Simulated Annealing Algorithm for Traveling Salesman Problem (TSP) Optimization

The Simulated Annealing algorithm is a heuristic optimization technique inspired by the thermal motion of atoms during metal annealing processes. It gradually converges toward optimal solutions by simulating physical annealing procedures, making it particularly effective for complex combinatorial optimization problems like the Traveling Salesman Problem (TSP).

### Core Algorithm Concept When solving TSP, Simulated Annealing starts with a randomly generated initial path and progressively optimizes it. The algorithm accepts temporarily "worse" solutions to escape local optima. As the temperature parameter decreases gradually, the probability of accepting inferior solutions reduces, ultimately converging to global or near-optimal solutions.

### Key Implementation Steps Initial Solution Generation: Create a random TSP route as starting point using permutation functions. Neighborhood Search: Generate new routes through operations like path swapping (using randperm in MATLAB) or reversal (flip function). Acceptance Criterion: Calculate objective function difference between new and current solutions, applying Metropolis criterion with probability exp(-ΔE/T). Cooling Schedule: Implement exponential (T = T0 * α^k) or linear cooling (T = T0 - k*β) to reduce search scope systematically.

### MATLAB Implementation Essentials Objective Function: Typically uses total path length calculated with distance matrix operations. Neighborhood Operations: Employ 2-opt optimization or segment reversal (circshift function) for candidate solutions. Temperature Control: Critical parameters include initial temperature (T0), final temperature (T_min), and cooling rate (α) requiring careful calibration. Stopping Criteria: Based on iteration counts (for-loop limits) or temperature thresholds (while T > T_min).

### Optimization Recommendations Balance convergence speed and accuracy by tuning initial temperature and cooling rates through parameter sweeping. Combine with local search techniques like greedy algorithms (min-function searches) to refine final solutions. Execute multiple algorithm runs (parfor parallelization) and select best results to mitigate randomness effects.

The algorithm performs well on medium-scale TSP instances, though very large problems may require hybrid approaches with other optimization strategies like genetic algorithms or ant colony optimization.