Optimization in Python with Multiple Functions and a Linear Constraint (Linear Programming Problem Query)

Question

I'm a little new to optimization in Python, particularly on Scipy. If I only have one function such as this:

def prod(x):
    return -x[0] * x[1]

Then I can easily know how to do maximization through:

linear_constraint = LinearConstraint([[1,1]], [20], [20])
x0 = np.array([15,5])
result = minimize(prod, x0, constraints=linear_constraint)
print(result['x'])

This is basically maximizing x0 * x1 given the constraint that x0 + x1 = 20. The answer here is array([10., 10.]).

So now, I wish to do something more sophisticated and not just optimizing one function. Suppose I would like to optimize multiple functions. Say I have this matrix P of size n x 3:

A     B     C
3     2     7
6     3     4
8     1     5
...

Is there a way to optimize (minimize) P[i] * x, where * here is a dot product for all i = 1..n? In short, I wish to optimize:

3*x[0] + 2*x[1] + 7*x[2]
6*x[0] + 3*x[1] + 4*x[2]
8*x[0] + 1*x[1] + 5*x[2]
...

and under the constraint that x[0] + x[1] + x[2] = 1. Anyone knows how to implement this correctly on Python (I'm using Scipy by the way)? I'm still not sure where to begin. Thanks in advance!

Erwin Kalvelagen · Accepted Answer · 2021-07-05 05:02:23Z

1

For standard optimization tools, the objective function must return a scalar. This is called a "single objective". There is a branch in optimization that deals with multiple objectives (with names like vector optimization, multiple objective optimization and multiple criteria optimization). There is a wide literature about this. I suggest doing some reading to get familiar with the concepts, ideas and algorithms for this.

answered Jul 5, 2021 at 5:02

Erwin Kalvelagen

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4 Comments

M. Fire Over a year ago

I thought about obtaining a vector by multiplying this array (or matrix) of size n x 3 by the 3 x 1 vector [x0, x1, x2]. As a result, I would get an n x 1 vector. And then find the L2-Norm of this vector. Then minimize on that norm (scalar); as opposed to trying to minimize multiple functions all at once. Not sure if that's the right way to do it... Thoughts?

Erwin Kalvelagen Over a year ago

I don't know. That is just a very different problem, You stated a multi-objective problem. If you do min ||Px|| you have a single objective problem.

M. Fire Over a year ago

Hmmm, but why does it have to be a multi-objective problem? Can't I just frame it into a single objective problem? So basically computing Px just gives me a vector. And if I want to "minimize" my vector, then I'll do so by find the minimum of the L2 norm of the resultant vector.

Erwin Kalvelagen Over a year ago

Sure. You can redefine the objective to anything you want. It would be one of the possible scalarizations.

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