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stncls | 9 months ago
In theory, the simplex method is not known to be polynomial-time, and it is likely that indeed it is not. Some variants of the simplex method have been proven to take exponential time in some worst cases (Klee-Minty cubes). What solvers implement could be said to be one such variant ("steepest-edge pricing"), but because solvers have tons of heuristics and engineering, and also because they work in floating-point arithmetic... it's difficult to tell for sure.
In practice, the main alternative is interior-point (aka. barrier) methods which, contrary to the simplex method, are polynomial-time in theory. They are usually (but not always) faster, and their advantage tends to increase for larger instances. The problem is that they are converging numerical algorithms, and with floating-point arithmetic they never quite 100% converge. By contrast, the simplex method is a combinatorial algorithm, and the numerical errors it faces should not accumulate. As a result, good solvers perform "crossover" after interior-point methods, to get a numerically clean optimal solution. Crossover is a combinatorial algorithm, like the simplex method. Unlike the simplex method though, crossover is polynomial-time in theory (strongly so, even). However, here, theory and practice diverge a bit, and crossover implementations are essentially simplified simplex methods. As a result, in my opinion, calling iterior-point + crossover polynomial-time would be a stretch.
Still, for large problems, we can expect iterior-point + crossover to be faster than the simplex method, by a factor 2x to 10x.
There is also first-order methods, which are getting much attention lately. However, in my experience, you should only use that if you are willing to tolerate huge constraint violations in the solution, and wildly suboptimal solutions. Their main use case is when other solvers need too much RAM to solve your instance.
underanalyzer|9 months ago