Gâteaux differential

The Gâteaux differential , named after René Gâteaux (1889–1914), represents a generalization of the common notion of differentiation in that it defines the directional derivation also in infinite-dimensional spaces. Usually one has an open set for a function that is differentiable at that point as the definition of the partial derivative ${\displaystyle f\colon G\to \mathbb {R} ,\ G\subset \mathbb {R} ^{n}}$ ${\displaystyle x_{0}\in G}$

${\displaystyle {\frac {\partial f(x_{0})}{\partial x_{i}}}=\lim _{h\to 0}{\frac {f(x_{0,1},x_{0,2},\ldots ,x_{0,i}+h,\ldots ,x_{0,n})-f(x_{0})}{h}},\ \ (i={1,2,...,n})}$.

In particular, it results for the known differential ${\displaystyle n=1}$

${\displaystyle {\frac {\mathrm {d} f(x_{0})}{\mathrm {d} x}}=\lim _{h\to 0}{\frac {f(x_{0}+h)-f(x_{0})}{h}}}$.

The Gâteaux differential generalizes these concepts to infinite-dimensional vector spaces.

Definitions

Weierstrasse's decomposition formula

Be with open and standardized spaces. Then in Gâteaux is called-differentiable if the Weierstrasse decomposition formula holds, i.e. if a linear function exists such that ${\displaystyle f\colon D\subset X\to Y}$ ${\displaystyle D}$ ${\displaystyle X,Y}$ ${\displaystyle f}$ ${\displaystyle x_{0}\in D}$ ${\displaystyle A\in L(X,Y)}$

${\displaystyle \lim _{t\rightarrow 0}{\frac {1}{t}}[f(x_{0}+th)-f(x_{0})-tAh]=0}$

for everyone with . This is equivalent to: ${\displaystyle h\in X}$ ${\displaystyle \lVert h\rVert =1}$

${\displaystyle f(x_{0}+th)-f(x_{0})=tAh+o(|t|)}$

Then it is called the Gâteaux derivation of in the point . ${\displaystyle A=:f'(x_{0})}$ ${\displaystyle f}$ ${\displaystyle x_{0}}$

1. variation; Variation derivative

Let the following situation be given for the Gâteaux differential: Let it be, as usual, a functional defined in; be a linear normed space (that is, a vector space , provided with a norm ) or a more general topological vector space with prerequisites, about which one has to think more closely in the concrete application; further be and . Then the Gâteaux differential is at the point in direction , if it exists, defined by the following derivative : ${\displaystyle f\colon D(f)\to \mathbb {R} }$ ${\displaystyle D(f)\subseteq \Omega }$ ${\displaystyle \Omega }$ ${\displaystyle \|\cdot \|}$ ${\displaystyle x_{0}\in D(f)}$ ${\displaystyle v\in \Omega }$ ${\displaystyle x_{0}}$ ${\displaystyle v}$ ${\displaystyle \epsilon }$

${\displaystyle \delta f(x_{0},v)=\lim _{\epsilon \to 0}{\frac {f(x_{0}+\epsilon \cdot v)-f(x_{0})}{\epsilon }}=\left.{\frac {\mathrm {d} f(x_{0}+\epsilon \cdot v)}{\mathrm {d} \epsilon }}\right|_{\epsilon =0}}$

or for by ${\displaystyle x_{1}\in D(f)}$

${\displaystyle \delta f(x_{0},x_{1}-x_{0})=\lim _{\epsilon \to 0}{\frac {f(x_{0}+\epsilon \cdot (x_{1}-x_{0}))-f(x_{0})}{\epsilon }}\,.\,}$

Note here , and also in it, but . ${\displaystyle x_{0}\in D(f)}$ ${\displaystyle v\in \Omega }$ ${\displaystyle x_{1}-x_{0}}$ ${\displaystyle \epsilon \in \mathbb {R} }$

The Gâteaux derivation according to is a functional with regard to size , which is also referred to as the 1st variation of at this point . ${\displaystyle \epsilon }$ ${\displaystyle h:=x_{1}-x_{0}}$ ${\displaystyle f}$ ${\displaystyle x_{0}}$

Another possibility is to use more general topological vector spaces with a corresponding concept of convergence instead of normalized vector spaces . In physics books in particular, functionals are usually denoted by letters , and instead of size one usually writes with distribution-valued quantities. Instead of the derivation , the so-called variation derivation is introduced in an additional step , which is closely related to the Gâteaux derivation. ${\displaystyle I}$ ${\displaystyle h:=x_{1}-x_{0}}$ ${\displaystyle \delta q(x)}$ ${\displaystyle {\tfrac {dI(x+\epsilon \cdot h)}{d\epsilon }}_{\,|\epsilon =0}}$

example

For

${\displaystyle f(\epsilon ):=\int \,{\rm {d}}t\,{\mathcal {L}}\left(t,q(t)+\epsilon \cdot \delta q(t),{\dot {q}}(t)+\epsilon \cdot {\frac {{\rm {d}}(\delta q(t))}{{\rm {d}}t}}\right)}$

after a partial integration with a vanishing, fully integrated part, a result of the form with the derivative of the variation is obtained ${\displaystyle \textstyle {\frac {{\rm {d}}f}{{\rm {d}}\epsilon }}(\epsilon \to 0)\,=\,\int \,{\rm {d}}t\,{\frac {\delta {\mathcal {L}}}{\delta q(t)}}\cdot \delta q(t)}$

${\displaystyle {\frac {\delta {\mathcal {L}}}{\delta q(t)}}\equiv {\frac {\partial {\mathcal {L}}}{\partial q(t)}}-{\frac {\rm {d}}{{\rm {d}}t}}{\frac {\partial {\mathcal {L}}}{\partial {\dot {q}}(t)}}\,.}$

(The variation derivative "at point q (t)" for continuous variables is therefore the generalization of the partial derivative of a function from n variables, for example for the fictitious case . Similar to the total differential of a function of n variables in the fictitious case , so here , too , the total differential of the functional has an invariant meaning. Further details in the chapter Lagrange formalism .) ${\displaystyle {\tfrac {\partial {\mathcal {L}}}{\partial x_{i}}}}$ ${\displaystyle {\mathcal {L}}={\mathcal {L}}(x_{1},...,x_{n})}$ ${\displaystyle \delta f}$

In the following, for the sake of simplicity, the vectors are not identified by "bold" letters.

2nd variation

${\displaystyle \delta ^{2}f(x_{0},v)=\left.{\frac {\mathrm {d} ^{2}f(x_{0}+\epsilon \cdot v)}{\mathrm {d} \epsilon ^{2}}}\right|_{\epsilon =0}}$

Half-sided differential and directional derivative

Under the same conditions as above, the one-sided Gâteaux differential is through

${\displaystyle \delta _{+}f(x_{0},v)=\lim _{\epsilon \to 0^{+}}{\frac {f(x_{0}+\epsilon \cdot v)-f(x_{0})}{\epsilon }}}$

respectively through

${\displaystyle \delta _{-}f(x_{0},v)=\lim _{\epsilon \to 0^{-}}{\frac {f(x_{0}+\epsilon \cdot v)-f(x_{0})}{\epsilon }}}$

Are defined. The one-sided Gâteaux differential is also called the directional differential of at the point . For the direction belonging to the vector , in the case of “continuous variables”, the one-sided Gâteaux differential (more precisely: the associated variation derivative) just generalizes the directional derivative from in the direction at the point . ${\displaystyle f}$ ${\displaystyle x_{0}}$ ${\displaystyle v}$ ${\displaystyle f}$ ${\displaystyle v}$ ${\displaystyle x_{0}}$

Gâteaux derivative

If an in continuous linear functional (i.e. the function mediated by is homogeneous, additive and continuous in the argument ), then it is called Gâteaux-derivative in that place and Gâteaux-differentiable in . ${\displaystyle \delta f(x_{0},v)}$ ${\displaystyle v}$ ${\displaystyle v\mapsto \delta f(x_{0},v)}$ ${\displaystyle v}$ ${\displaystyle f'(x_{0})}$ ${\displaystyle x_{0}}$ ${\displaystyle f}$ ${\displaystyle x_{0}}$

Properties of the 1st variation

• The Gâteaux differential is homogeneous, which means for everyone . The same applies to the one-sided Gâteaux differential.
  ${\displaystyle \delta f(x_{0},k\cdot v)=k\cdot \delta f(x_{0},v)}$
  ${\displaystyle k\in \mathbb {R} }$

• The Gâteaux differential is not linear. In the general case, for an example that the Gâteaux differential is not linear, consider for and , where , then is . The function is not linear. It applies, for example .
  ${\displaystyle \delta f(x_{0},v)+\delta f(x_{0},w)\neq \delta f(x_{0},v+w).}$
  ${\displaystyle f(x)={\tfrac {x_{1}^{2}x_{2}}{x_{1}^{2}+x_{2}^{2}}}}$ ${\displaystyle x\neq 0}$ ${\displaystyle f(0)=0}$ ${\displaystyle x=(x_{1},x_{2})}$
  ${\displaystyle \delta f(0,v)=\lim _{\epsilon \to 0}{\frac {f(0+\epsilon \cdot v)-f(0)}{\epsilon }}=\lim _{\epsilon \to 0}{\frac {f(\epsilon \cdot v)}{\epsilon }}=\lim _{\epsilon \to 0}{\frac {\epsilon ^{2}v_{1}^{2}\cdot \epsilon v_{2}}{(\epsilon ^{2}v_{1}^{2}+\epsilon ^{2}v_{2}^{2})\epsilon }}={\frac {v_{1}^{2}v_{2}}{v_{1}^{2}+v_{2}^{2}}}=f(v)}$
  ${\displaystyle f}$ ${\displaystyle \delta f(0,(0,1))+\delta f(0,(1,0))=0+0\neq {\tfrac {1}{2}}=\delta f(0,(1,1))}$

Examples

1. ${\displaystyle f(x_{1},x_{2})=1}$if , or otherwise . ${\displaystyle x_{2}=x_{1}^{2}}$ ${\displaystyle x_{1}\neq 0}$ ${\displaystyle 0}$ ${\displaystyle \delta f((0,0),v)=\lim _{t\to 0}{\frac {0-0}{t}}=0}$
2. ${\displaystyle f(x)=|x|,\ x\in \mathbb {R} ^{n}}$ ${\displaystyle \delta _{+}f(0,v)=\lim _{\epsilon \to 0^{+}}{\frac {|0+\epsilon \cdot v|-0}{\epsilon }}=|v|}$
3. ${\displaystyle f(x_{1},x_{2})=x_{1}^{2}\left(1+{\frac {1}{x_{2}}}\right)}$for and for , ${\displaystyle x_{2}\neq 0}$ ${\displaystyle -{\frac {x_{1}^{2}}{x_{2}^{2}}}}$ ${\displaystyle x_{2}=0}$ ${\displaystyle \nabla f(x_{1},x_{2})=\left(2\cdot x_{1}\cdot \left(1+{\frac {1}{x_{2}}}\right),-{\frac {x_{1}^{2}}{x_{2}^{2}}}\right)^{T}}$

${\displaystyle \delta f((0,0),v)=\lim _{\epsilon \to 0}{\frac {(\epsilon \cdot v_{1})^{2}\cdot \left(1+{\frac {1}{\epsilon \cdot v_{2}}}\right)}{\epsilon }}={\frac {v_{1}^{2}}{v_{2}}}}$ (where ) ${\displaystyle v=(v_{1},v_{2})^{T}}$

Applications

Like the usual derivative , the Gâteaux differential is useful for determining extrema and therefore in optimization . Be open, linear normalized space, (the interior of the set ), and the open ball around with radius . Necessary optimality condition: Let be a local minimum of on , then if the one-sided Gâteaux differential in exists. Sufficient optimality condition: have in a 2nd variation and . If and for one and , then there is a strict local minimum of on . ${\displaystyle f\colon X\to \mathbb {R} ,\ X\subset D(f)\subset \Omega }$ ${\displaystyle \Omega }$ ${\displaystyle x_{0}\in \operatorname {int} (X)}$ ${\displaystyle X}$ ${\displaystyle \operatorname {int} (X)\neq \emptyset }$ ${\displaystyle B_{\varepsilon }(x_{0})}$ ${\displaystyle x_{0}}$ ${\displaystyle \varepsilon }$ ${\displaystyle x_{0}}$ ${\displaystyle f}$ ${\displaystyle X}$ ${\displaystyle \delta _{+}f(x_{0},v)\geq 0\ \forall v\in \Omega }$ ${\displaystyle x_{0}}$ ${\displaystyle f}$ ${\displaystyle B_{\varepsilon }(x_{0})}$ ${\displaystyle \forall v\in \Omega }$ ${\displaystyle \forall x\in B_{\varepsilon }(x_{0})}$ ${\displaystyle \delta f(x_{0},v)=0\ \forall v\in \Omega }$ ${\displaystyle c>0}$ ${\displaystyle \delta ^{2}f(x_{0},v)\geq c\cdot \|v\|^{2}\ \forall v\in \Omega }$ ${\displaystyle \forall x\in B_{\varepsilon }(x_{0})}$ ${\displaystyle x_{0}}$ ${\displaystyle f}$ ${\displaystyle \operatorname {int} (X)}$

See also

• Fréchet derivation