Convex Conjugate - Definition

Definition

Let be a real normed vector space, and let be the dual space to . Denote the dual pairing by

For a functional

taking values on the extended real number line, the convex conjugate

is defined in terms of the supremum by

or, equivalently, in terms of the infimum by

This definition can be interpreted as an encoding of the convex hull of the function's epigraph in terms of its supporting hyperplanes.

Read more about this topic:  Convex Conjugate

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