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).
Read more about this topic: Duality (optimization)
Famous quotes containing the word principle:
“Now, what I want is, Facts. Teach these boys and girls nothing but Facts. Facts alone are wanted in life. Plant nothing else, and root out everything else. You can only form the minds of reasoning animals upon Facts: nothing else will ever be of any service to them. This is the principle on which I bring up my own children, and this is the principle on which I bring up these children. Stick to Facts, sir!”
—Charles Dickens (1812–1870)