Hardest optimization problems in NC When learning optimization problems, we usually consider linear programming (or more generally: convex optimization) as the simplest example. It is solvable in polynomial time, and has relatively easy to understand algorithms. However, the decision version of LP is $\mathsf{P}$-complete. This suggests that it is one of the hardest problems we can solve in polynomial time.
Under the assumption that $\mathsf{NC} \neq \mathsf{P}$. What is the 'hardest' natural kind of optimization problems with decision problems in $\mathsf{NC}$?
If this is too vague, then we can restrict to restrictions. What is the minimal set of restrictions we need to place on linear programs (or more generally: convex programs) to allow the decision problem associated with the restricted programs to be solvable in $\mathsf{NC}$?

Motivation
To a large extent, this is idle curiosity. However, it was brought about by Cosma Shalizi's &quot;In Soviet Union, Optimization Problem Solves You&quot;. In particular, if LP is too difficult to solve to have a centralized economy (i.e. optimizing in $\mathsf{P}$ is asking too much), then any decentralized system must be doing some sort of parallel processing faster than poly-time (for me: $\mathsf{NC}$).
Are you aware of the work on positive LPs/SDPs? There are a bunch of results in the area, mostly along the lines of "if the constraints of the LP/SDP are positive, then the problem can be solved in NC."

Some important references in this line of work are Luby-Nisan 93 and Jain-Yao 11. Another excellent resource is this slide of Rahul Jain's talk at the "Recent Progress in Quantum Algorithms" conference at IQC. The entire talk is available on youtube.