Pymoo Algorithms Reference

Reference for pymoo 0.6.2. Selection guidance is heuristic; compare seeds at matched evaluation budgets. Continuous defaults do not automatically support every variable type.

Single-Objective Optimization Algorithms

Genetic Algorithm (GA)

Purpose: General-purpose single-objective evolutionary optimization Best for: Continuous problems with default operators; discrete types require compatible operators Algorithm type: (μ+λ) genetic algorithm

Key parameters:

• pop_size: Population size (default: 100)
• sampling: Initial population generation strategy
• selection: Parent selection mechanism (default: Tournament)
• crossover: Recombination operator (default: SBX)
• mutation: Variation operator (default: Polynomial)
• eliminate_duplicates: Remove redundant solutions (default: True)
• n_offsprings: Offspring per generation

Usage:

from pymoo.algorithms.soo.nonconvex.ga import GA
algorithm = GA(pop_size=100, eliminate_duplicates=True)

Differential Evolution (DE)

Purpose: Single-objective continuous optimization Best for: Continuous parameter optimization with good global search Algorithm type: Population-based differential evolution

Variants: Multiple DE strategies available (rand/1/bin, best/1/bin, etc.)

Particle Swarm Optimization (PSO)

Purpose: Single-objective optimization through swarm intelligence Best for: Continuous problems, fast convergence on smooth landscapes

CMA-ES

Purpose: Covariance Matrix Adaptation Evolution Strategy Best for: Continuous optimization, particularly for noisy or ill-conditioned problems

Pattern Search

Purpose: Direct search method Best for: Problems where gradient information is unavailable

Nelder-Mead

Purpose: Simplex-based optimization Best for: Local optimization of continuous functions

MixedVariableGA

Purpose: Single-objective optimization with mixed variable types Best for: Problems with continuous, integer, binary, and categorical variables

Usage:

from pymoo.core.mixed import MixedVariableGA
from pymoo.core.variable import Real, Integer, Choice, Binary

# Define problem with vars dict (see mixed-variable docs)
algorithm = MixedVariableGA(pop_size=20)

For multi-objective mixed-variable problems, pass a survival operator:

from pymoo.operators.survival.rank_and_crowding import RankAndCrowding
algorithm = MixedVariableGA(pop_size=20, survival=RankAndCrowding())

Optuna (Mixed-Variable SOO)

Purpose: Single-objective mixed-variable search via Optuna wrapper Best for: Hyperparameter-style mixed search when Optuna's TPE/samplers are preferred

Usage:

from pymoo.algorithms.soo.nonconvex.optuna import Optuna
algorithm = Optuna()

Requires Optuna installed separately: uv pip install optuna

Multi-Objective Optimization Algorithms

NSGA-II (Non-dominated Sorting Genetic Algorithm II)

Purpose: Multi-objective optimization with 2-3 objectives Best for: Bi- and tri-objective problems requiring well-distributed Pareto fronts Selection strategy: Non-dominated sorting + crowding distance

Key features:

• Fast non-dominated sorting
• Crowding distance for diversity
• Elitist approach
• Binary tournament mating selection

Key parameters:

• pop_size: Population size (default: 100)
• sampling: Initial population strategy
• crossover: Default SBX for continuous
• mutation: Default Polynomial Mutation
• survival: RankAndCrowding

Usage:

from pymoo.algorithms.moo.nsga2 import NSGA2
algorithm = NSGA2(pop_size=100)

When to use:

• 2-3 objectives
• Need for distributed solutions across Pareto front
• Standard multi-objective benchmark

SPEA2 (Strength Pareto Evolutionary Algorithm 2)

Purpose: Multi-objective optimization using strength and density survival Best for: Bi- and tri-objective problems; alternative to NSGA-II for strength/density selection Selection strategy: Strength-based fitness + k-nearest-neighbor density estimation

Key features:

• Strength/density survival in the current population; no separate persistent external archive is configured by this constructor
• Strength value measures how many solutions a point dominates
• Improved in pymoo 0.6.1.6

Usage:

from pymoo.algorithms.moo.spea2 import SPEA2
algorithm = SPEA2(pop_size=100)

When to use:

• 2-3 objectives
• Prefer strength/density survival over crowding distance
• Compare against NSGA-II on benchmark problems

NSGA-III

Purpose: Many-objective optimization (4+ objectives) Best for: Problems with 4 or more objectives requiring uniform Pareto front coverage Selection strategy: Reference direction-based diversity maintenance

Key features:

• Reference directions guide population
• Maintains diversity in high-dimensional objective spaces
• Niche preservation through reference points
• Underrepresented reference direction selection

Key parameters:

• ref_dirs: Reference directions (REQUIRED)
• pop_size: Defaults to number of reference directions
• crossover: Default SBX
• mutation: Default Polynomial Mutation

Usage:

from pymoo.algorithms.moo.nsga3 import NSGA3
from pymoo.util.ref_dirs import get_reference_directions

ref_dirs = get_reference_directions("das-dennis", 4, n_partitions=3)  # n_dim is positional
algorithm = NSGA3(ref_dirs=ref_dirs)

NSGA-II vs NSGA-III:

• Use NSGA-II for 2-3 objectives
• Use NSGA-III for 4+ objectives
• NSGA-III aims to spread candidates over chosen directions; coverage is not guaranteed
• NSGA-II has lower computational overhead

R-NSGA-II (Reference Point Based NSGA-II)

Purpose: Multi-objective optimization with preference articulation Best for: When decision maker has preferred regions of Pareto front

U-NSGA-III (Unified NSGA-III)

Purpose: Improved version handling various scenarios Best for: Many-objective problems with additional robustness

MOEA/D (Multi-Objective Evolutionary Algorithm based on Decomposition)

Purpose: Decomposition-based multi-objective optimization Best for: Unconstrained problems where decomposition into scalar subproblems is effective

Supply ref_dirs explicitly (the constructor uses their count). The 0.6.2 implementation rejects constraints. Choose n_neighbors no greater than the direction count and compatible with parent selection. Transforming constraints into objectives changes the problem and requires final feasibility checks.

AGE-MOEA

Purpose: Adaptive geometry estimation Best for: Multi and many-objective problems with adaptive mechanisms

RVEA (Reference Vector guided Evolutionary Algorithm)

Purpose: Reference vector-based many-objective optimization Best for: Many-objective problems with adaptive reference vectors

SMS-EMOA

Purpose: S-Metric Selection Evolutionary Multi-objective Algorithm Best for: Problems where hypervolume indicator is critical Selection: Uses dominated hypervolume contribution

Dynamic Multi-Objective Algorithms

D-NSGA-II

Purpose: Dynamic multi-objective problems Best for: Time-varying objective functions or constraints

KGB-DMOEA

Purpose: Knowledge-guided dynamic multi-objective optimization Best for: Dynamic problems leveraging historical information

Constrained Optimization

SRES (Stochastic Ranking Evolution Strategy)

Purpose: Single-objective constrained optimization Best for: Heavily constrained problems

ISRES (Improved SRES)

Purpose: Enhanced constrained optimization Best for: Complex constraint landscapes

Algorithm Selection Guidelines

For single-objective problems:

• Start with GA for general problems
• Use DE for continuous optimization
• Try PSO for faster convergence on smooth problems
• Use CMA-ES for difficult/noisy landscapes

For multi-objective problems:

• 2-3 objectives: NSGA-II or SPEA2
• 4+ objectives: NSGA-III
• Preference articulation: R-NSGA-II
• Decomposition-friendly: MOEA/D
• Hypervolume focus: SMS-EMOA

For constrained problems:

• Check algorithm support: feasibility-based survival is common, but MOEA/D rejects constrained problems
• Heavy constraints: SRES/ISRES
• Penalty methods for algorithm compatibility

For dynamic problems:

• Time-varying: D-NSGA-II
• Historical knowledge useful: KGB-DMOEA

For Das-Dennis directions, get_reference_directions("das-dennis", m, n_partitions=p) creates C(p+m-1,m-1) points on the simplex. With five objectives, p=3 gives 35 directions whereas p=12 gives 1820; NSGA-III defaults its population to this count. Reference directions are not feasible decision vectors or guaranteed Pareto points. NSGA-III also works with two/three objectives; four objectives is a routing heuristic, not an API minimum.

Reviewed 2026-10-01 against algorithm catalogue, NSGA-III, MOEA/D, mixed variables, and released source. GA/DE/PSO, mixed/Optuna, NSGA-II/III, and MOEA/D received bounded native checks; the other algorithms are documentation/source-reviewed choices, not performance-validated recommendations.