# Perturbation function

In mathematical optimization, the **perturbation function** is any function which relates to primal and dual problems. 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.[1]

In some texts the value function is called the perturbation function, and the perturbation function is called the **bifunction**.[2]

## Definition

Given two dual pairs separated locally convex spaces and . Then given the function , we can define the primal problem by

If there are constraint conditions, these can be built into the function by letting where is the indicator function. Then is a *perturbation function* if and only if .[1][3]

## Use in duality

The duality gap is the difference of the right and left hand side of the inequality

where is the convex conjugate in both variables.[3][4]

For any choice of perturbation function *F* weak duality holds. There are a number of conditions which if satisfied imply strong duality.[3] For instance, if *F* is proper, jointly convex, lower semi-continuous with (where is the algebraic interior and is the projection onto *Y* defined by ) and *X*, *Y* are Fréchet spaces then strong duality holds.[1]

## Examples

### Lagrangian

Let and be dual pairs. Given a primal problem (minimize *f*(*x*)) and a related perturbation function (*F*(*x*,*y*)) then the **Lagrangian** is the negative conjugate of *F* with respect to *y* (i.e. the concave conjugate). That is the Lagrangian is defined by

In particular the weak duality minmax equation can be shown to be

If the primal problem is given by

where . Then if the perturbation is given by

then the perturbation function is

Thus the connection to Lagrangian duality can be seen, as *L* can be trivially seen to be

### Fenchel duality

Let and be dual pairs. Assume there exists a linear map with adjoint operator . Assume the primal objective function (including the constraints by way of the indicator function) can be written as such that . Then the perturbation function is given by

In particular if the primal objective is then the perturbation function is given by , which is the traditional definition of Fenchel duality.[5]

## References

- Radu Ioan Boţ; Gert Wanka; Sorin-Mihai Grad (2009).
*Duality in Vector Optimization*. Springer. ISBN 978-3-642-02885-4. - J. P. Ponstein (2004).
*Approaches to the Theory of Optimization*. Cambridge University Press. ISBN 978-0-521-60491-8. - Zălinescu, C. (2002).
*Convex analysis in general vector spaces*. River Edge, NJ,: World Scientific Publishing Co., Inc. pp. 106–113. ISBN 981-238-067-1. MR 1921556.CS1 maint: extra punctuation (link) - Ernö Robert Csetnek (2010).
*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*. Logos Verlag Berlin GmbH. ISBN 978-3-8325-2503-3. - Radu Ioan Boţ (2010).
*Conjugate Duality in Convex Optimization*. Springer. p. 68. ISBN 978-3-642-04899-9.