You may be interested in some of the recent work of Bill Helton and his collaborators. The idea (very roughly) is to study convex problems which can be defined in some intrinsic way without reference to the dimension of the problem, usually in terms of matrix equations and inequalities (semidefiniteness constraints). For example the existence of a quadratic Lyapunov function for a linear system reduces to such a form: the matrix equations and inequalities you write down look the same regardless of the dimension of the system if you're representing each of these matrices with a symbol rather than an $n\times n$ array. This means there is some "uniformity" to the problem.
They study this by viewing the matrices involved as noncommuting indeterminates. They go on to define convexity and related ideas for noncommutative polynomials. The algebraic structure of this "noncommutative geometry" is very rigid (much more than the commutative case), so you can say a lot about how these polynomials must look. This in turn tells you things about which problems can be cast in such a uniform way, and perhaps why so many of them are semidefinite programs.
I don't know enough to even call myself a novice in this area, let alone an expert, so I'm sure I am not doing this work justice. I was hoping someone more qualified might come along and give a better explanation, but no one has mentioned it so far and it may be what you are looking for, so I gave it a shot.