Sentence examples for be a convex functional from inspiring English sources

Let f : C → R be a convex functional with L-Lipschitz continuous gradient ∇f.

Let H be a Hilbert space and (f:Htomathbb{R}) be a convex functional.

Let E be a real normed space with dual space, (E^) and (f:Erightarrowmathbb{R}) be a convex functional.

Let C be a nonempty closed convex subset of a real Hilbert space H. Let f : C → R be a convex functional with L-Lipschitz continuous gradient ∇f.

Let C be a nonempty closed convex subset of a real Hilbert space H and f : C → R be a convex functional with L-Lipschitz-continuous gradient ∇f.

Corollary 3.1 Let C be a nonempty closed convex subset of a real Hilbert space H. Let f : C → R be a convex functional with L-Lipschitz-continuous gradient ∇f.

Note that for each is a convex functional, that is Hence, (3.4).

It is seen that Φ is a convex functional and that the following result holds.

For any, If is a convex functional, then coincides with the usual subdifferential of in the sense of convex analysis.

(4) For any,   (5) If is a convex functional, then coincides with the usual subdifferential of in the sense of convex analysis.

Let F ( u ) = 1 2 ∫ Ω | u | 2 d x, then it is easy to prove that F ( u ) is a convex functional on L 2.