Low-rank approximation: Difference between revisions

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{{Merge to|Duality (optimization)|date=March 2012}}
'''Strong duality''' is a concept in [[optimization]] such that the [[dual problem|primal and dual solutions]] are equivalent. This is as opposed to [[weak duality]] (the primal problem has optimal value greater than the dual problem, in other words the [[duality gap]] is positive). Strong duality holds if and only if the [[duality gap]] is equal to 0.


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There are many conditions for strong duality to hold such as:
* <math>F = F^{**}</math> where <math>F</math> is the [[perturbation function]] relating the primal and dual problems and <math>F^{**}</math> is the [[convex conjugate|biconjugate]] of <math>F</math>;{{Citation needed|date=January 2012}}
* the primal problem is a [[linear optimization|linear optimization problem]];
* [[Slater's condition]] for a [[convex optimization|convex optimization problem]].<ref name="borwein">{{cite book |last1=Borwein |first1=Jonathan |last2=Lewis |first2=Adrian |title=Convex Analysis and Nonlinear Optimization: Theory and Examples| edition=2 |year=2006 |publisher=Springer |isbn=978-0-387-29570-1}}</ref><ref name="boyd">{{cite book |last1=Boyd |first1=Stephen |last2=Vandenberghe |first2=Lieven |title=Convex Optimization |publisher=Cambridge University Press |year=2004 |isbn=978-0-521-83378-3 |url=http://www.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf |format=pdf |accessdate=October 3, 2011}}</ref>
 
==References==
{{Reflist}}
 
[[Category:Mathematical optimization]]
[[Category:Linear programming]]
[[Category:Convex optimization]]
 
 
{{mathapplied-stub}}

Revision as of 23:55, 6 August 2013



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There are many conditions for strong duality to hold such as:

References

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