[aquafox]

Could someone maybe give a high-level explanation into why commercial ILP solvers (e.g. Gurobi) are that much better than free/open-source ones? Is it because ILP is inherently that difficult to solve (I know it's NP-hard), that the best solvers are just a large ensemble of heuristics for very specific sub-problems and thus no general "good" strategy has made it's way into the public domain?

[christina97]

It’s mostly that they work closely with clients in a very hands on way to implement problem-specific speedups. And they’ve been doing this for 10-20 years. In the MILP world this means good heuristics (to find good starting points for branch & bound, and to effectively prune the B&B tree), as well as custom cuts (to cut off fractional solutions in a way that effectively improves the objective and solution integrality).

It’s common that when researchers in Operations Research pick a problem, they can often beat Gurobi and other solvers pretty easily by writing their own cuts & heuristics. The solver companies just do this consistently (by hiring teams of PhDs and researchers) and have a battery of client problems to track improvements and watch for regressions.

[cpgxiii]

> the best solvers are just a large ensemble of heuristics for very specific sub-problems

The big commercial solvers have the resources (and the clients interested in helping) to have invested a lot of time in tuning everything in their solves to real-world problems. Heuristics are part of that; recognizing simpler sub-problems or approximations that can be fed back into the full problem is also part.

I think a big part is that the OSS solvers are somewhat hamstrung by the combination of several issues: (1) the barrier to entry in SoTA optimizer development is very high, meaning that there are very few researchers/developers capable of usefully contributing both the mathematical and programming needed in the first place, (2) if you are capable of (1), the career paths that make lots money lead you away from OSS contribution, and (3) the nature of OSS projects means that "customers" are unlikely to contribute back to kind of examples, performance data, and/or profiling that is really needed to improve the solvers.

There are some exceptions to (2), although being outside of traditional commercial solver development doesn't guarantee being OSS (e.g. SNOPT, developed at Stanford, is still commercially licensed). A lot of academic solver work happens in the context of particular applications (e.g. Clarabel) and so tends to be more narrowly focused on particular problem classes. A lot of other fields have gotten past this bottleneck by having a large tech company acquire an existing commercial project (e.g. Mujoco) or fund an OSS project as a means of undercutting competitors. There are narrow examples of this for solvers (e.g. Ceres) but I suspect the investment to develop an entire general-purpose solver stack from scratch has been considered prohibitive.

[whatever1]

Commercial solvers have a huge bag of tricks & good pattern detection mechanisms to detect which tricks will likely help the problem at hand.

If you know your problem structure then you can exploit it and it is possible to surpass commercial solver performance. But for a random problem, we stand 0 chance.

[zozbot234]

> solvers are just a large ensemble of heuristics for very specific sub-problems

Isn't that statement trivially applicable to anything NP-Hard (which ILP is, since it's equivalent to SAT)?

[graycat]

NP-hard is really hard, but it is hard for (a) polynomial running time, (b) for exact solutions, (c) on worst case problems.

One might suspect that fast enough on specific problems for approximate solutions that still make/save a lot of money might also be welcome. Ah, perhaps not!

E.g., in NYC, two guys had a marketing resource allocation problem, tried simulated annealing, and ran for days before giving up.

They sent me the problem statement via email, and in one week I had the software written and in the next week used the IBM OSL (Optimization Subroutine Library) and some Lagrangian relaxation. In 500 primal-dual iterations with

600,000 variables

40,000 constraints

found a feasible solution within 0.025% of optimality.

So, I'd solved their problem (for practical purposes, the 0.025% has to count as a solving) for free.

They were so embarrassed they wanted nothing to do with me. We never got to where I set a price for my work.

The problem those two guys had was likely that, if they worked with me, then I would understand their customers and, then, beat the guys and take their customers. There in NYC, that happened a second time.

If a guy is in, say, the auto business, and needs a lawyer, the guy might want the best lawyer but will not fear that the lawyer will enter the auto business as a powerful competitor. Similarly for a good medical doctor.

For an optimization guy saving, say, 5% of the operating costs of a big business, say, $billion in revenue a year, all the management suite will be afraid of the guy getting too much power and work to get him out -- Goal Subordination 101 or just fighting to retain position in the tribe.

After having some grand successes in applied math where other people had the problem but then being afraid that I would be too powerful, I formulated:

If some technical, computing, math, etc. idea you have is so valuable, then start your own business exploiting that idea -- of course, need a suitable business for the idea to be powerful.

[aleph_minus_one]

> If a guy is in, say, the auto business, and needs a lawyer, the guy might want the best lawyer but will not fear that the lawyer will enter the auto business as a powerful competitor. Similarly for a good medical doctor.

> For an optimization guy saving, say, 5% of the operating costs of a big business, say, $billion in revenue a year, all the management suite will be afraid of the guy getting too much power and work to get him out -- *Goal Subordination 101 or just fighting to retain position in the tribe.

The optimization guy will also not have the infrastructure to compete with the big business. Additionally, the optimization guy will likely not fight for the management position (not every great applied mathematician is a great manager (in my opinion in particular because leadership of employees and office politics are very different skills)).

So, there is no competition: simply pay the optimization guy a great salary and somewhat isolate him from the gory office politics - problem solved, everybody will live in peace.

But this is not what happens in your example; so the only reason that I can imagine is the usual, irrational bullying of nerds that many nerds know from the schoolyard.

[loehnsberg]

I would argue that not spending money on it and showing to upper mgmt that the folks they hired can actually get the job done often contributes to an external contractor not getting hired.

You might want to question, why didn‘t you ask the guys for money before starting to work for them. True, but I guess they were of the kind, show me results and then maybe we move on. On the other side, a 30-40k pilot project in this area is not difficult to negotiate if you‘re patient.

It takes so much more to running a business than lower cost with clever math that this step often comes at a later stage when larger companies look for ways to stay competitive, which is when they start to take a look at their accounts and figure that certain cost really stack up. Then you come in. The only real power you would have gotten over that company would have been for those guys getting fired and replaced by a vendor - ideally that‘s you!

[fooker]

No, good algorithms for NP hard problems can be more than just heuristics.

Modern SAT solvers are a good example of this. CDCL is elegant.

[sirwhinesalot]

A SAT solver without any preprocessing won't be competitive with SoTA SAT solver.

CDCL is core to the problem, but it is not sufficient. You even have SAT solvers like CryptoMiniSAT that try to detect clauses that encode xor gates so they can use Gaussian Elimination.

This is also true of ILP solvers. Simplex + Branch & Cut is elegant. But that's not how you get to the top.

[lukebuehler]

scale and speed. for example, most quant trading firms run huge optimizations as often as possible. open-source solver often cannot even solve the problems (OOM exceptions, etc)

[FilosofumRex]

In most MILP domains, the underlying engineering know-how is more critical than mathematical formulations or CS coding: (that's why most OR groups operate independently of math or CS departments).

OSS never took off among professional engineers because they've have "skin in the game", unlike math and CS folks who just reboot, and pretend nothing is wrong.