Question

Summary: How would I go about solving this problem?

Hi there,

I’m working on a mixture-style maximization problem where my variables are going to be bounded by minima and maxima. A representative example of my problem might be:

maximize: (2x-3y+4z)/(x^2+y^2+z^2+3x+4y+5z+10)
subj. to: x+y+z=1
          1 < x < 2
         -2 < y < 3
          5 < z < 8
where numerical coefficients and the minima/maxima are given.

My final project is involving a more complicated problem similar to the one above. The structure of the problems won’t change- only the coefficients and inputs will change. So with the example above, I would be looking for a set of functions that might allow a C# program to quickly determine x, then y, then z like:

x = f(given inputs)
y = f(given inputs,x)
z = f(given inputs,x,y)

Would love to hear your thoughts on this one!

Thanks!

Answer 1

The standard optimization approach for your type of problem, non-linear minimization, is the Levenberg-Marquardt algorithm:

• Levenberg–Marquardt algorithm

but unfortunately it does not directly support the linear constraints you have added. Many different approaches have been tried to add linear constraints to Levenberg-Marquardt with varying success.

Another algorithm I can recommend in this situation is the Simplex algorithm:

• Nelder–Mead method

Like the Levenberg-Marquardt, it also works with non-linear equations but handles linear constraints which act like discontinuities. This could work well for your case above.

In either case, this is not so much a programming problem as an algorithm selection problem. The literature is rife with algorithms and you can find C# implementations of either of the above with a little searching.

You can also combine algorithms. For example, you can do a preliminary search with Simplex with the constraints and the refine it with Levenberg-Marquardt without the constraints.