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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 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.
In applied mathematics, weak duality is a concept in optimization which states that the duality gap is always greater than or equal to 0. This means that for any minimization problem, called the primal problem, the solution to the primal problem is always greater than or equal to the solution to the dual maximization problem.
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.
In optimization problems in applied mathematics, the duality gap is the difference between the primal and dual solutions. If is the optimal dual value and is the optimal primal value then the duality gap is equal to . This value is always greater than or equal to 0 (for minimization problems).
In theoretical physics, S-duality (short for strong–weak duality, or Sen duality) is an equivalence of two physical theories, which may be either quantum field theories or string theories. S-duality is useful for doing calculations in theoretical physics because it relates a theory in which calculations are difficult to a theory in which they ...
Whilst weak duality holds in conic linear programming, strong duality does not necessarily hold. [1] Semidefinite Program The dual of a semidefinite program in ...
Montonen–Olive duality or electric–magnetic duality is the oldest known example of strong–weak duality [note 1] or S-duality according to current terminology. [note 2] It generalizes the electro-magnetic symmetry of Maxwell's equations by stating that magnetic monopoles, which are usually viewed as emergent quasiparticles that are "composite" (i.e. they are solitons or topological ...