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the equality from (i) holds.
A sufficient condition for strong duality to hold for a SDP problem (and in general, for any convex optimization problem) is the [[Slater's condition]]. It is also possible to attain strong duality for SDPs without additional regularity conditions by using an extended dual problem proposed by Ramana.<ref>{{Cite journal |last=Ramana |first=Motakuri V. |date=1997 |title=An exact duality theory for semidefinite programming and its complexity implications |url=http://link.springer.com/10.1007/BF02614433 |journal=Mathematical Programming |language=en |volume=77 |issue=1 |pages=129–162 |doi=10.1007/BF02614433 |s2cid=12886462 |issn=0025-5610}}</ref><ref>{{Cite journal |
== Examples ==
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== Applications ==
Semidefinite programming has been applied to find approximate solutions to combinatorial optimization problems, such as the solution of the [[max cut]] problem with an [[approximation ratio]] of 0.87856. SDPs are also used in geometry to determine tensegrity graphs, and arise in control theory as [[Linear matrix inequality|LMIs]], and in inverse elliptic coefficient problems as convex, non-linear, semidefiniteness constraints.<ref>{{citation|last1=Harrach|first1=Bastian|date=2021|title=Solving an inverse elliptic coefficient problem by convex non-linear semidefinite programming|journal=Optimization Letters|volume=16 |issue=5 |pages=1599–1609 |language=en|doi=10.1007/s11590-021-01802-4|arxiv=2105.11440|s2cid=235166806}}</ref>
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