# 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
``` default
pip install optipoly
```
[Full description of the module is available](https://mazenalamir.github.io/optipoly/).
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 `xmin`and `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, `optipoly`can be used as well to find a maximum or to find a root of the polynomial.
>
[Full description of the module is available](https://mazenalamir.github.io/optipoly/).
## 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:
```python
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:
```python
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 `psi`to `psi=lambda v:abs(v)` asks the solver to zero the polynomial and hence, leads to the following results:
```python
>> 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:
```python
>> xopt = [-1. -1. 2.]
>> fopt = -6.0
>> computation time = 0.005150318145751953
```
[Full description of the module is available](https://mazenalamir.github.io/optipoly/).
## Citing optipoly
``` bibtex
@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.
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"description": "# optipoly\n\n`optipoly` is a non conventional optimizer dedicated to the optimization\nof multi-variate polynomial cost functions on admissible hypercubes. It\nleverages the extraordinarily fast computation of scalar polynomials by\nthe `scipy.interpolation.interp1d` method in oder to derive a robust to\nlocal minima solution of the polynomial optimization problem.\n\nInterested reader can refer to the citation provided \nfor a complete description and comparison with some existing solvers.\n\nHere, the main elements are explained briefly for an easy and quick use\nof the solver.\n\n## Installation\n\n``` default\npip install optipoly\n```\n\n[Full description of the module is available](https://mazenalamir.github.io/optipoly/).\n\nHere a very short presentation is given. \n\n## The problem addressed\n\nGiven a polynomial in several variables of the form:\n\n$$\nP(x)=\\sum_{i=1}^{n_c}c_i\\phi_i(x)\\quad \\text{where} \\quad \\phi_i(x):=\\prod_{j=1}^n x_j^{p_ij}\n$$\n\nAssume that we want to find the minimum value of the polynomial over some hyper box defined by `xmin`and `xmax`, namely\n\n$$\n\\min_{x} P(x) \\quad \\text{$\\vert\\quad x\\in [x_\\text{min}, x_\\text{max}]\\subset \\mathbb R^n$}\n$$\n\nThis is the kind of porblem `optipoly` is done for. \n\n> **Tip**\n> \n> Using specific choice of the call inputs, `optipoly`can be used as well to find a maximum or to find a root of the polynomial.\n>\n\n[Full description of the module is available](https://mazenalamir.github.io/optipoly/).\n\n## Example of use\n\nConsider the polynomial in three variables defined by:\n\n$$\nP(x) = x_1x_3^2+2x_2^3\n$$\n\nAn instance of the class `Pol` that represent this polynomial can be created via the following script:\n\n```python \nfrom optipoly import Pol\n\n# Define the matrix of powers and c.\n \npowers = [[1, 0, 2], [0,3,0]] \ncoefs = [1.0, 2.0] \n\n# Create an instance of the class.\n\npol = Pol(powers, coefs) \n```\n\nThe 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:\n\n```python\nnx = 3\nx0 = np.zeros(nx)\nntrials = 6\nngrid = 1000\nxmin = -1*np.ones(nx)\nxmax = 2*np.ones(nx)\n\nsolution, cpu = pol.solve(x0=x0, \n xmin=xmin, \n xmax=xmax, \n ngrid=ngrid, \n Ntrials=ntrials, \n psi=lambda v:-v\n )\n \nprint(f'xopt = {solution.x}')\nprint(f'fopt = {solution.f}')\nprint(f'computation time = {solution.cpu}')\n\n>> xopt = [-1. 2. 0.]\n>> fopt = 16.0\n>> computation time = 0.0046999454498291016\n```\n\nChanging the argument `psi`to `psi=lambda v:abs(v)` asks the solver to zero the polynomial and hence, leads to the following results:\n\n```python\n>> xopt = [-0.996997 0.58858859 0.63963964]\n>> fopt = -9.305087356087371e-05\n>> computation time = 0.003011941909790039\n```\n\nFinally, using the default definition leads to `solve` trying to find a minimum of the polynomial leading to:\n\n```python \n>> xopt = [-1. -1. 2.]\n>> fopt = -6.0\n>> computation time = 0.005150318145751953\n```\n\n[Full description of the module is available](https://mazenalamir.github.io/optipoly/).\n\n## Citing optipoly\n\n``` bibtex\n@misc{optipoly2024,\n title={optipoly: A Python package for boxed-constrained multi-variable polynomial cost functions optimization}, \n author={Mazen Alamir},\n year={2024},\n eprint={5987757},\n archivePrefix={arXiv},\n primaryClass={eess.SY},\n url={https://arxiv.org/submit/5987757}, \n}\n```\n\n\n> **Tip**\n>\n> The above reference contains some comparison with alternative solver\n> that underlines the performance of `optipoly` in terms of the achieved\n> cost as well as in terms of the compuation time.\n",
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