Hi Shirley,
By dualize, I meant find the Fenchel-Legendre conjugate. Often you can find this very quickly if you know a few tricks, e.g., the conjugate function of a norm is the indicator function of the dual norm (so, the proximity operator of the l1 norm is the projection operator onto the l_infinity ball, and vice-versa).
Unfortunately I don’t have time to help with individual codes these days, but for more references on the conjugate function, you could try the books by Amir Beck (his '17 book) or Combettes & Bauschke, or try these:
- lecture notes from my class; see page 51 at https://github.com/stephenbeckr/convex-optimization-class/blob/master/TypedNotes/lecture_notes.pdf
- SIGNAL RECOVERY BY PROXIMAL FORWARD-BACKWARD SPLITTING, https://pcombet.math.ncsu.edu/mms1.pdf
- Proximal Splitting Methods in Signal Processing, https://link.springer.com/chapter/10.1007/978-1-4419-9569-8_10
Best,
Stephen