Numerical stability of Simplex Algorithm

Question

Edit : Simplex the mathematical optimization algorithm, not to be confused with simplex noise or triangulation.

I'm implementing my own linear programming solver and I would like to do so using 32bit floats. I know Simplex is very sensitive to the precision of the numbers because it performs lots of calculations and if too little precision is used, rounding errors may occur. But still, I would like to implement it using 32bit floats so I can make the instructions 4-wide, that is, so I can use SIMD to perform 4 calculations at a time. I'm aware that I could use doubles and make instructions 2-wide, but 4 is greater than 2 :)

I have had problems with my floating point implementation where the solution was suboptimal or the problem was said to be unfeasible. This happens especially with mixed integer linear programs, which I solve with the branch and bound method.

So my question is: how can I prevent as much as possible having rounding errors resulting in unfeasible, unbounded or suboptimal solutions?

I know one thing I can do is to scale the input values so that they are close to one ( http://lpsolve.sourceforge.net/5.5/scaling.htm ). Is there something else I can do?

Answer 1

Yes, I tried to implement an algorithm for the Extended Knapsack problem using the Branch and bound method and Greedy Algorithm as a heuristic, is the exact analogue of the simplex running with a pivoting strategy that chooses the largest objective increase.

I had problems with the numerical stabilities of the algorithm too.

Personally, I don't think there is an easy way to eliminate the issues if we keep using the floating-point, but there is a way to detect the instability during the branching process.

I think, via experiment instead of rigorous maths on Stability Analysis, the majority of errors propagate through an integer solution that is extremely close to the constraints of the system.

Given any integer solution, we figure out the slack for that solution, and if the elements of the vector are extremely small, or on the magnitude of 1e-14 to 1e-15, then stop the branching and report instability.