gh-k-dense-ai-claude-scientific-skills-scientific-skills/skills/pymoo/references/problems.md

Pymoo Test Problems Reference

Comprehensive reference for benchmark optimization problems in pymoo.

Single-Objective Test Problems

Ackley Function

Characteristics:

• Highly multimodal
• Many local optima
• Tests algorithm's ability to escape local minima
• Continuous variables

Griewank Function

Characteristics:

• Multimodal with regularly distributed local minima
• Product term introduces interdependencies between variables
• Global minimum at origin

Rastrigin Function

Characteristics:

• Highly multimodal with regularly spaced local minima
• Challenging for gradient-based methods
• Tests global search capability

Rosenbrock Function

Characteristics:

• Unimodal but narrow valley to global optimum
• Tests algorithm's convergence in difficult landscape
• Classic benchmark for continuous optimization

Zakharov Function

Characteristics:

• Unimodal
• Single global minimum
• Tests basic convergence capability

Multi-Objective Test Problems (2-3 objectives)

ZDT Test Suite

Purpose: Standard benchmark for bi-objective optimization Construction: f₂(x) = g(x) · h(f₁(x), g(x)) where g(x) = 1 at Pareto-optimal solutions

ZDT1

• Variables: 30 continuous
• Bounds: [0, 1]
• Pareto front: Convex
• Purpose: Basic convergence and diversity test

ZDT2

• Variables: 30 continuous
• Bounds: [0, 1]
• Pareto front: Non-convex (concave)
• Purpose: Tests handling of non-convex fronts

ZDT3

• Variables: 30 continuous
• Bounds: [0, 1]
• Pareto front: Disconnected (5 separate regions)
• Purpose: Tests diversity maintenance across discontinuous front

ZDT4

• Variables: 10 continuous (x₁ ∈ [0,1], x₂₋₁₀ ∈ [-10,10])
• Pareto front: Convex
• Difficulty: 21⁹ local Pareto fronts
• Purpose: Tests global search with many local optima

ZDT5

• Variables: 11 discrete (bitstring)
• Encoding: x₁ uses 30 bits, x₂₋₁₁ use 5 bits each
• Pareto front: Convex
• Purpose: Tests discrete optimization and deceptive landscapes

ZDT6

• Variables: 10 continuous
• Bounds: [0, 1]
• Pareto front: Non-convex with non-uniform density
• Purpose: Tests handling of biased solution distributions

Usage:

from pymoo.problems.multi import ZDT1, ZDT2, ZDT3, ZDT4, ZDT5, ZDT6
problem = ZDT1()  # or ZDT2(), ZDT3(), etc.

BNH (Binh and Korn)

Characteristics:

• 2 objectives
• 2 variables
• Constrained problem
• Tests constraint handling in multi-objective context

OSY (Osyczka and Kundu)

Characteristics:

• 6 objectives
• 6 variables
• Multiple constraints
• Real-world inspired

TNK (Tanaka)

Characteristics:

• 2 objectives
• 2 variables
• Disconnected feasible region
• Tests handling of disjoint search spaces

Truss2D

Characteristics:

• Structural engineering problem
• Bi-objective (weight vs displacement)
• Practical application test

Welded Beam

Characteristics:

• Engineering design problem
• Multiple constraints
• Practical optimization scenario

Omni-test

Characteristics:

• Configurable test problem
• Various difficulty levels
• Systematic testing

SYM-PART

Characteristics:

• Symmetric problem structure
• Tests specific algorithmic behaviors

Many-Objective Test Problems (4+ objectives)

DTLZ Test Suite

Purpose: Scalable many-objective benchmarks Objectives: Configurable (typically 3-15) Variables: Scalable

DTLZ1

• Pareto front: Linear (hyperplane)
• Difficulty: 11^k local Pareto fronts
• Purpose: Tests convergence with many local optima

DTLZ2

• Pareto front: Spherical (concave)
• Difficulty: Straightforward convergence
• Purpose: Basic many-objective diversity test

DTLZ3

• Pareto front: Spherical
• Difficulty: 3^k local Pareto fronts
• Purpose: Combines DTLZ1's multimodality with DTLZ2's geometry

DTLZ4

• Pareto front: Spherical with biased density
• Difficulty: Non-uniform solution distribution
• Purpose: Tests diversity maintenance with bias

DTLZ5

• Pareto front: Degenerate (curve in M-dimensional space)
• Purpose: Tests handling of degenerate fronts

DTLZ6

• Pareto front: Degenerate curve
• Difficulty: Harder convergence than DTLZ5
• Purpose: Challenging degenerate front

DTLZ7

• Pareto front: Disconnected regions
• Difficulty: 2^(M-1) disconnected regions
• Purpose: Tests diversity across disconnected fronts

Usage:

from pymoo.problems.many import DTLZ1, DTLZ2
problem = DTLZ1(n_var=7, n_obj=3)  # 7 variables, 3 objectives

WFG Test Suite

Purpose: Walking Fish Group scalable benchmarks Features: More complex than DTLZ, various front shapes and difficulties

Variants: WFG1-WFG9 with different characteristics

• Non-separable
• Deceptive
• Multimodal
• Biased
• Scaled fronts

Constrained Multi-Objective Problems

MW Test Suite

Purpose: Multi-objective problems with various constraint types Features: Different constraint difficulty levels

DAS-CMOP

Purpose: Difficulty-adjustable and scalable constrained multi-objective problems Features: Tunable constraint difficulty

MODAct

Purpose: Multi-objective optimization with active constraints Features: Realistic constraint scenarios

Dynamic Multi-Objective Problems

DF Test Suite

Purpose: CEC2018 Competition dynamic multi-objective benchmarks Features:

• Time-varying objectives
• Changing Pareto fronts
• Tests algorithm adaptability

Variants: DF1-DF14 with different dynamics

Custom Problem Definition

Define custom problems by extending base classes:

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
            n_ieq_constr=0,    # inequality constraints
            n_eq_constr=0,     # equality constraints
            xl=np.array([0, 0]),   # lower bounds
            xu=np.array([1, 1])    # 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]**2

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

        # Optional: constraints
        # out["G"] = constraint_values  # <= 0
        # out["H"] = equality_constraints  # == 0

Problem Selection Guidelines

For algorithm development:

• Simple convergence: DTLZ2, ZDT1
• Multimodal: ZDT4, DTLZ1, DTLZ3
• Non-convex: ZDT2
• Disconnected: ZDT3, DTLZ7

For comprehensive testing:

• ZDT suite for bi-objective
• DTLZ suite for many-objective
• WFG for complex landscapes
• MW/DAS-CMOP for constraints

For real-world validation:

• Engineering problems (Truss2D, Welded Beam)
• Match problem characteristics to application domain

Variable types:

• Continuous: Most problems
• Discrete: ZDT5
• Mixed: Define custom problem