermite-Hadamard type inequality is obtained. Some applications for some generalized special means are also given.

متن کامل

Coupled fixed points of generalized Kanann contraction and its applications

The purpose of this paper is to find of the theoretical results of fixed point theorems for a mixed monotone mapping in a metric space endowed with partially order by using the generalized Kanann type contractivity of assumption. Also, as an application, we prove the existence and uniqueness of solution for a first-order ordinary differential equation with periodic bou...

متن کامل

Applications of quadratic D-forms to generalized quadratic forms

In this paper, we study generalized quadratic forms over a division algebra with involution of the first kind in characteristic two. For this, we associate to every generalized quadratic from a quadratic form on its underlying vector space. It is shown that this form determines the isotropy behavior and the isometry class of generalized quadratic forms.

متن کامل

The Generalized Reparameterization Gradient

The reparameterization gradient has become a widely used method to obtain Monte Carlo gradients to optimize the variational objective. However, this technique does not easily apply to commonly used distributions such as beta or gamma without further approximations, and most practical applications of the reparameterization gradient fit Gaussian distributions. In this paper, we introduce the gene...

متن کامل

ذخیره در منابع من


  با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

عنوان ژورنال:

دوره   شماره 

صفحات  -

تاریخ انتشار 2010