In optimization theory, the duality principle states that optimization problems may be viewed from either of two perspectives, the primal problem or the dual problem.
In general given two dual pairs of separated locally convex spaces and . Then given the function, we can define the primal problem as finding such that
In other words, is the infimum (greatest lower bound) of the function .
If there are constraint conditions, these can be built in to the function by letting where is the indicator function. Then let be a perturbation function such that .
The duality gap is the difference of the right and left hand sides of the inequality
where is the convex conjugate in both variables and denotes the supremum (least upper bound).
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“The principle of subordination is the great bond of union and harmony through the universe.”
—Catherine E. Beecher (1800–1878)