Perturbation function

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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[edit]

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[edit]

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[edit]

Lagrangian[edit]

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[edit]

Main article: 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[edit]

  1. ^ a b c Radu Ioan Boţ; Gert Wanka; Sorin-Mihai Grad (2009). Duality in Vector Optimization. Springer. ISBN 978-3-642-02885-4. 
  2. ^ J. P. Ponstein (2004). Approaches to the Theory of Optimization. Cambridge University Press. ISBN 978-0-521-60491-8. 
  3. ^ a b c 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. 
  4. ^ 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. 
  5. ^ Radu Ioan Boţ (2010). Conjugate Duality in Convex Optimization. Springer. p. 68. ISBN 978-3-642-04899-9.