Set "not equal to" constraint for a linear system of equations solver:

Hi, I'm looking to set a constraint for the solution to my linear system of equations

Cx=D

such that

x~=0

or

|x|>eps

I'm currently looking at lsqlin() but it seems that I am not able to do this. Would you please let me know how can I do this with lsqlin() or another function, if it is possible?

Thank you.

[Torsten]

Do you mean that at least one component of the x-vector should be different from 0 or all components should be different from 0 ?

[Ash Ash]

I'm looking to have all my components to be different from 0

[Walter Roberson]

What is your situation such that it would not be permitted for any component to be 0, but it would be acceptable for the component to be 4.94065645841247e-324 ?

Answer 1

[Walter Roberson]

There is no direct way to do that.

You need to run your solver multiple 2^N times where N is the length of your x vector. Each position in the vector needs to be considered with a lower bound of -inf and an upper bound of -eps(realmin), and then again with a lower bound of +eps(realmin) and an upper bound of +inf . Then out of all of those 2^N solutions, you have to somehow figure out which one is "best".

[Ash Ash]

I see, thank you.

Answer 2

[Matt J]

You may as well just find the solution x without this constraint and then add random noise.

x=x + eps.*(x==0).*rand(size(x));

You are not going to get a better solution by attempting rigorous numerical optimization, subject to |x(i)|>=eps.

Remember that these solvers are iterative approximators - you would have to iterate for ages to get within eps of the actual solution you are looking for, even assuming that floating point arithmetic errors would allow such precise convergence at all. And since you can't get within eps of the ideal solution, you may as well accept any solution that is eps or more away from the ideal solution, e.g., the solution without the |x(i)|>=eps constraint or some random perturbation of it.