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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.
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.
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]
A set C (blue) and its dual cone C * (red).. A duality in geometry is provided by the dual cone construction. Given a set of points in the plane (or more generally points in ), the dual cone is defined as the set consisting of those points (,) satisfying + for all points (,) in , as illustrated in the diagram.
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 idea of mathematical duality was first noticed as projective duality. There it appears as the idea of interchanging dimension k and codimension k+1 in propositions of projective geometry. A large number of duality theories have now been created in mathematics, ranging as far as optimization theory and theoretical physics.
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Farkas's lemma can be varied to many further theorems of alternative by simple modifications, [5] such as Gordan's theorem: Either < has a solution x, or = has a nonzero solution y with y ≥ 0. Common applications of Farkas' lemma include proving the strong duality theorem associated with linear programming and the Karush–Kuhn–Tucker ...