I've been reading Boyd & Vandenberghe and it says that conic programming is a subclass of convex optimization. I haven't been able to find an example of a convex optimization problem that I cannot translate to an equivalent conic optimization problem (equivalent in the sense that I can find a solution to my original problem given a solution to the conic program). Do you have any examples?
If so, how can it be solved automatically? My understanding is that the best solvers out there such as SDPT3, MOSEK or Sedumi work with conic programs.
Let $C \subset \mathbb R^n$ and $f \colon \mathbb R^n \to \mathbb R$ be convex. We consider $$ \text{minimize} \; f(x) \; \text{s.t.} \; x \in C. $$ In order to simplify the objective, we introduce $D := (C \times \mathbb R) \cap \operatorname{epi} f$ and the problem is equivalent to $$ \text{minimize} \; t \; \text{s.t.} \; (x,t) \in D. $$ It remains to transform $D$ into a cone. To this end, we set $E := \operatorname{cl\,cone}( D \times \{1\})$ and one can check that $(x,t) \in D$ is equivalent to $(x,t,1) \in E$. Thus, our original problem is equivalent to the conic problem $$ \text{minimize} \; t \; \text{s.t.} \; (x,t,s) \in E\;\text{and}\; s = 1. $$