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

bash
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:

python
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:

python
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:

python
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:

1. Normalize objectives when scales differ significantly
2. Set random seed for reproducibility
3. Save history to analyze convergence: save_history=True
4. Visualize results to understand solution quality
5. Compare with true Pareto front when available
6. Use appropriate termination criteria (generations, evaluations, tolerance)
7. 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:

bash
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)