2
$\begingroup$

It is well known in convex analysis that when a closed, proper, function $f$ is Legendre-type, that is, essentially strictly convex and essentially smooth, the Legendre transform yields a dual function $g$ which has the same properties, whereby the gradient is a bijection from the domain of $f$ to the domain of $g$.

Does there exist a similar set of results for strongly convex $f$? or for essentially strongly convex $f$?

$\endgroup$
1
$\begingroup$

Yes, strong convexity is conjugate to uniform smoothness or Lipschitz-continuous differentiability (where the Lipschitz constant is the reciprocal of the modulus of strong continuity), see, e.g.,

Azé, Dominique; Penot, Jean-Paul, Uniformly convex and uniformly smooth convex functions, Ann. Fac. Sci. Toulouse, VI. Sér., Math. 4, No. 4, 705-730 (1995). ZBL0870.49010.

$\endgroup$
  • $\begingroup$ Does the gradient remain a bijection? If not, what guarantees it? $\endgroup$ – The man of your dream Sep 6 at 1:37
  • $\begingroup$ Yes, strongly convex functions are in particular strictly convex, which is the property that guarantees the bijection (of the subgradient as a set-valued operator, which reduces to the gradient if you also have smoothness). $\endgroup$ – Christian Clason Sep 6 at 6:34

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.