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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.
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.
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 board also owns a stadium in Latifabad used for league, inter-club and -city hockey tournaments. But sporting events The District Hockey Association (DHA) for the Hyderabad District, Pakistan had allocated a budget of 1.6 million rupees for renovations for the betterment of hockey arenas but were reluctant to give the board their share.
He is best known for two results in economics, now known as Shephard's lemma and the Shephard duality theorem. Shephard proved these results in his book Theory of Cost and Production Functions (Princeton University Press, 1953), which Dale W. Jorgenson, in the preface of a reprint, called "the most original contribution to economic theory of ...
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.
Schur–Weyl duality is a mathematical theorem in representation theory that relates irreducible finite-dimensional representations of the general linear and symmetric groups. . Schur–Weyl duality forms an archetypical situation in representation theory involving two kinds of symmetry that determine each oth
In mathematics, Tannaka–Krein duality theory concerns the interaction of a compact topological group and its category of linear representations.It is a natural extension of Pontryagin duality, between compact and discrete commutative topological groups, to groups that are compact but noncommutative.