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Conic optimization

From Wikipedia, the free encyclopedia

Conic optimization is a subfield of convex optimization that studies problems consisting of minimizing a convex function over the intersection of an affine subspace and a convex cone.

The class of conic optimization problems includes some of the most well known classes of convex optimization problems, namely linear and semidefinite programming.

Definition

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Given a real vector space X, a convex, real-valued function

defined on a convex cone , and an affine subspace defined by a set of affine constraints , a conic optimization problem is to find the point in for which the number is smallest.

Examples of include the positive orthant , positive semidefinite matrices , and the second-order cone . Often is a linear function, in which case the conic optimization problem reduces to a linear program, a semidefinite program, and a second order cone program, respectively.

Duality

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Certain special cases of conic optimization problems have notable closed-form expressions of their dual problems.

Conic LP

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The dual of the conic linear program

minimize
subject to

is

maximize
subject to

where denotes the dual cone of .

Whilst weak duality holds in conic linear programming, strong duality does not necessarily hold.[1]

Semidefinite Program

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The dual of a semidefinite program in inequality form

minimize
subject to

is given by

maximize
subject to

References

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  1. ^ "Duality in Conic Programming" (PDF).
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