Do all convex optimisation problems have a closed form solution?

Question

Is convexity of the objective function a sufficient criterion for having a closed-form solution? If not, is there any specific condition that will definitely lead to a closed-form solution?

Answer 1

I think what you are thinking of is the following. For some families of functions there are explicit closed-form formulas for the minimizer in terms of the parameters of the family. For example, if we write

$$f_{a,b,c}(x) = ax^2 + bx + c$$

then the special family of functions

$$\mathcal{F} = \{f_{a,b,c} | a,b,c \in \mathbb{R}, a>0\}$$

has a closed-form formula for the minimizers: $$\operatorname{argmin}_x f_{a,b,c}(x) = -b/2a.$$ That is, there is a compact description of $f$ given by the parameters $a,b,c$; and if we know this description, we can give you the minimizer.

But if you consider an arbitrary convex function, there is no reason to expect it to have any description other than itself. The only way to describe is the set of order pairs $(x,y): f(x) = y$. So it's not clear what a closed-form solution would even mean.