Melvin Avarez
496 lines
13 KiB
Markdown
496 lines
13 KiB
Markdown
# Differential Evolution: Mathematical Theory
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## Introduction
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Differential Evolution (DE) is a population-based metaheuristic optimization algorithm introduced by Storn and Price (1997). It is particularly effective for continuous, non-convex, multimodal optimization problems where gradient information is unavailable or unreliable.
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## Problem Formulation
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### Objective
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Minimize $f: \mathbb{R}^D \rightarrow \mathbb{R}$:
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$$\min_{\mathbf{x} \in \mathbb{R}^D} f(\mathbf{x})$$
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subject to box constraints:
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$$x_j \in [l_j, u_j], \quad j = 1, ..., D$$
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### Characteristics
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**DE is suitable when**:
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- $f$ is continuous but non-differentiable
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- Multiple local minima exist
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- Gradient information is unavailable or expensive
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- Problem dimension is moderate ($D < 100$)
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## Algorithm Overview
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### Population
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Maintain a population of $N_P$ candidate solutions:
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$$P_g = \{\mathbf{x}_{1,g}, \mathbf{x}_{2,g}, ..., \mathbf{x}_{N_P,g}\}$$
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where $g$ is the generation number and $\mathbf{x}_{i,g} \in \mathbb{R}^D$.
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### Main Loop
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For each generation $g = 0, 1, 2, ...$:
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1. **Mutation**: Create mutant vectors
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2. **Crossover**: Create trial vectors
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3. **Selection**: Keep better solutions
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## Mutation Strategies
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### DE/rand/1 (Classic)
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For each target vector $\mathbf{x}_{i,g}$, create mutant:
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$$\mathbf{v}_{i,g+1} = \mathbf{x}_{r_1,g} + F \cdot (\mathbf{x}_{r_2,g} - \mathbf{x}_{r_3,g})$$
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where:
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- $r_1, r_2, r_3 \in \{1, ..., N_P\}$ are randomly chosen, distinct, and $\neq i$
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- $F \in (0, 2]$ is the **mutation factor** (typically 0.5-1.0)
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**Interpretation**:
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- Start from a random population member $\mathbf{x}_{r_1}$
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- Move in direction given by difference $(\mathbf{x}_{r_2} - \mathbf{x}_{r_3})$
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- Scale movement by $F$
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### DE/best/1
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$$\mathbf{v}_{i,g+1} = \mathbf{x}_{\text{best},g} + F \cdot (\mathbf{x}_{r_1,g} - \mathbf{x}_{r_2,g})$$
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**Advantage**: Faster convergence
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**Disadvantage**: More likely to get stuck in local minima
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### DE/current-to-best/1
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$$\mathbf{v}_{i,g+1} = \mathbf{x}_{i,g} + F \cdot (\mathbf{x}_{\text{best},g} - \mathbf{x}_{i,g}) + F \cdot (\mathbf{x}_{r_1,g} - \mathbf{x}_{r_2,g})$$
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**Interpretation**: Move current solution toward best while exploring
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### DE/rand/2
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$$\mathbf{v}_{i,g+1} = \mathbf{x}_{r_1,g} + F \cdot (\mathbf{x}_{r_2,g} - \mathbf{x}_{r_3,g}) + F \cdot (\mathbf{x}_{r_4,g} - \mathbf{x}_{r_5,g})$$
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More disruptive, better for highly multimodal problems.
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## Crossover
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### Binomial Crossover
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For each component $j = 1, ..., D$:
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$$u_{i,j,g+1} = \begin{cases}
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v_{i,j,g+1} & \text{if } \text{rand}(0,1) \leq CR \text{ or } j = j_{\text{rand}} \\
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x_{i,j,g} & \text{otherwise}
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\end{cases}$$
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where:
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- $CR \in [0, 1]$ is the **crossover probability**
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- $j_{\text{rand}} \in \{1, ..., D\}$ ensures at least one component is from mutant
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**Effect**: Controls how much of the mutant vector is used
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### Exponential Crossover
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Copy consecutive components from mutant with probability $CR$.
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Less common, similar performance to binomial.
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## Selection
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Greedy selection (for minimization):
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$$\mathbf{x}_{i,g+1} = \begin{cases}
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\mathbf{u}_{i,g+1} & \text{if } f(\mathbf{u}_{i,g+1}) \leq f(\mathbf{x}_{i,g}) \\
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\mathbf{x}_{i,g} & \text{otherwise}
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\end{cases}$$
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**Property**: Population quality never decreases:
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$$f(\mathbf{x}_{\text{best},g+1}) \leq f(\mathbf{x}_{\text{best},g})$$
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## Complete Algorithm
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```
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1. Initialize population:
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For i = 1 to N_P:
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x_{i,0} = l + rand(0,1) · (u - l)
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2. Evaluate fitness:
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f_i = f(x_{i,0}) for all i
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3. While stopping criterion not met:
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a. For i = 1 to N_P:
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i. Mutation:
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Select r_1, r_2, r_3 distinct and ≠ i
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v_{i,g+1} = x_{r_1,g} + F · (x_{r_2,g} - x_{r_3,g})
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ii. Crossover:
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j_rand = randint(1, D)
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For j = 1 to D:
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if rand(0,1) ≤ CR or j = j_rand:
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u_{i,j,g+1} = v_{i,j,g+1}
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else:
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u_{i,j,g+1} = x_{i,j,g}
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iii. Boundary handling:
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Clip u_{i,g+1} to [l, u]
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iv. Selection:
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if f(u_{i,g+1}) ≤ f(x_{i,g}):
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x_{i,g+1} = u_{i,g+1}
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else:
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x_{i,g+1} = x_{i,g}
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b. g = g + 1
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4. Return x_best and f(x_best)
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```
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## Parameter Selection
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### Population Size ($N_P$)
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**Rule of thumb**: $N_P = 10D$ where $D$ is problem dimension
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**Small population** (< 4D):
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- Faster convergence
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- Risk premature convergence
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- Use for: simple unimodal problems
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**Large population** (> 20D):
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- Better exploration
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- Slower convergence
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- Use for: highly multimodal problems
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**Minimum**: $N_P \geq 4$ (needed for mutation)
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### Mutation Factor ($F$)
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**Typical range**: $F \in [0.4, 1.0]$
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**Low F** (0.4-0.6):
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- Fine-tuning, local search
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- Use near end of optimization
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- Safer, less disruptive
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**High F** (0.8-1.2):
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- Exploration, global search
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- Escape local minima
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- More aggressive
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**Adaptive F**: Some variants adjust $F$ during optimization
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### Crossover Probability ($CR$)
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**Typical range**: $CR \in [0.1, 0.9]$
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**Low CR** (0.1-0.3):
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- Less information exchange
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- Slower convergence
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- Use for: separable problems
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**High CR** (0.7-0.9):
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- More information exchange
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- Faster convergence
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- Use for: non-separable problems
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**Special cases**:
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- $CR = 0$: Pure mutation (except $j_{\text{rand}}$)
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- $CR = 1$: Full crossover
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### Stopping Criteria
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1. **Maximum generations**: $g_{\max}$
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2. **Function evaluations**: $FE_{\max}$
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3. **Target fitness**: $f(\mathbf{x}_{\text{best}}) \leq f_{\text{target}}$
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4. **Stagnation**: No improvement for $G_{\text{stag}}$ generations
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5. **Diversity loss**: Population variance below threshold
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## Convergence Analysis
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### Theoretical Results
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**Theorem** (Zaharie, 2002): Under certain conditions on $F$ and $CR$, DE converges to a stationary point.
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**Conditions**:
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- Bounded search space
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- Continuous objective function
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- Appropriate parameter settings
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### Convergence Rate
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**Empirical observations**:
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- Linear convergence in early stages
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- Slows down near optimum
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- Faster than genetic algorithms for many problems
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- Slower than gradient methods (when gradients available)
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### No Free Lunch
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DE is not universally optimal. Performance depends on:
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- Problem landscape
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- Parameter settings
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- Population size
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## Variants and Extensions
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### Self-Adaptive DE (jDE)
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Parameters $F$ and $CR$ evolve with the population:
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$$F_{i,g+1} = \begin{cases}
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F_l + \text{rand}(0,1) \cdot (F_u - F_l) & \text{if } \text{rand}(0,1) < \tau_1 \\
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F_{i,g} & \text{otherwise}
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\end{cases}$$
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$$CR_{i,g+1} = \begin{cases}
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\text{rand}(0,1) & \text{if } \text{rand}(0,1) < \tau_2 \\
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CR_{i,g} & \text{otherwise}
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\end{cases}$$
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### SHADE (Success-History Adaptive DE)
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Uses historical information about successful parameters.
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### L-SHADE
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SHADE with linear population size reduction.
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### CoDE (Composite DE)
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Uses multiple mutation strategies simultaneously.
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### Opposition-Based DE
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Initialize with both random solutions and their opposites.
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### Constraint Handling
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For constrained optimization:
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1. **Penalty method**: Add penalty to objective
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2. **Feasibility rules**: Prefer feasible solutions
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3. **ε-constrained**: Relax constraints gradually
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## Theoretical Properties
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### Global Convergence
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**Sufficient conditions** (Lampinen, 2001):
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- Population size $N_P > 3$
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- Mutation factor $F > 0$
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- At least one component crossed over ($j_{\text{rand}}$)
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Then DE is a **global optimization method**: Can reach any point with positive probability.
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### Diversity Maintenance
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Mutation creates diversity, selection reduces it. Balance determines exploration vs. exploitation.
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**Diversity measure**:
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$$D_g = \frac{1}{N_P D} \sum_{i=1}^{N_P} \sum_{j=1}^D |x_{i,j,g} - \bar{x}_{j,g}|$$
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High diversity → exploration
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Low diversity → exploitation
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### Convergence Speed
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**Expected number of generations** to reach near-optimum depends on:
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- Problem difficulty (number of local minima, basin sizes)
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- Population size
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- Parameter settings
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**Empirical rule**: Budget $10^4 D$ function evaluations for moderately difficult problems.
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## Comparison with Other Algorithms
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| Algorithm | Gradient | Global | Constraints | Speed | Best For |
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|-----------|----------|--------|-------------|-------|----------|
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| **DE** | No | Yes | Penalty | Medium | Non-convex, continuous |
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| Gradient Descent | Yes | No | Yes | Fast | Smooth, convex |
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| Genetic Algorithm | No | Yes | Yes | Slow | Discrete, combinatorial |
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| Particle Swarm | No | Yes | Penalty | Fast | Continuous, many dims |
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| Simulated Annealing | No | Yes | Penalty | Slow | Small problems |
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| CMA-ES | No | Yes | Penalty | Fast | Continuous, noisy |
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## Applications
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### 1. Engineering Design
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**Example**: Antenna design
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- Objective: Maximize gain, minimize side lobes
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- Constraints: Physical realizability
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- High-dimensional, non-convex
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### 2. Machine Learning
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**Example**: Neural network hyperparameter tuning
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- Objective: Validation accuracy
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- Parameters: Learning rate, regularization, architecture
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- Noisy, expensive evaluations
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### 3. Chemical Engineering
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**Example**: Reactor optimization
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- Objective: Maximize yield, minimize cost
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- Constraints: Safety, temperature, pressure
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- Nonlinear dynamics
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### 4. Portfolio Optimization
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**Example**: Asset allocation
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- Objective: Maximize Sharpe ratio
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- Constraints: Budget, diversification
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- Non-convex risk measures
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### 5. System Identification
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**Example**: Parameter estimation
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- Objective: Minimize prediction error
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- Parameters: Model coefficients
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- Multimodal likelihood surface
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## Computational Complexity
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### Time Complexity
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Per generation: $O(N_P \cdot D \cdot T_f)$
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where $T_f$ is cost of evaluating $f$.
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Total: $O(G_{\max} \cdot N_P \cdot D \cdot T_f)$
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### Space Complexity
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$O(N_P \cdot D)$ for population storage.
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### Parallelization
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**Embarrassingly parallel**: Each trial vector evaluation is independent.
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**Speedup**: Near-linear with number of processors (up to $N_P$ processors).
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## Practical Tips
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### 1. Start Simple
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Use default parameters: $N_P = 10D$, $F = 0.8$, $CR = 0.7$
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### 2. Scale Variables
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Normalize parameters to similar ranges for better performance.
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### 3. Warm Start
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If you have a good initial guess, seed population around it.
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### 4. Hybrid Approach
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Use DE for global search, then local optimizer for refinement:
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```
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1. Run DE for G_global generations
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2. Take best solution x_best
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3. Run local optimizer starting from x_best
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```
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### 5. Monitor Convergence
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Plot:
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- Best fitness vs. generation
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- Average fitness vs. generation
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- Population diversity vs. generation
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### 6. Restarts
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If premature convergence detected, restart with new random population.
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## Advantages and Limitations
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### Advantages
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✅ No gradient information needed
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✅ Handles non-convex, multimodal functions well
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✅ Few parameters to tune
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✅ Simple to implement
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✅ Robust across problem types
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✅ Naturally handles box constraints
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✅ Population maintains diversity
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### Limitations
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❌ Slower than gradient methods (when gradients available)
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❌ Scales poorly to high dimensions ($D > 100$)
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❌ No convergence guarantees for finite time
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❌ Requires many function evaluations
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❌ Performance sensitive to parameters
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❌ Difficult to handle complex constraints
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❌ No theoretical optimal parameter settings
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## Key References
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1. **Storn, R., & Price, K.** (1997). *Differential evolution - A simple and efficient heuristic for global optimization over continuous spaces*. Journal of Global Optimization, 11(4), 341-359.
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- Original DE paper
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2. **Price, K., Storn, R. M., & Lampinen, J. A.** (2005). *Differential Evolution: A Practical Approach to Global Optimization*. Springer.
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- Comprehensive book on DE
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3. **Das, S., & Suganthan, P. N.** (2011). *Differential evolution: A survey of the state-of-the-art*. IEEE Transactions on Evolutionary Computation, 15(1), 4-31.
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- Survey of DE variants and applications
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4. **Brest, J., Greiner, S., Bošković, B., Mernik, M., & Žumer, V.** (2006). *Self-adapting control parameters in differential evolution: A comparative study on numerical benchmark problems*. IEEE Transactions on Evolutionary Computation, 10(6), 646-657.
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- jDE algorithm
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5. **Tanabe, R., & Fukunaga, A.** (2013). *Success-history based parameter adaptation for differential evolution*. In IEEE Congress on Evolutionary Computation (pp. 71-78).
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- SHADE algorithm
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6. **Qin, A. K., Huang, V. L., & Suganthan, P. N.** (2009). *Differential evolution algorithm with strategy adaptation for global numerical optimization*. IEEE Transactions on Evolutionary Computation, 13(2), 398-417.
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- Self-adaptive DE
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## Summary
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Differential Evolution is a powerful metaheuristic for global optimization:
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**Key Features**:
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- Population-based search
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- Mutation, crossover, selection operators
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- Self-organizing behavior
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**Best suited for**:
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- Non-convex, multimodal problems
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- Moderate dimensions (< 100)
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- When gradients unavailable
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- Robust optimization needed
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**Success factors**:
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- Appropriate parameter settings
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- Sufficient population size
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- Adequate function evaluation budget
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## See Also
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- [Differential Evolution API Documentation](../differential_evolution.md) - Implementation and usage
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- [Grid Search Theory](../grid_search.md) - Alternative for small spaces
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- [MCMC Theory](mcmc.md) - Sampling-based inference
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