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In [[mathematical optimization]], the '''perturbation function''' is any [[function (mathematics)|function]] which relates to primal and [[dual problem]]s. The name comes from the fact that any such function defines a perturbation of the initial problem. In many cases this takes the form of shifting the constraints.<ref name="BWG">{{cite book|title=Duality in Vector Optimization|author1=Radu Ioan Boţ|author2=Gert Wanka|author3=Sorin-Mihai Grad|year=2009|publisher=Springer|isbn=978-3-642-02885-4}}</ref>

In some texts the [[value function]] is called the perturbation function, and the perturbation function is called the '''bifunction'''.<ref>{{cite book|title=Approaches to the Theory of Optimization|author=J. P. Ponstein|publisher=Cambridge University Press|year=2004|isbn=978-0-521-60491-8}}</ref>

== Definition ==
Given two [[dual pair]]s [[separated space|separated]] [[locally convex space]]s <math>\left(X,X^*\right)</math> and <math>\left(Y,Y^*\right)</math>. Then given the function <math>f: X \to \mathbb{R} \cup \{+\infty\}</math>, we can define the primal problem by

:<math>\inf_{x \in X} f(x). \, </math>

If there are constraint conditions, these can be built into the function <math>f</math> by letting <math>f = f + I_\mathrm{constraints}</math> where <math>I</math> is the [[Characteristic function (convex analysis)|indicator function]]. Then <math>F: X \times Y \to \mathbb{R} \cup \{+\infty\}</math> is a ''perturbation function'' if and only if <math>F(x,0) = f(x)</math>.<ref name="BWG" /><ref name="Zalinescu">{{cite book|last=Zălinescu|first=C.|title=Convex analysis in general vector spaces|publisher=World Scientific Publishing  Co., Inc|location = River Edge, NJ, |year=2002|pages=106–113|isbn=981-238-067-1|mr=1921556}}</ref>

== Use in duality ==
The [[duality gap]] is the difference of the right and left hand side of the inequality
:<math>\sup_{y^* \in Y^*} -F^*(0,y^*) \le \inf_{x \in X} F(x,0),</math>
where <math>F^*</math> is the [[convex conjugate]] in both variables.<ref name="Zalinescu" /><ref>{{cite book|title=Overcoming the failure of the classical generalized interior-point regularity conditions in convex optimization. Applications of the duality theory to enlargements of maximal monotone operators|author=Ernö Robert Csetnek|year=2010|publisher=Logos Verlag Berlin GmbH|isbn=978-3-8325-2503-3}}</ref>

For any choice of perturbation function ''F'' [[weak duality]] holds. There are a number of conditions which if satisfied imply [[strong duality]].<ref name="Zalinescu" /> For instance, if ''F'' is [[proper convex function|proper]], jointly [[convex function|convex]], [[lower semi-continuous]] with <math>0 \in \operatorname{core}(\operatorname{Pr}_Y(\operatorname{dom}F))</math> (where <math>\operatorname{core}</math> is the [[algebraic interior]] and <math>\operatorname{Pr}_Y</math> is the [[projection (set theory)|projection]] onto ''Y'' defined by <math>\operatorname{Pr}_Y(x,y) = y</math>) and ''X'', ''Y'' are [[Fréchet space]]s then strong duality holds.<ref name="BWG" />

== Examples ==

=== Lagrangian ===
Let <math>(X,X^*)</math> and <math>(Y,Y^*)</math> be dual pairs. Given a primal problem (minimize ''f(x)'') and a related perturbation function (''F(x,y)'') then the '''Lagrangian''' <math>L: X \times Y^* \to \mathbb{R} \cup \{+\infty\}</math> is the negative conjugate of ''F'' with respect to ''y'' (i.e. the concave conjugate). That is the Lagrangian is defined by
:<math>L(x,-y^*) = \inf_{y \in Y} \left\{F(x,y) - y^*(y)\right\}.</math>
In particular the [[weak duality]] minmax equation can be shown to be
:<math>\sup_{y^* \in Y^*} -F^*(0,y^*) = \sup_{y^* \in Y^*} \inf_{x \in X} L(x,y^*) \leq \inf_{x \in X} \sup_{y^* \in Y^*} L(x,y^*) = \inf_{x \in X} F(x,0).</math>

If the primal problem is given by
:<math>\inf_{x: g(x) \leq 0} f(x) = \inf_{x \in X} \tilde{f}(x)</math>
where <math>\tilde{f}(x) = f(x) + I_{\mathbb{R}^d_+}(-g(x))</math>. Then if the perturbation is given by
:<math>\inf_{x: g(x) \leq y} f(x)</math>
then the perturbation function is
:<math>F(x,y) = f(x) + I_{\mathbb{R}^d_+}(y - g(x))</math>.
Thus the connection to Lagrangian duality can be seen, as ''L'' can be trivially seen to be
:<math>L(x,y^*) = \begin{cases}
f(x) + y^*(g(x)) & \text{if } y^* \in \mathbb{R}^d_+\\
-\infty & \text{else}
\end{cases}</math>.

=== Fenchel duality ===
{{main|Fenchel duality}}
Let <math>(X,X^*)</math> and <math>(Y,Y^*)</math> be dual pairs. Assume there exists a [[linear map]] <math>T: X \to Y</math> with [[adjoint operator]] <math>T^*: Y^* \to X^*</math>. Assume the primal [[objective function]] <math>f(x)</math> (including the constraints by way of the indicator function) can be written as <math>f(x) = J(x,Tx)</math> such that <math>J: X \times Y \to \mathbb{R} \cup \{+\infty\}</math>. Then the perturbation function is given by
: <math>F(x,y) = J(x,Tx - y)</math>.

In particular if the primal objective is <math>f(x) + g(Tx)</math> then the perturbation function is given by <math>F(x,y) = f(x) + g(Tx - y)</math>, which is the traditional definition of [[Fenchel duality]].<ref>{{cite book|title=Conjugate Duality in Convex Optimization|author=Radu Ioan Boţ|publisher=Springer|year=2010|isbn=978-3-642-04899-9|page=68}}</ref>

== References ==
{{Reflist}}

[[Category:Mathematical optimization]]
[[Category:Linear programming]]
[[Category:Convex optimization]]

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