pymoo

What this skill does

# 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), 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.

## 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 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
**Dynamic:** Time-varying objectives or constraints

## Quick Start Workflows

### Workflow 1: Single-Objective Optimization

**When:** Optimizing one objective function

**Steps:**
1. Define or select problem
2. Choose single-objective algorithm (GA, DE, PSO, CMA-ES)
3. Configure termination criteria
4. Run optimization
5. 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:**
1. Define multi-objective problem
2. Configure NSGA-II
3. Run optimization to obtain Pareto front
4. Visualize trade-offs
5. 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:**
1. Define many-objective problem
2. Generate reference directions
3. Configure NSGA-III with reference directions
4. Run optimization
5. 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_dim=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:**
1. Extend `ElementwiseProblem` class
2. Define `__init__` with problem dimensions and bounds
3. Implement `_evaluate` method for objectives (and constraints)
4. 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 han