Simulated Annealing

Simulated annealing is a probabilistic optimization technique inspired by the annealing process in metallurgy. It's particularly effective for finding global optima in complex, non-convex landscapes.

The Physical Inspiration

When metal is heated and then slowly cooled, atoms move toward lower energy configurations. This process:

1. Starts with high temperature (random exploration)
2. Gradually cools (focused exploitation)
3. Results in a low-energy crystalline structure

Algorithm Overview

def simulated_annealing(objective, initial_solution, T_initial, T_final, alpha):
    current = initial_solution
    T = T_initial
    
    while T > T_final:
        # Generate neighbor solution
        neighbor = perturb(current)
        
        # Calculate energy difference
        delta_E = objective(neighbor) - objective(current)
        
        # Accept or reject
        if delta_E < 0 or random() < exp(-delta_E / T):
            current = neighbor
        
        # Cool down
        T = T * alpha
    
    return current

Cooling Schedules

The cooling schedule significantly impacts performance:

Linear Cooling

T(k) = T_0 - k * alpha

Simple but may cool too quickly.

Exponential Cooling

T(k) = T_0 * alpha^k

Most commonly used, with alpha typically between 0.8 and 0.99.

Logarithmic Cooling

T(k) = T_0 / log(1 + k)

Theoretically guarantees convergence but very slow.

Applications

Traveling Salesman Problem

Finding the shortest route visiting all cities is a classic application where simulated annealing excels.

Neural Network Training

Alternative to gradient-based methods for specific architectures or non-differentiable objectives.

Image Processing

Used in image segmentation and restoration tasks.

My Research

In my work on cellular spheroid optimization, I use simulated annealing to:

• Optimize grid-based placement problems
• Fine-tune detection parameters
• Balance multiple competing objectives

Advantages and Limitations

Advantages

• Can escape local minima
• Simple to implement
• Works with any objective function
• No gradient computation required

Limitations

• Many hyperparameters to tune
• Can be slow for high-dimensional problems
• No guarantee of finding global optimum

Tips for Effective Use

1. Start with high temperature: Ensure adequate initial exploration
2. Tune the cooling rate: Problem-specific adjustment needed
3. Use restart strategies: Multiple independent runs improve results
4. Combine with local search: Hybrid approaches often work best

This research was presented as part of my studies in Mathematical Modeling at Mohammed V University.