# Computational complexity and reductions to linear programming

My question is rather general, but I have not been able to find a satisfactory answer for quite a long time, and I wonder if there has been a research concerning these issues.

I have come over a relatively large number of optimization problems, that can be expressed as a Linear Programming problem, or as an Integer Linear Programming problem. Also decision problems could be expressed as LP or ILP, with a constant objective function (i.e., concerning only (in)feasibility). If a problem can be expressed as LP, then it is certainly in $\mathbf{P}$, since Linear Programming problems can be solved in polynomial time. On the other hand, problems that cannot be expressed as LP, but only as ILP, usually are $\mathbf{NP}$-complete, despite the fact that this is not a valid proof of $\mathbf{NP}$-completeness (there could be a polynomial algorithm not based on LP solving the problem).

So my question is: is there any known relation between the complexity of a problem and it's expressibility in the form of (integer) Linear Programming (except the obvious facts stated in the previous paragraph)? Moreover, I believe that there are inherently nonlinear problems that cannot be expressed in the form of LP nor ILP. Is there some known relation between their complexity and their expressibility in the form of some type of Nonlinear Programming problems? Are there any references dealing with similar problems?