python - Is this a unconstrained multivariable optimization problem, and best way to solve?

Question

I have a dataset that looks like this:

x1	x2	...	xn	y1	y2	...	yn	z1	z2	...	zn
50	40	3	80	-0.9	0.1	0.9	-0.3	0	0.5	0.9	0.8
20	10	8	20	-0.1	-0.5	0.8	-0.2	0.1	0.4	0.3	0.1

question from:https://stackoverflow.com/questions/66065417/is-this-a-unconstrained-multivariable-optimization-problem-and-best-way-to-solv

Answer 1

Not really a complete answer, but this is how I would start looking at a model for solving this.

I would start with introducing decision variables:

s(i) = 1 if row i is selected
       0 otherwise

w(i) ≥ 0 weight of row i  (not sure about the role of the weights)

The objective can look like:

min/max sum( (i,j)|j∈C, s(i)*w(i)*a(i,j) )

Here we could sum over a subset of all columns. a(i,j) is our data matrix.

The constraints can look like:

s(i)*20 ≤ s(i)*a(i,'x2') ≤ s(i)*40 
s(i)*(-.1) ≤ s(i)*a(i,'y') ≤ s(i)*.4 

This can be solved with a Mixed-Integer Programming Solver (as long as we keep things linear). The objective looks quadratic right now (s(i)*w(i)) but that can be linearized or we can use a non-convex MIQP solver.