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

Workflow 1: Single-Objective Optimization

When: Optimizing one objective function

Steps:

Define or select problem
Choose single-objective algorithm (GA, DE, PSO, CMA-ES)
Configure termination criteria
Run optimization
Extract best solution

Example:

python

from pymoo.algorithms.soo.nonconvex.ga import GA
from pymoo.problems import get_problem
from pymoo.optimize import minimize

# Built-in problem
problem = get_problem("rastrigin", n_var=10)

# Configure Genetic Algorithm
algorithm = GA(
    pop_size=100,
    eliminate_duplicates=True
)

# Optimize
result = minimize(
    problem,
    algorithm,
    ('n_gen', 200),
    seed=1,
    verbose=True
)

print(f"Best solution: {result.X}")
print(f"Best objective: {result.F[0]}")

See: scripts/single_objective_example.py for complete example

Workflow 2: Multi-Objective Optimization (2-3 objectives)

When: Optimizing 2-3 conflicting objectives, need Pareto front

Algorithm choice: NSGA-II (standard for bi/tri-objective)

Steps:

Define multi-objective problem
Configure NSGA-II
Run optimization to obtain Pareto front
Visualize trade-offs
Apply decision making (optional)

Example:

python

from pymoo.algorithms.moo.nsga2 import NSGA2
from pymoo.problems import get_problem
from pymoo.optimize import minimize
from pymoo.visualization.scatter import Scatter

# Bi-objective benchmark problem
problem = get_problem("zdt1")

# NSGA-II algorithm
algorithm = NSGA2(pop_size=100)

# Optimize
result = minimize(problem, algorithm, ('n_gen', 200), seed=1)

# Visualize Pareto front
plot = Scatter()
plot.add(result.F, label="Obtained Front")
plot.add(problem.pareto_front(), label="True Front", alpha=0.3)
plot.show()

print(f"Found {len(result.F)} Pareto-optimal solutions")

See: scripts/multi_objective_example.py for complete example

Workflow 3: Many-Objective Optimization (4+ objectives)

When: Optimizing 4 or more objectives

Algorithm choice: NSGA-III (designed for many objectives)

Key difference: Must provide reference directions for population guidance

Steps:

Define many-objective problem
Generate reference directions
Configure NSGA-III with reference directions
Run optimization
Visualize using Parallel Coordinate Plot

Example:

python

from pymoo.algorithms.moo.nsga3 import NSGA3
from pymoo.problems import get_problem
from pymoo.optimize import minimize
from pymoo.util.ref_dirs import get_reference_directions
from pymoo.visualization.pcp import PCP

# Many-objective problem (5 objectives)
problem = get_problem("dtlz2", n_obj=5)

# Generate reference directions (required for NSGA-III)
ref_dirs = get_reference_directions("das-dennis", n_obj=5, n_partitions=12)

# Configure NSGA-III
algorithm = NSGA3(ref_dirs=ref_dirs)

# Optimize
result = minimize(problem, algorithm, ('n_gen', 300), seed=1)

# Visualize with Parallel Coordinates
plot = PCP(labels=[f"f{i+1}" for i in range(5)])
plot.add(result.F, alpha=0.3)
plot.show()

See: scripts/many_objective_example.py for complete example

Workflow 4: Custom Problem Definition

When: Solving domain-specific optimization problem

Steps:

Extend ElementwiseProblem class
Define __init__ with problem dimensions and bounds
Implement _evaluate method for objectives (and constraints)
Use with any algorithm

Unconstrained example:

python

from pymoo.core.problem import ElementwiseProblem
import numpy as np

class MyProblem(ElementwiseProblem):
    def __init__(self):
        super().__init__(
            n_var=2,              # Number of variables
            n_obj=2,              # Number of objectives
            xl=np.array([0, 0]),  # Lower bounds
            xu=np.array([5, 5])   # Upper bounds
        )

    def _evaluate(self, x, out, *args, **kwargs):
        # Define objectives
        f1 = x[0]**2 + x[1]**2
        f2 = (x[0]-1)**2 + (x[1]-1)**2

        out["F"] = [f1, f2]

Constrained example:

python

class ConstrainedProblem(ElementwiseProblem):
    def __init__(self):
        super().__init__(
            n_var=2,
            n_obj=2,
            n_ieq_constr=2,        # Inequality constraints
            n_eq_constr=1,         # Equality constraints
            xl=np.array([0, 0]),
            xu=np.array([5, 5])
        )

    def _evaluate(self, x, out, *args, **kwargs):
        # Objectives
        out["F"] = [f1, f2]

        # Inequality constraints (g <= 0)
        out["G"] = [g1, g2]

        # Equality constraints (h = 0)
        out["H"] = [h1]

Constraint formulation rules:

Inequality: Express as g(x) <= 0 (feasible when ≤ 0)
Equality: Express as h(x) = 0 (feasible when = 0)
Convert g(x) >= b to -(g(x) - b) <= 0

See: scripts/custom_problem_example.py for complete examples

Workflow 5: Constraint Handling

When: Problem has feasibility constraints

Approach options:

1. Feasibility First (Default - Recommended)

python

from pymoo.algorithms.moo.nsga2 import NSGA2

# Works automatically with constrained problems
algorithm = NSGA2(pop_size=100)
result = minimize(problem, algorithm, termination)

# Check feasibility
feasible = result.CV[:, 0] == 0  # CV = constraint violation
print(f"Feasible solutions: {np.sum(feasible)}")

2. Penalty Method

python

from pymoo.constraints.as_penalty import ConstraintsAsPenalty

# Wrap problem to convert constraints to penalties
problem_penalized = ConstraintsAsPenalty(problem, penalty=1e6)

3. Constraint as Objective

python

from pymoo.constraints.as_obj import ConstraintsAsObjective

# Treat constraint violation as additional objective
problem_with_cv = ConstraintsAsObjective(problem)

4. Specialized Algorithms

python

from pymoo.algorithms.soo.nonconvex.sres import SRES

# SRES has built-in constraint handling
algorithm = SRES()

See: references/constraints_mcdm.md for comprehensive constraint handling guide

Workflow 6: Decision Making from Pareto Front

When: Have Pareto front, need to select preferred solution(s)

Steps:

Run multi-objective optimization
Normalize objectives to [0, 1]
Define preference weights
Apply MCDM method
Visualize selected solution

Example using Pseudo-Weights:

python

from pymoo.mcdm.pseudo_weights import PseudoWeights
import numpy as np

# After obtaining result from multi-objective optimization
# Normalize objectives
F_norm = (result.F - result.F.min(axis=0)) / (result.F.max(axis=0) - result.F.min(axis=0))

# Define preferences (must sum to 1)
weights = np.array([0.3, 0.7])  # 30% f1, 70% f2

# Apply decision making
dm = PseudoWeights(weights)
selected_idx = dm.do(F_norm)

# Get selected solution
best_solution = result.X[selected_idx]
best_objectives = result.F[selected_idx]

print(f"Selected solution: {best_solution}")
print(f"Objective values: {best_objectives}")

Other MCDM methods:

Compromise Programming: Select closest to ideal point
Knee Point: Find balanced trade-off solutions
Hypervolume Contribution: Select most diverse subset

See:

scripts/decision_making_example.py for complete example
references/constraints_mcdm.md for detailed MCDM methods

Workflow 7: Visualization

Choose visualization based on number of objectives:

2 objectives: Scatter Plot

python

from pymoo.visualization.scatter import Scatter

plot = Scatter(title="Bi-objective Results")
plot.add(result.F, color="blue", alpha=0.7)
plot.show()

3 objectives: 3D Scatter

python

plot = Scatter(title="Tri-objective Results")
plot.add(result.F)  # Automatically renders in 3D
plot.show()

4+ objectives: Parallel Coordinate Plot

python

from pymoo.visualization.pcp import PCP

plot = PCP(
    labels=[f"f{i+1}" for i in range(n_obj)],
    normalize_each_axis=True
)
plot.add(result.F, alpha=0.3)
plot.show()

Solution comparison: Petal Diagram

python

from pymoo.visualization.petal import Petal

plot = Petal(
    bounds=[result.F.min(axis=0), result.F.max(axis=0)],
    labels=["Cost", "Weight", "Efficiency"]
)
plot.add(solution_A, label="Design A")
plot.add(solution_B, label="Design B")
plot.show()

See: references/visualization.md for all visualization types and usage

Workflow 8: Parallel Evaluation

When: Each _evaluate call is expensive (simulations, ML models, external solvers)

Approach: Pass an elementwise_runner to ElementwiseProblem using StarmapParallelization or JoblibParallelization.

Example (thread pool):

python

from multiprocessing.pool import ThreadPool
from pymoo.algorithms.soo.nonconvex.ga import GA
from pymoo.core.problem import ElementwiseProblem
from pymoo.optimize import minimize
from pymoo.parallelization.starmap import StarmapParallelization

class MyProblem(ElementwiseProblem):
    def __init__(self, elementwise_runner=None, **kwargs):
        super().__init__(
            n_var=10, n_obj=1, xl=-5, xu=5,
            elementwise_runner=elementwise_runner, **kwargs,
        )

    def _evaluate(self, x, out, *args, **kwargs):
        out["F"] = (x ** 2).sum()  # Replace with expensive evaluation

pool = ThreadPool(4)
runner = StarmapParallelization(pool.starmap)
problem = MyProblem(elementwise_runner=runner)

result = minimize(problem, GA(), ("n_gen", 50), seed=1)
pool.close()

See: references/parallelization.md for process pools, joblib, and pickling notes

Workflow 9: Mixed-Variable Optimization

When: Decision variables include continuous, integer, binary, and/or categorical types

Approach: Define a vars dict with typed variables; use MixedVariableGA (SOO) or add MOO survival.

Example:

python

from pymoo.core.problem import ElementwiseProblem
from pymoo.core.variable import Real, Integer, Choice, Binary
from pymoo.core.mixed import MixedVariableGA
from pymoo.optimize import minimize

class MixedProblem(ElementwiseProblem):
    def __init__(self, **kwargs):
        vars = {
            "b": Binary(),
            "x": Choice(options=["nothing", "multiply"]),
            "y": Integer(bounds=(0, 2)),
            "z": Real(bounds=(0, 5)),
        }
        super().__init__(vars=vars, n_obj=1, **kwargs)

    def _evaluate(self, X, out, *args, **kwargs):
        b, x, z, y = X["b"], X["x"], X["z"], X["y"]
        f = z + y
        if b:
            f = 100 * f
        if x == "multiply":
            f = 10 * f
        out["F"] = f

algorithm = MixedVariableGA(pop_size=20)
result = minimize(MixedProblem(), algorithm, ("n_evals", 1000), seed=1)

For multi-objective mixed-variable problems, use MixedVariableGA(pop_size=20, survival=RankAndCrowdingSurvival()). For single-objective mixed search, pymoo also wraps Optuna via pymoo.algorithms.soo.nonconvex.optuna.Optuna.

See: references/algorithms.md for MixedVariableGA and Optuna details

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:

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:

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)