optipoly

optipoly is a non conventional optimizer dedicated to the optimization of multi-variate polynomial cost functions on admissible hypercubes. It leverages the extraordinarily fast computation of scalar polynomials by the scipy.interpolation.interp1d method in oder to derive a robust to local minima solution of the polynomial optimization problem.

Interested reader can refer to the citation provided for a complete description and comparison with some existing solvers.

Here, the main elements are explained briefly for an easy and quick use of the solver.

Installation

pip install optipoly

Full description of the module is available.

Here a very short presentation is given.

The problem addressed

Given a polynomial in several variables of the form:

$$ P(x)=\sum_{i=1}^{n_c}c_i\phi_i(x)\quad \text{where} \quad \phi_i(x):=\prod_{j=1}^n x_j^{p_ij} $$

Assume that we want to find the minimum value of the polynomial over some hyper box defined by xminand xmax, namely

$$ \min_{x} P(x) \quad \text{$\vert\quad x\in [x_\text{min}, x_\text{max}]\subset \mathbb R^n$} $$

This is the kind of porblem optipoly is done for.

Tip

Using specific choice of the call inputs, optipolycan be used as well to find a maximum or to find a root of the polynomial.

Full description of the module is available.

Example of use

Consider the polynomial in three variables defined by:

$$ P(x) = x_1x_3^2+2x_2^3 $$

An instance of the class Pol that represent this polynomial can be created via the following script:

from optipoly import Pol

# Define the matrix of powers and c.
 
powers = [[1, 0, 2], [0,3,0]] 
coefs = [1.0, 2.0]            

# Create an instance of the class.

pol = Pol(powers, coefs)      

The following script gives an example of a call that asks for the maximization of the polynomial defined earlier (see @eq-examplePx) then prints the results so obtained:

nx = 3
x0 = np.zeros(nx)
ntrials = 6
ngrid = 1000
xmin = -1*np.ones(nx)
xmax = 2*np.ones(nx)

solution, cpu = pol.solve(x0=x0, 
                          xmin=xmin, 
                          xmax=xmax, 
                          ngrid=ngrid, 
                          Ntrials=ntrials, 
                          psi=lambda v:-v
                          )
                          
print(f'xopt = {solution.x}')
print(f'fopt = {solution.f}')
print(f'computation time = {solution.cpu}')

>> xopt = [-1.  2.  0.]
>> fopt = 16.0
>> computation time = 0.0046999454498291016

Changing the argument psito psi=lambda v:abs(v) asks the solver to zero the polynomial and hence, leads to the following results:

>> xopt = [-0.996997    0.58858859  0.63963964]
>> fopt = -9.305087356087371e-05
>> computation time = 0.003011941909790039

Finally, using the default definition leads to solve trying to find a minimum of the polynomial leading to:

>> xopt = [-1. -1.  2.]
>> fopt = -6.0
>> computation time = 0.005150318145751953

Full description of the module is available.

Citing optipoly

@misc{optipoly2024,
      title={optipoly: A Python package for boxed-constrained multi-variable polynomial cost functions optimization}, 
      author={Mazen Alamir},
      year={2024},
      eprint={5987757},
      archivePrefix={arXiv},
      primaryClass={eess.SY},
      url={https://arxiv.org/submit/5987757}, 
}

Tip

The above reference contains some comparison with alternative solver that underlines the performance of optipoly in terms of the achieved cost as well as in terms of the compuation time.