Pymoo

Pymoo - Multi-Objective Optimization in Python

Overview

Pymoo is a comprehensive Python framework for optimization with emphasis on multi-objective problems. Solve single and multi-objective optimization using state-of-the-art algorithms (NSGA-II/III, MOEA/D, SPEA2), benchmark problems (ZDT, DTLZ), customizable genetic operators, and multi-criteria decision making methods. Excels at finding trade-off solutions (Pareto fronts) for problems with conflicting objectives. Current stable release: pymoo 0.6.1.6 (November 2025).

Installation

uv pip install pymoo

For reproducible environments, pin a version: uv pip install "pymoo==0.6.1.6".

Dependencies: NumPy (2.x compatible since 0.6.1.3), SciPy, matplotlib (visualization). Autograd is optional for gradient-based features (since 0.6.1.3).

Documentation: https://pymoo.org/ — LLM-friendly index: https://pymoo.org/llms.txt

When to Use This Skill

This skill should be used when:

• Solving optimization problems with one or multiple objectives
• Finding Pareto-optimal solutions and analyzing trade-offs
• Implementing evolutionary algorithms (GA, DE, PSO, NSGA-II/III)
• Working with constrained optimization problems
• Benchmarking algorithms on standard test problems (ZDT, DTLZ, WFG)
• Customizing genetic operators (crossover, mutation, selection)
• Visualizing high-dimensional optimization results
• Making decisions from multiple competing solutions
• Handling binary, discrete, continuous, or mixed-variable problems

Core Concepts

The Unified Interface

Pymoo uses a consistent minimize() function for all optimization tasks:

from pymoo.optimize import minimize

result = minimize(
    problem,        # What to optimize
    algorithm,      # How to optimize
    termination,    # When to stop
    seed=1,
    verbose=True
)

Result object contains:

• result.X: Decision variables of optimal solution(s)
• result.F: Objective values of optimal solution(s)
• result.G: Constraint violations (if constrained)
• result.algorithm: Algorithm object with history

Problem Definition Styles

Pymoo supports three problem definition styles:

• Problem: Vectorized — _evaluate receives a batch of solutions (matrix)
• ElementwiseProblem: One solution per call — recommended for custom problems and parallel evaluation
• FunctionalProblem: Define objectives and constraints as separate functions without subclassing

Problem Types

Single-objective: One objective to minimize/maximize

Multi-objective: 2-3 conflicting objectives → Pareto front

Many-objective: 4+ objectives → High-dimensional Pareto front

Constrained: Objectives + inequality/equality constraints

Mixed-variable: Continuous, integer, binary, and categorical variables in one problem

Dynamic: Time-varying objectives or constraints

Quick Start Workflows

Nine runnable workflows are in

references/quick_start_workflows.md:

| # | Workflow | Use when |

| --- | --- | --- |

| 1 | Single-objective optimization | one objective, GA or DE |

| 2 | Multi-objective (2-3 objectives) | NSGA-II and a Pareto front |

| 3 | Many-objective (4+ objectives) | NSGA-III or reference-direction methods |

| 4 | Custom problem definition | subclassing Problem / ElementwiseProblem |

| 5 | Constraint handling | inequality and equality constraints |

| 6 | Decision making from a Pareto front | scalarization and MCDM selection |

| 7 | Visualization | scatter, PCP, radviz, and heatmap views |

| 8 | Parallel evaluation | threads, processes, or Dask for expensive objectives |

| 9 | Mixed-variable optimization | integer, binary, and categorical variables |

Algorithm Selection Guide

Single-Objective Problems

| Algorithm | Best For | Key Features |

|-----------|----------|--------------|

| GA | General-purpose | Flexible, customizable operators |

| DE | Continuous optimization | Good global search |

| PSO | Smooth landscapes | Fast convergence |

| CMA-ES | Difficult/noisy problems | Self-adapting |

Multi-Objective Problems (2-3 objectives)

| Algorithm | Best For | Key Features |

|-----------|----------|--------------|

| NSGA-II | Standard benchmark | Fast, reliable, well-tested |

| SPEA2 | Archive-based MOO | Strength-based fitness, external archive |

| R-NSGA-II | Preference regions | Reference point guidance |

| MOEA/D | Decomposable problems | Scalarization approach |

Many-Objective Problems (4+ objectives)

| Algorithm | Best For | Key Features |

|-----------|----------|--------------|

| NSGA-III | 4-15 objectives | Reference direction-based |

| RVEA | Adaptive search | Reference vector evolution |

| AGE-MOEA | Complex landscapes | Adaptive geometry |

Constrained Problems

| Approach | Algorithm | When to Use |

|----------|-----------|-------------|

| Feasibility-first | Any algorithm | Large feasible region |

| Specialized | SRES, ISRES | Heavy constraints |

| Penalty | GA + penalty | Algorithm compatibility |

See: references/algorithms.md for comprehensive algorithm reference

Benchmark Problems

Quick problem access:

from pymoo.problems import get_problem

# Single-objective
problem = get_problem("rastrigin", n_var=10)
problem = get_problem("rosenbrock", n_var=10)

# Multi-objective
problem = get_problem("zdt1")        # Convex front
problem = get_problem("zdt2")        # Non-convex front
problem = get_problem("zdt3")        # Disconnected front

# Many-objective
problem = get_problem("dtlz2", n_obj=5, n_var=12)
problem = get_problem("dtlz7", n_obj=4)

See: references/problems.md for complete test problem reference

Genetic Operator Customization

Standard operator configuration:

from pymoo.algorithms.soo.nonconvex.ga import GA
from pymoo.operators.crossover.sbx import SBX
from pymoo.operators.mutation.pm import PM

algorithm = GA(
    pop_size=100,
    crossover=SBX(prob=0.9, eta=15),
    mutation=PM(eta=20),
    eliminate_duplicates=True
)

Operator selection by variable type:

Continuous variables:

• Crossover: SBX (Simulated Binary Crossover)
• Mutation: PM (Polynomial Mutation)

Binary variables:

• Crossover: TwoPointCrossover, UniformCrossover
• Mutation: BitflipMutation

Permutations (TSP, scheduling):

• Crossover: OrderCrossover (OX)
• Mutation: InversionMutation

See: references/operators.md for comprehensive operator reference

Performance and Troubleshooting

Common issues and solutions:

Problem: Algorithm not converging

• Increase population size
• Increase number of generations
• Check if problem is multimodal (try different algorithms)
• Verify constraints are correctly formulated

Problem: Poor Pareto front distribution

• For NSGA-III: Adjust reference directions
• Increase population size
• Check for duplicate elimination
• Verify problem scaling

Problem: Few feasible solutions

• Use constraint-as-objective approach
• Apply repair operators
• Try SRES/ISRES for constrained problems
• Check constraint formulation (should be g <= 0)

Problem: High computational cost

• Reduce population size
• Decrease number of generations
• Use simpler operators
• Enable parallel evaluation via elementwise_runner (see Workflow 8)

Best practices:

• Normalize objectives when scales differ significantly
• Set random seed for reproducibility
• Save history to analyze convergence: save_history=True
• Visualize results to understand solution quality
• Compare with true Pareto front when available
• Use appropriate termination criteria (generations, evaluations, tolerance)
• Tune operator parameters for problem characteristics

Resources

This skill includes comprehensive reference documentation and executable examples:

references/

Detailed documentation for in-depth understanding:

• algorithms.md: Complete algorithm reference with parameters, usage, and selection guidelines
• problems.md: Benchmark test problems (ZDT, DTLZ, WFG) with characteristics
• operators.md: Genetic operators (sampling, selection, crossover, mutation) with configuration
• visualization.md: All visualization types with examples and selection guide
• constraints_mcdm.md: Constraint handling techniques and multi-criteria decision making methods
• parallelization.md: Parallel evaluation with StarmapParallelization and JoblibParallelization

Search patterns for references:

• Algorithm details: grep -r "NSGA-II\|NSGA-III\|MOEA/D" references/
• Constraint methods: grep -r "Feasibility First\|Penalty\|Repair" references/
• Visualization types: grep -r "Scatter\|PCP\|Petal" references/

scripts/

Executable examples demonstrating common workflows:

• single_objective_example.py: Basic single-objective optimization with GA
• multi_objective_example.py: Multi-objective optimization with NSGA-II, visualization
• many_objective_example.py: Many-objective optimization with NSGA-III, reference directions
• custom_problem_example.py: Defining custom problems (constrained and unconstrained)
• decision_making_example.py: Multi-criteria decision making with different preferences

Run examples:

python3 scripts/single_objective_example.py
python3 scripts/multi_objective_example.py
python3 scripts/many_objective_example.py
python3 scripts/custom_problem_example.py
python3 scripts/decision_making_example.py

Additional Notes

Common patterns:

• Use ElementwiseProblem for custom problems (or FunctionalProblem for function-based definitions)
• Use vars dict with typed variables for mixed-variable problems
• Constraints formulated as g(x) <= 0 and h(x) = 0
• Reference directions required for NSGA-III
• Normalize objectives before MCDM
• Use appropriate termination: ('n_gen', N) or get_termination("f_tol", tol=0.001)