Generalized Evolution Strategies for Mixed-Variable Optimization

An experimental Bachelor's Thesis (TFG) in Computer Science at the Facultat d'Informàtica de Barcelona (FIB - UPC).

This framework formalizes and implements a Generalized $(\mu, \lambda)$-Evolution Strategy designed for black-box optimization over heterogeneous search spaces. It co-evolves decision variables and internal strategy parameters with linear $\mathcal{O}(n)$ per-generation complexity, featuring native support for 6 distinct variable domains, dual discrete/intermediate recombination, and modular benchmark suites.

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Decision Variable Genome & Mutation Kernels

The framework natively co-evolves 6 decision variable domains alongside their respective strategy parameters within a unified genome representation:

Variable Type	Domain	Strategy Parameter	Mutation Operator	Mathematical Dynamics
Real ($\mathbb{R}$)	$\mathbb{R}^{N_\text{real}}$	$\boldsymbol{\sigma}_\text{real} \in \mathbb{R}_{>0}^{N_\text{real}}$	Gaussian	Specular reflection: $x' = \text{reflect}(x + \sigma \cdot \mathcal{N}(0,1), [L, U])$
Count ($\mathbb{Z}$)	$\mathbb{Z}_{\ge 0}^{N_\text{count}}$	$\boldsymbol{\lambda}_\text{count} \in \mathbb{R}_{>0}^{N_\text{count}}$	Symmetric Skellam	Difference of Poissons: $x' = \text{clip}(x + \text{Pois}_1(\lambda) - \text{Pois}_2(\lambda), [L, U])$
Binary ($\mathbb{B}$)	$\{0, 1\}^{N_\text{binary}}$	$\mathbf{z}_\text{bin} \in [-4.6, 0.0]^{N_\text{binary}}$	Bit-Flip (XOR)	Guided by Bernoulli($p$), where $p = \sigma(z) \in [0.01, 0.50]$
Categorical ($\mathbb{C}$)	$\prod_k \{0, \dots, K_k - 1\}$	$\mathbf{z}_\text{cat} \in [-4.6, 0.0]^{N_\text{cat}}$	Modular Jump	Jump offset: $(c + \Delta) \pmod K$ with prob. $p = \sigma(z)$, inertia $1-p$
Fuzzy ($\mathbb{F}$)	$([0, 1] \times [\epsilon, d_\text{max}])^{N_\text{fuzzy}}$	$\boldsymbol{\sigma}_\text{fuzzy} \in \mathbb{R}_{>0}^{2 \cdot N_\text{fuzzy}}$	Dual Gaussian Reflection	Perturb $(m, d)$: reflect $m' \in [0, 1]$, reflect & clamp $d' \in [\epsilon, d_\text{max}]$
Circular ($\mathbb{O}$)	$\prod_k \mathbb{Z}_{K_k}$	$\boldsymbol{\lambda}_\text{circ} \in \mathbb{R}_{>0}^{N_\text{circ}}$	Modular Skellam	Ring offset: $x' = (x + \text{Pois}_1(\lambda) - \text{Pois}_2(\lambda)) \pmod{K_k}$

Self-Adaptation Dynamics

Global & Coordinate Learning Rates:

$$ \tau' = \frac{1}{\sqrt{2 \cdot N_\text{total}}}, \quad \tau = \frac{1}{\sqrt{2 \cdot \sqrt{N_\text{total}}}} $$

Log-Normal Scale Adaptation ($\boldsymbol{\sigma}_\text{real}, \boldsymbol{\lambda}_\text{count}, \boldsymbol{\sigma}_\text{fuzzy}, \boldsymbol{\lambda}_\text{circ}$):

$$ \sigma_i' = \max\left(\sigma_\text{floor}, , \sigma_i \cdot \exp\left(\tau' \mathcal{N}(0, 1) + \tau \mathcal{N}_i(0, 1)\right)\right), \quad \sigma_\text{floor} = 10^{-6} $$

Logit-Normal Probability Adaptation ($\mathbf{z}_\text{bin}, \mathbf{z}_\text{cat}$):

$$ \mathbf{z}_i' = \mathbf{z}_i + \tau \mathcal{N}_i(0, \mathbf{I}) $$

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Quick Start with uv

The project is fully managed with uv for fast, reproducible, and lockfile-pinned builds.

1. Install uv (if needed)

# Windows (PowerShell)
irm https://astral.sh/uv/install.ps1 | iex

# Linux / macOS
curl -LsSf https://astral.sh/uv/install.sh | sh

2. Synchronize Environment

From the project root:

uv sync

This single command automatically resolves Python 3.12, creates .venv, installs all dependencies (numpy, matplotlib, cma, cmaes, scipy, pandas, pymoo, pytest, ipykernel), and registers the console scripts es-model and generalized-es.

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Running the Optimizer

Execute commands from the project root using uv run:

Interactive Menu (Recommended)

uv run es-model
# or: uv run generalized-es

Launches an interactive terminal menu with options:

1. model: Runs the Generalized ES optimization loop on the default mixed configuration.
2. benchmarks: Evaluates baseline optimizers (Isotropic ES, sep-CMA-ES, Standard CMA-ES).
3. smoke: Fast sanity verification with minimal budget (dim=5, evals=200).
4. all: Executes model followed by benchmarks.

Direct CLI Workflows

# Fast smoke validation
uv run es-model smoke

# Model execution with custom generations and logging frequency
uv run es-model model --generations 200 --log-interval 20

# Run continuous benchmarks (dim=20, 50,000 evaluations across 5 seeds)
uv run es-model benchmarks --dim 20 --max-evals 50000 --seeds 5

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Experiments & Notebooks

In-depth empirical notebooks are available under experiments/:

• experiment_1.ipynb: Equal-budget continuous benchmark comparisons against CMA-ES and Isotropic ES.
• experiment_2.ipynb: Target Shift (TS) tracking and mutation self-adaptation across all 6 variable types in isolation.
• experiment_3.ipynb: Simultaneous co-evolution across all 6 variable types on synthetic heterogeneous landscapes (Sphere, Coupled, Rastrigin, Multiplicative), including 4-way baseline comparisons against Uniform Random Search, Continuous Relaxation CMA-ES with rounding, and State-of-the-Art CatCMA with Margin (CatCMAwM, 2025).

To launch a Jupyter session:

uv run jupyter lab

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Testing & Verification

Execute the test suite with pytest:

uv run pytest

Statistical Hypothesis Testing

Execute the paired two-sided Wilcoxon signed-rank tests with Holm-Bonferroni correction ($\alpha = 0.05$) and matched-pairs rank-biserial effect sizes:

# Experiment 3: Evaluate against external baselines (Random Search, CMA-ES, CatCMAwM)
uv run python experiments/experiment_3/runners/run_experiment_3.py --load-precomputed

# Experiment 1: Evaluate against canonical continuous baselines
uv run python experiments/experiment_1/runners/run_experiment_1.py

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Architecture & Technical Documentation

For detailed architectural specifications, execution flowcharts, candidate genome diagrams, the full directory structure, and the complete module dependency matrix, see:

👉 Architecture & Technical Documentation

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License

This project is licensed under the MIT License - see the LICENSE file for details.