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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.
Strong duality is a condition in mathematical optimization in which the primal optimal ... is the perturbation function relating the primal and dual ...
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).
The duality gap is the difference of the right and left hand side of the inequality (,) (,),where is the convex conjugate in both variables. [3] [4]For any choice of perturbation function F weak duality holds.
In logic, functions or relations A and B are considered dual if A(¬x) = ¬B(x), where ¬ is logical negation. The basic duality of this type is the duality of the ∃ and ∀ quantifiers in classical logic. These are dual because ∃x.¬P(x) and ¬∀x.
In mathematical optimization, Wolfe duality, named after Philip Wolfe, is type of dual problem in which the objective function and constraints are all differentiable functions. Using this concept a lower bound for a minimization problem can be found because of the weak duality principle.
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.