.. _user_guide_optimization:

============
Optimization
============

.. include:: ../../global_refs.rst

📖 :ref:`Optimization API Reference <api_optimization>`

Given that the main purpose of the SigmaEpsilon ecosystem is to facilitate structural optimization, this submodule needs no further elaboration, as having a capable optimization toolkit is essential for the project. The overarching approach is to utilize third-party solutions whenever feasible, though we also provide essential solutions when necessary or fill in the gaps wherever required.

.. note::

   Most of the classes for optimization rely on other classes like :class:`~sigmaepsilon.math.function.function.Function` or :class:`~sigmaepsilon.math.function.relation.Relation`. Before you proceed forward, it might be a good idea to check them out first :ref:`here <user_guide_function>`.

Linear Programming (LP)
========================

📖 :ref:`Linear Programming API Reference <api_optimization_LP>`

The library offers a solver for a wide range of linear optimization problems, including continuous, integer and mixed-integer problems, built around `SciPy`_'s `linprog` and `SymPy`_. See :doc:`lp` for the basic API, a 2d example visualized with Matplotlib, and mixed-integer programs.

Nonlinear Programming (NLP)
=============================

📖 :ref:`Nonlinear Programming API Reference <api_optimization_NLP>`

Nonlinear programming deals with problems where the objective function, the constraints, or both are nonlinear. The library provides a small family of genetic algorithms (GAs) for this purpose:

* :doc:`bga` -- an introduction to genetic algorithms in general, and a full walkthrough of the Binary Genetic Algorithm (BGA).
* :doc:`ga_variants` -- the Real-Valued (RGA) and Integer (IGA) genetic algorithms.
* :doc:`ga_tuning` -- pluggable selection strategies and monitoring population diversity.
* :doc:`ga_performance` -- vectorized and parallel objective evaluation, choosing a representation, and posing objectives symbolically.
* :doc:`custom_ga` -- a full, worked example of subclassing the framework for a custom representation.

Ant Colony Optimization (ACO)
===============================

📖 :ref:`Ant Colony Optimization API Reference <api_optimization_ACO>`

**Ant Colony Optimization (ACO)** is another nature-inspired metaheuristic, this time based
on the pheromone-trail-following behavior of foraging ants. Unlike the GA family, which
evolves a population of candidate solutions directly, ACO builds solutions incrementally
(step by step) and steers the construction process with a shared memory (the "pheromones")
that gets reinforced along good solutions and decays over time. See :doc:`aco` for how the
continuous (ACOR) and combinatorial (Ant System / TSP) variants work, when to reach for them
instead of a GA or `linprog`, and worked examples of both.

.. toctree::
    :maxdepth: 1
    :hidden:

    lp
    bga
    ga_variants
    ga_tuning
    ga_performance
    custom_ga
    aco