Generalizations of linear programming Linear problems can be solved in polynomial time. So can semidefinite programs and, presumably, many other useful classes of optimization programs.

Is there a survey/lecture notes describing generalizations of LP that can be solved in poly time?

This is related to this question, but the questions in that post are quite different and most of the questions have not been answered.
Fixed Point Logic $+$ Counting (FPC) is believed to capture most of the $P$ solvable problems. Anderson, Dawar and Holm 2015 [1]showed that optimization of linear programs is expressible in FPC. Dawar and Wang 2016 [2]showed that The FPC implementation of the ellipsoid method extends to semidefinite programs (subject to some technical conditions).