Exercice 25 - strong convexity

Hi,

I think I am missing something for solving exercise 25. How do we know that \(\psi(y)\) is strongly convex?

Thanks in advance.

Top comment

Hello,

Screenshot 2021-07-02 at 14.58.28.jpg

\(\psi(y)\) 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.
JB

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.

Top comment

Hello,

Screenshot 2021-07-02 at 14.58.28.jpg

\(\psi(y)\) 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.
JB

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