Subderivative

Subderivative

In mathematics, the concepts of subderivative, subgradient, and subdifferential arise in convex analysis, that is, in the study of convex functions, often in connection to convex optimization.

Let "f":"I"→R be a real-valued convex function defined on an open interval of the real line. Such a function may not necessarily be differentiable at all points, as for example, the absolute value, "f"("x")=|"x"|. However, as seen in the picture on the right (and which can be proved rigorously), for any "x"0 in the domain of the function one can draw a line which goes through the point ("x"0, "f"("x"0)) and which is everywhere either touching or below the graph of "f". The slope of such a line is called a "subderivative" (because the line is under the graph of "f").

Definition

Rigorously, a "subderivative" of a convex function "f":"I"→R at a point "x"0 in the open interval "I" is a real number "c" such that :f(x)-f(x_0)ge c(x-x_0)for all "x" in "I". One may show that the set of subderivatives at "x"0 is a nonempty closed interval ["a", "b"] , where "a" and "b" are the one-sided limits

:a=lim_{x o x_0^-}frac{f(x)-f(x_0)}{x-x_0}

:b=lim_{x o x_0^+}frac{f(x)-f(x_0)}{x-x_0}

which are guaranteed to exist and satisfy "a" ≤ "b".

The set ["a", "b"] of all subderivatives is called the subdifferential of the function "f" at "x"0.

Examples

Consider the function "f"("x")=|"x"| which is convex. Then, the subdifferential at the origin is the interval [−1, 1] . The subdifferential at any point "x"0<0 is the singleton set {−1}, while the subdifferential at any point "x"0>0 is the singleton {1}.

Properties

* A convex function "f":"I"→R is differentiable at "x"0 if and only if the subdifferential is made up of only one point, which is the derivative at "x"0.

* A point "x"0 is a global minimum of a convex function "f" if and only if zero is contained in the subdifferential, that is, in the figure above, one may draw a horizontal "subtangent line" to the graph of "f" at ("x"0, "f"("x"0)). This last property is a generalization of the fact that the derivative of a function differentiable at a local minimum is zero.

The subgradient

The concepts of subderivative and subdifferential can be generalized to functions of several variables. If "f":"U"→ R is a real-valued convex function defined on a convex open set in the Euclidean space R"n", a vector "v" in that space is called a subgradient at a point "x"0 in "U" if for any "x" in "U" one has:f(x)-f(x_0)ge vcdot (x-x_0)where the dot denotes the dot product. The set of all subgradients at "x"0 is called the subdifferential at "x"0 and is denoted &part;"f"("x"0). The subdifferential is always a nonempty convex compact set.

These concepts generalize further to convex functions "f":"U"→ R on a convex set in a locally convex space "V". A functional "v"&lowast; in the dual space V&lowast; is called "subgradient" at "x"0 in "U" if:f(x)-f(x_0)ge v^*(x-x_0).The set of all subgradients at "x"0 is called the subdifferential at "x"0 and is again denoted &part;"f"("x"0). The subdifferential is always a convex closed set. It can be an empty set; consider for example an unbounded operator, which is convex, but has no subgradient. If "f" is continuous, the subdifferential is nonempty.

ee also

* Weak derivative

References

* Jean-Baptiste Hiriart-Urruty, Claude Lemaréchal, "Fundamentals of Convex Analysis", Springer, 2001. ISBN 3-540-42205-6.


Wikimedia Foundation. 2010.

Игры ⚽ Нужна курсовая?

Look at other dictionaries:

  • Subderivative — Sub de*riv a*tive, n. A word derived from a derivative, and not directly from the root; as, friendliness is a subderivative, being derived from friendly , which is in turn a derivative from friend. [1913 Webster] …   The Collaborative International Dictionary of English

  • subderivative — “+ noun Etymology: sub + derivative : a word derived from a derivative friendliness is a subderivative from friendly which is derived from friend * * * /sub di riv euh tiv/, n. a word derived from a derivative. [SUB + DERIVATIVE] …   Useful english dictionary

  • subderivative — /sub di riv euh tiv/, n. a word derived from a derivative. [SUB + DERIVATIVE] * * * …   Universalium

  • subderivative — noun a) A word derived indirectly from a derivative of another b) The slope of a line that either touches, or is below the line of a convex function …   Wiktionary

  • subderivative — sub·derivative …   English syllables

  • Absolute value — For the philosophical term, see Value (ethics). For the Akrobatik album, see Absolute Value (album). In mathematics, the absolute value (or modulus) |a| of a real number a is the numerical value of a without regard to its sign. So, for example,… …   Wikipedia

  • Convex function — on an interval. A function (in black) is convex if and only i …   Wikipedia

  • List of numerical analysis topics — This is a list of numerical analysis topics, by Wikipedia page. Contents 1 General 2 Error 3 Elementary and special functions 4 Numerical linear algebra …   Wikipedia

  • Weak derivative — In mathematics, a weak derivative is a generalization of the concept of the derivative of a function ( strong derivative ) for functions not assumed differentiable, but only integrable, i.e. to lie in the Lebesgue space L^1( [a,b] ). See… …   Wikipedia

  • List of mathematics articles (S) — NOTOC S S duality S matrix S plane S transform S unit S.O.S. Mathematics SA subgroup Saccheri quadrilateral Sacks spiral Sacred geometry Saddle node bifurcation Saddle point Saddle surface Sadleirian Professor of Pure Mathematics Safe prime Safe… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”