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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 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 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.
Duality gap — difference between primal and dual solution; Fenchel's duality theorem — relates minimization problems with maximization problems of convex conjugates; Perturbation function — any function which relates to primal and dual problems; Slater's condition — sufficient condition for strong duality to hold in a convex ...
In mathematics, Slater's condition (or Slater condition) is a sufficient condition for strong duality to hold for a convex optimization problem, named after Morton L. Slater. [1] Informally, Slater's condition states that the feasible region must have an interior point (see technical details below).
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.
Hints About Today's NYT Connections Categories on Thursday, December 12. 1. All of these words sound like a specific letter in the alphabet. 2. These items are known for their notched edges. 3 ...
Sufficiency: the solution pair , (,) satisfies the KKT conditions, thus is a Nash equilibrium, and therefore closes the duality gap. Necessity: any solution pair x ∗ , ( μ ∗ , λ ∗ ) {\displaystyle x^{*},(\mu ^{*},\lambda ^{*})} must close the duality gap, thus they must constitute a Nash equilibrium (since neither side could do any ...