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Paref - using and building problem tailored MOO

A multi-objective optimization (MOO) problem comes with an idea of what properties identified (Pareto) points must satisfy. The fact that these characteristics are fulfilled is what makes a MOO successful in the first place. Why not construct a MOO algorithm which searches exactly for those properties? With the language of PAreto REFlections this is now possible. This package contains

• a series of ready-to-use [MOO algorithms](table below) corresponding to frequently targeted properties
• a framework for you to implement your problem specific MOO algorithm
• generic and intuitive interfaces for MOO algorithms, blackbox functions and more such that solving an MOO problem with Paref requires minimal effort
• documentation including how-to guides, examples, check sheets and Parefs' architecture so that getting started with and using Paref becomes easy

See the official documentation for more information.

Content

• Installation
• Usage
• What are properties of Pareto points?

Installation

The official release is available at PyPi:

pip install paref

You can clone this repository by running the following command:

git clone https://github.com/nicolaipalm/paref
cd paref
pip install

In your python terminal run (hello world where left side of w is pareto front)

from paref.

Usage

💡 See the how-to guides for a more detailled description.

Solving an MOO problem with Paref consists of the following steps

0. Define design and target space
1. Define desired properties of Pareto points
2. Initialize corresponding [MOO algorithm](link moo algos)
3. [Implement and initialize bbf](link how to implement bbf)
4. [Apply problem tailored MOO algorithm to blackbox function](link how to apply moo)

This may look as follows:

0. We use a mathematical test function with three input dimensions all between zero and one (i.e. design space is given by three-dimensional unit cube) and with two output dimensions (i.e. target space is the real plane)
1. We want to have an idea of the "dimension" of the Pareto front (i.e. the Pareto points representing the minima in components) with minimum number of evaluations
2. Accordingly, we choose the FindEdgePoints algorithm:

from paref.moo_algorithms.multi_dimensional.find_edge_points import FindEdgePoints
moo = FindEdgePoints()

3. We implement and initialize the blackbox function in the Paref blackbox function interface

import numpy as np
from paref.black_box_functions.design_space.bounds import Bounds
from paref.interfaces.moo_algorithms.blackbox_function import BlackboxFunction

class TestFunction(BlackboxFunction):
    def __call__(self, x: np.ndarray) -> np.ndarray:
        y = np.array([np.sum(x**2),x[0]])
        self._evaluations.append([x, y])
        return y

    @property
    def dimension_design_space(self) -> int:
        return 3

    @property
    def dimension_target_space(self) -> int:
        return 2

    @property
    def design_space(self) -> Bounds:
        return Bounds(upper_bounds=np.ones(self.dimension_design_space),
                      lower_bounds=-np.zeros(self.dimension_design_space))

blackbox_function = TestFunction()

4. We apply the MOO algorithm to the blackbox function with a maximum number of five iterations and print the so found Pareto front:

from paref.moo_algorithms.stopping_criteria.max_iterations_reached import MaxIterationsReached
moo(blackbox_function = blackbox_function,
    stopping_criteria = MaxIterationsReached(max_iterations=5))
print(f"Calculated Pareto front: {blackbox_function.pareto_front}")

What are properties of Pareto points?

A MOO problem comes with an idea of what properties identified (Pareto) points must satisfy. The fact that these characteristics are fulfilled is what makes a MOO successful in the first place.

In mathematical terms, we understand properties of Pareto points as being element of a (mostly implicit defined) subset of the Pareto front.

They include but are certainly not limited to the following:

Property	Graphic	Example	Algorithm(s)	Sequence	Pareto reflection
Being an edge point
Filling a gap
Being evenly distributed
Being constrained to a defined area