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By definition, strong duality holds if and only if the duality gap is equal to 0. This is opposed to weak duality (the primal problem has optimal value smaller than or equal to the dual problem, in other words the duality gap is greater than or equal to zero).
The max-flow min-cut theorem is a special case of the strong duality theorem: flow-maximization is the primal LP, and cut-minimization is the dual LP. See Max-flow min-cut theorem#Linear program formulation. Other graph-related theorems can be proved using the strong duality theorem, in particular, Konig's theorem. [9]
In mathematics, a duality, generally speaking, translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a one-to-one fashion, often (but not always) by means of an involution operation: if the dual of A is B, then the dual of B is A.
For example, Desargues' theorem is self-dual in this sense under the standard duality in projective geometry. In mathematical contexts, duality has numerous meanings. [1] It has been described as "a very pervasive and important concept in (modern) mathematics" [2] and "an important general theme that has manifestations in almost every area of ...
o o o s. c: o thO 00 . Created Date: 9/20/2007 3:37:18 PM
In projective geometry, duality or plane duality is a formalization of the striking symmetry of the roles played by points and lines in the definitions and theorems of projective planes. There are two approaches to the subject of duality, one through language ( § Principle of duality ) and the other a more functional approach through special ...
In mathematics, Slater's condition (or Slater condition) is a sufficient condition for strong duality to hold for a convex optimization problem, named after Morton L. Slater. [1] Informally, Slater's condition states that the feasible region must have an interior point (see technical details below).
A function that is not lower semi-continuous.By the Fenchel-Moreau theorem, this function is not equal to its biconjugate.. In convex analysis, the Fenchel–Moreau theorem (named after Werner Fenchel and Jean Jacques Moreau) or Fenchel biconjugation theorem (or just biconjugation theorem) is a theorem which gives necessary and sufficient conditions for a function to be equal to its biconjugate.
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