Furthermore, depending on publishing site, a paper may also be available as HTML rendered from the LaTeX source, in addition to PDF. (If the page does not now, it may in the future.)
The purpose of a [PDF] tag is to warn about possible unsuitability of the linked resource for mobile consumption (which isn’t the case for the article page linked here), possible download size (though maybe not anymore, nowadays), and possible brightness shock when using dark mode.
There is a result that say that if you can solve general ILP problems then you can solve general G3C.
Satisfiability is NP and NP-Hard, therefore NP-Complete (NPC). It is therefore equivalent (under some definition of "equivalent") to G3C.
There is a result that say that if you can solve general ILP problems then you can solve general G3C.
There is a known result that if you can solve arbitrary G3C problems then you can factor integers. While the problem of factoring integers (FAC) is not NPC, clearly factoring integers is very important in today's computing environment.
So if you can solve arbitrary ILP problems you can break several current encryption algorithms that are thought to be secure.
So we can deduce that ILP is a fairly tricky problem to solve.
The thing that fools a lot of people is that random instances of NPC problems tend to be easy. The core of difficult instances gets smaller (in relative terms) as the problems get bigger, so if, say, you pick a random graph, it's probably trivial to find a vertex 3-colouring, or show that none such exists.
> if we needed two months of running time to solve an LP in the early 1990s, we would need less than one second today. Recently, Bixby compared the machine-independent performance of two MILP solvers, CPLEX and Gurobi, between 1990 and 2020 and reported speed-ups of almost 4×10^6.
As I recall it the raw computer power had increased by a factor of around a thousand and the algorithms had improved by about the same, giving us a factor of a million improvement.
Worth pondering when trying to predict the future!
The "resources" in question were diamonds by the way...
Isn't that statement trivially applicable to anything NP-Hard (which ILP is, since it's equivalent to SAT)?
Modern SAT solvers are a good example of this. CDCL is elegant.
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.
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.
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.
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.
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.
The EU electricity spot price is set each day in a single giant solver run, look up Euphemia for some write ups of how that works.
Most any field where there is a clear goal to optimise and real money on the line will be riddled with solvers
1. Salesman & delivery travel plan
2. Machine, Human and material resource scheduling for production
3. Inventory level for warehouse distribution center. This one isn't fully automatic because demand forecasting is hard
gurobi case studies: https://www.gurobi.com/case_studies/
some cplex case studies: https://www.ibm.com/products/ilog-cplex-optimization-studio/...
hexaly (formerly localsolver) case studies: https://www.hexaly.com/customers
"The authors observed a speedup of 1000 between [the commercial MILP solvers of] 2001 and 2020 (50 due to algorithms, 20 due to faster computers)."
I wonder if we can collect these speedup factors across computing subfields, decomposed by the contribution of algorithmic improvements, and faster computers.
In compilers, there's "Proebsting's Law": compiler advances double computing power every 18 years.
djoldman•17h ago
Chio•16h ago
https://highs.dev/ https://www.scipopt.org/
nrclark•15h ago
sirwhinesalot•12h ago
7thaccount•7h ago
antman•14h ago
wombatpm•11h ago
If you have a problem that needs Gurobi, it’s worth paying for it. Talk with their sales team. They are happy to help you get started. They know once you know how to use it, and how it can solve problems you will be inclined to use it in the future.
RainyDayTmrw•15h ago
almostgotcaught•15h ago
0cf8612b2e1e•10h ago
edot•14h ago
cschmidt•14h ago
quanto•13h ago
I want to add that, for many in the industry, it is well worth the price.
__alexs•12h ago
7thaccount•6h ago
jwr•4h ago