Simplex algorithm should never work given its exponential worst case performance. Yet it solves the vast majority of LP problems, including ones with billions of variables. It is the default algorithm for every solver despite the fact that we have a polynomial interior point method in our disposal. Go figure.
Worst case performance or NP completeness is not a predictor of how fast we can solve an optimization problem.
> Simplex algorithm should never work given its exponential worst case performance.
Note that it has been proven however that if you slightly perturbate the inputs then the expected complexity becomes polynomial, meaning the inputs for which it's exponential are actually just pathological cases and it is polynomial in practice.
We started understanding of why it works so well like 30 years after its first implementation (in the 1950's)? The paper you mention is written in the 2000s while the Simplex dominated the field from at least the 70's.
Meanwhile the GP would have discarded the simplex algorithm due to its worst case performance.
Honestly, the parent is pretty accurate. No one is claiming that P = NP. However, the technology to solved mixed integer programs has improved dramatically over the last 30 years and that improvement in performance has outpaced computational speed by multiple orders of magnitude. It's the algorithms.
I just went to pull up some numbers. The following comes from a talk that Bob Bixby gave at ISMP in 2012. He is the original author of CPLEX and one of the current owners of Gurobi. Between 1991 and 2007, CPLEX achieved a 29530x speedup in their solvers. Their 1997/98 breakthrough year attributes the following speedups, Cutting planes: 33.3x, presolve: 7.7x, variable selection: 2.7x, node presolve 1.3x, heuristics: 1.1x, dive probing 1.1x. I don't have a paper reference for these numbers and I don't think he has published them, but I was at the talk.
The point is that integer programming solvers perform unreasonably well. There is theory as to why. Yes, there is still a lot of searching. However, search in-and-of-itself is not sufficient to solve the problems that we regularly solve now. Further, that increase in performance is not just heuristics.
Laugh. Probably! I gave a talk at that conference titled, "Software Abstractions for Matrix-Free PDE Optimization with Cone Constraints." I still work in the field, so you want to talk algorithms sometime, feel free to send me an email. I keep my email off of HN to limit spam, but if you search for the lead author on that presentation, it should list my website.
> which what gave us the exponential speed ups we enjoy today
You're just patently wrong because
1. P=?NP is still open
2. You can consult the implementation of any solver
There's really nothing more to be said about it because both of these bullets are formal proofs that you are wrong.