I think I am missing something for solving exercise 25. How do we know that \(\psi(y)\) is strongly convex?
Thanks in advance.
\(\psi(y)\) is a sum of a constant function of \(y\) (convex), a linear function (convex), a quadratic function (strongly convex) and a convex function (by Theorem 3.14 assumption). Hence, you can show that it is strongly convex. There is a little stretch from the course as you saw the definition of strong convexity when \(f\) is differentiable but here we would need a definition of strong convexity with sub-gradients as \(h\) is not necessarily differentiable.
I hope this clarify the solution.
Are you referring to Exercise 25 of the lecture notes? I don't see a \(\psi(y)\) there?
Yes, but I am referring to the solution.