Just because it is NP-hard in the worst-case doesn't mean it is not practical. As can be seen in the many theorems under which conditions the regular polynomial-time LP algorithm provides an integer solution.
> It's surprising how many problems can be formulated as linear optimization.
i.e., all problems in NP (which is most problems you're likely to encounter on a day-to-day basis) can be solved with ILP, and many of them can be solved or well-approximated quickly.
To interpret the observation a bit more meaningfully:
It's surprising how many problems can be formulated as continuous (!) linear optimisation.
And it is surprising how many problems can be formulated somewhat naturally as mixed-integer linear optimisation. And 'many of them can be solved or well-approximated quickly', exactly as you say.
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I seem to remember that continuous linear optimisation is to P what integer linear optimisation is to NP. In the sense that there's some natural reduction of many problems in P to continuous linear optimisation.
(I don't remember if that's just an informal observation, or whether there's some formal way to reduce problems in P in eg linear time to linear optimisation? https://en.wikipedia.org/wiki/P-complete#P-complete_problems mentions Linear Optimisation as being P-complete, but I haven't vetted all the fine-print, eg about what specific reduction they are using.)
First of all, you can't solve a neoclassical economy using LP, because equilibrium constraints can only be represented as complementarity constraints. You would have to give up on some aspects, like dynamic prices.
The linear complementarity problem in itself is NP hard. So you're screwed from the get go, because your problems are now LPCC problems.
Good luck finding an LPCC solver. I can confirm that an open source QPCC solver exists though, which should be even slower.
Next is the fact that if you wanted to build a neoclassical economy model, only global optimization will do.
This means that you need to simulate every time step in one large LPCC model, instead of using a finite horizon. Due to the perfect information assumption, you must know about the state of every person on the planet. You're going to need millions of variables due to simple combinatorial explosion.
It's kind of startling how these assumptions, which are supposed to make analytical solutions tractable by the way, also make non-analytical solutions literal hell.
And before you say that prices can be determined iteratively, as I mentioned, you would run into the problem that future prices are unknown to you, so how are you going to plug them into the second time step? The very thing you want to calculate depends on it's future value.
Economics is a weird science, where experienced reality works much better than the theory.
Computational economics is a relatively new field where intelligent agents are used with lots of runs instead of general optimization solvers I believe. Pretty nifty. One of my colleagues publishes a good bit on it.
Huh? Are you replying to the wrong comment? I never made any claims about 'solving a neoclassical economy'.
I'm not quite sure who cares about solving a neoclassical economic model like that?
As you indirectly suggest, neoclassical assumption of the type you suggested are not computationally tractable. So the kind of computations real economic agents actually do are likely to be different. (Whether that flavour of neoclassical economics is still useful after taking this caveat into account, is a different question.)
In any case: yes, not all NP-hard or NP-complete problems are easy to solve in practice. Even worse, many problems widely believed to be neither NP-hard nor NP-complete, like factoring integers or computing discrete logarithms, are also hard for many practical instances. (And they have to be, if cryptography is supposed to work.)
