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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 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.
Duality (optimization) Weak duality — dual solution gives a bound on the primal solution; Strong duality — primal and dual solutions are equivalent; Shadow price; Dual cone and polar cone; Duality gap — difference between primal and dual solution; Fenchel's duality theorem — relates minimization problems with maximization problems of ...
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.
The strong duality theorem states that if the primal has an optimal solution, x *, then the dual also has an optimal solution, y *, and c T x * =b T y *. A linear program can also be unbounded or infeasible. Duality theory tells us that if the primal is unbounded then the dual is infeasible by the weak duality theorem.
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