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Zhongwei Li
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skills/pymoo/references/problems.md
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skills/pymoo/references/problems.md
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# Pymoo Test Problems Reference
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Comprehensive reference for benchmark optimization problems in pymoo.
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## Single-Objective Test Problems
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### Ackley Function
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**Characteristics:**
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- Highly multimodal
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- Many local optima
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- Tests algorithm's ability to escape local minima
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- Continuous variables
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### Griewank Function
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**Characteristics:**
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- Multimodal with regularly distributed local minima
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- Product term introduces interdependencies between variables
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- Global minimum at origin
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### Rastrigin Function
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**Characteristics:**
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- Highly multimodal with regularly spaced local minima
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- Challenging for gradient-based methods
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- Tests global search capability
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### Rosenbrock Function
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**Characteristics:**
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- Unimodal but narrow valley to global optimum
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- Tests algorithm's convergence in difficult landscape
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- Classic benchmark for continuous optimization
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### Zakharov Function
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**Characteristics:**
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- Unimodal
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- Single global minimum
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- Tests basic convergence capability
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## Multi-Objective Test Problems (2-3 objectives)
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### ZDT Test Suite
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**Purpose:** Standard benchmark for bi-objective optimization
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**Construction:** f₂(x) = g(x) · h(f₁(x), g(x)) where g(x) = 1 at Pareto-optimal solutions
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#### ZDT1
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- **Variables:** 30 continuous
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- **Bounds:** [0, 1]
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- **Pareto front:** Convex
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- **Purpose:** Basic convergence and diversity test
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#### ZDT2
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- **Variables:** 30 continuous
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- **Bounds:** [0, 1]
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- **Pareto front:** Non-convex (concave)
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- **Purpose:** Tests handling of non-convex fronts
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#### ZDT3
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- **Variables:** 30 continuous
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- **Bounds:** [0, 1]
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- **Pareto front:** Disconnected (5 separate regions)
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- **Purpose:** Tests diversity maintenance across discontinuous front
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#### ZDT4
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- **Variables:** 10 continuous (x₁ ∈ [0,1], x₂₋₁₀ ∈ [-10,10])
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- **Pareto front:** Convex
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- **Difficulty:** 21⁹ local Pareto fronts
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- **Purpose:** Tests global search with many local optima
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#### ZDT5
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- **Variables:** 11 discrete (bitstring)
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- **Encoding:** x₁ uses 30 bits, x₂₋₁₁ use 5 bits each
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- **Pareto front:** Convex
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- **Purpose:** Tests discrete optimization and deceptive landscapes
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#### ZDT6
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- **Variables:** 10 continuous
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- **Bounds:** [0, 1]
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- **Pareto front:** Non-convex with non-uniform density
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- **Purpose:** Tests handling of biased solution distributions
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**Usage:**
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```python
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from pymoo.problems.multi import ZDT1, ZDT2, ZDT3, ZDT4, ZDT5, ZDT6
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problem = ZDT1() # or ZDT2(), ZDT3(), etc.
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```
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### BNH (Binh and Korn)
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**Characteristics:**
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- 2 objectives
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- 2 variables
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- Constrained problem
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- Tests constraint handling in multi-objective context
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### OSY (Osyczka and Kundu)
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**Characteristics:**
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- 6 objectives
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- 6 variables
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- Multiple constraints
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- Real-world inspired
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### TNK (Tanaka)
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**Characteristics:**
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- 2 objectives
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- 2 variables
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- Disconnected feasible region
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- Tests handling of disjoint search spaces
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### Truss2D
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**Characteristics:**
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- Structural engineering problem
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- Bi-objective (weight vs displacement)
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- Practical application test
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### Welded Beam
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**Characteristics:**
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- Engineering design problem
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- Multiple constraints
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- Practical optimization scenario
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### Omni-test
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**Characteristics:**
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- Configurable test problem
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- Various difficulty levels
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- Systematic testing
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### SYM-PART
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**Characteristics:**
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- Symmetric problem structure
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- Tests specific algorithmic behaviors
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## Many-Objective Test Problems (4+ objectives)
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### DTLZ Test Suite
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**Purpose:** Scalable many-objective benchmarks
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**Objectives:** Configurable (typically 3-15)
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**Variables:** Scalable
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#### DTLZ1
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- **Pareto front:** Linear (hyperplane)
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- **Difficulty:** 11^k local Pareto fronts
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- **Purpose:** Tests convergence with many local optima
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#### DTLZ2
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- **Pareto front:** Spherical (concave)
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- **Difficulty:** Straightforward convergence
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- **Purpose:** Basic many-objective diversity test
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#### DTLZ3
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- **Pareto front:** Spherical
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- **Difficulty:** 3^k local Pareto fronts
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- **Purpose:** Combines DTLZ1's multimodality with DTLZ2's geometry
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#### DTLZ4
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- **Pareto front:** Spherical with biased density
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- **Difficulty:** Non-uniform solution distribution
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- **Purpose:** Tests diversity maintenance with bias
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#### DTLZ5
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- **Pareto front:** Degenerate (curve in M-dimensional space)
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- **Purpose:** Tests handling of degenerate fronts
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#### DTLZ6
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- **Pareto front:** Degenerate curve
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- **Difficulty:** Harder convergence than DTLZ5
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- **Purpose:** Challenging degenerate front
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#### DTLZ7
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- **Pareto front:** Disconnected regions
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- **Difficulty:** 2^(M-1) disconnected regions
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- **Purpose:** Tests diversity across disconnected fronts
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**Usage:**
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```python
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from pymoo.problems.many import DTLZ1, DTLZ2
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problem = DTLZ1(n_var=7, n_obj=3) # 7 variables, 3 objectives
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```
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### WFG Test Suite
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**Purpose:** Walking Fish Group scalable benchmarks
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**Features:** More complex than DTLZ, various front shapes and difficulties
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**Variants:** WFG1-WFG9 with different characteristics
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- Non-separable
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- Deceptive
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- Multimodal
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- Biased
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- Scaled fronts
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## Constrained Multi-Objective Problems
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### MW Test Suite
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**Purpose:** Multi-objective problems with various constraint types
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**Features:** Different constraint difficulty levels
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### DAS-CMOP
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**Purpose:** Difficulty-adjustable and scalable constrained multi-objective problems
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**Features:** Tunable constraint difficulty
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### MODAct
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**Purpose:** Multi-objective optimization with active constraints
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**Features:** Realistic constraint scenarios
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## Dynamic Multi-Objective Problems
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### DF Test Suite
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**Purpose:** CEC2018 Competition dynamic multi-objective benchmarks
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**Features:**
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- Time-varying objectives
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- Changing Pareto fronts
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- Tests algorithm adaptability
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**Variants:** DF1-DF14 with different dynamics
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## Custom Problem Definition
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Define custom problems by extending base classes:
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```python
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from pymoo.core.problem import ElementwiseProblem
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import numpy as np
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class MyProblem(ElementwiseProblem):
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def __init__(self):
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super().__init__(
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n_var=2, # number of variables
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n_obj=2, # number of objectives
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n_ieq_constr=0, # inequality constraints
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n_eq_constr=0, # equality constraints
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xl=np.array([0, 0]), # lower bounds
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xu=np.array([1, 1]) # upper bounds
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)
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def _evaluate(self, x, out, *args, **kwargs):
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# Define objectives
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f1 = x[0]**2 + x[1]**2
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f2 = (x[0]-1)**2 + x[1]**2
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out["F"] = [f1, f2]
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# Optional: constraints
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# out["G"] = constraint_values # <= 0
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# out["H"] = equality_constraints # == 0
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```
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## Problem Selection Guidelines
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**For algorithm development:**
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- Simple convergence: DTLZ2, ZDT1
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- Multimodal: ZDT4, DTLZ1, DTLZ3
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- Non-convex: ZDT2
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- Disconnected: ZDT3, DTLZ7
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**For comprehensive testing:**
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- ZDT suite for bi-objective
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- DTLZ suite for many-objective
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- WFG for complex landscapes
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- MW/DAS-CMOP for constraints
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**For real-world validation:**
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- Engineering problems (Truss2D, Welded Beam)
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- Match problem characteristics to application domain
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**Variable types:**
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- Continuous: Most problems
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- Discrete: ZDT5
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- Mixed: Define custom problem
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