We definitely do study specific instances. For example, the other day an article was posted here on Support Vector Machines, which are a convex optimization pro

We definitely do study specific instances. For example, the other day an article was posted here on Support Vector Machines, which are a convex optimization problem having a special structure that is exploitable for fast solutions. For more examples, see Convex Optimization by Boyd and Vandenberghe, and the course notes for EE364a/b at Stanford. Modern Convex Optimization by Bertsekas is also good. I had the privelege of taking the convex optimization sequence at Stanford from Prof. Boyd, and one of the major take aways is that many practical problems can be solved in quadratic or even linear time. A general purpose solver typically runs in cubic time. For large problems (lots of variables), a special purpose solver can solve problems in seconds that would take a general purpose solver hours! The book posted here is very much an introduction; even the EE364 sequence was merely a jumping off point for being able to explore the literature. Mathematical optimization, and especially convex optimization, is a deep and beautiful field!