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The duality gap is zero if and only if strong duality holds. Otherwise the gap is strictly positive and weak duality holds. [5] In computational optimization, another "duality gap" is often reported, which is the difference in value between any dual solution and the value of a feasible but suboptimal iterate for the primal problem.
The strong duality theorem says that if one of the two problems has an optimal solution, so does the other one and that the bounds given by the weak duality theorem are tight, i.e.: max x c T x = min y b T y. The strong duality theorem is harder to prove; the proofs usually use the weak duality theorem as a sub-routine.
A duality that respects the orderings in question is known as a Galois connection. An example is the standard duality in Galois theory mentioned in the introduction: a bigger field extension corresponds—under the mapping that assigns to any extension L ⊃ K (inside some fixed bigger field Ω) the Galois group Gal (Ω / L) —to a smaller ...
In mathematics and mathematical optimization, the convex conjugate of a function is a generalization of the Legendre transformation which applies to non-convex functions. It is also known as Legendre–Fenchel transformation, Fenchel transformation, or Fenchel conjugate (after Adrien-Marie Legendre and Werner Fenchel).
Theorem — (sufficiency) If there exists a solution to the primal problem, a solution (,) to the dual problem, such that together they satisfy the KKT conditions, then the problem pair has strong duality, and , (,) is a solution pair to the primal and dual problems.
In mathematics, Fenchel's duality theorem is a result in the theory of convex functions named after Werner Fenchel. Let ƒ be a proper convex function on R n and let g be a proper concave function on R n. Then, if regularity conditions are satisfied,
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.
Strong duality is a condition in mathematical optimization in which the primal optimal objective and the dual optimal objective are equal. By definition, strong duality holds if and only if the duality gap is equal to 0.
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