Python Implementation

When optimization problems are trapped in the maze of local optima, deterministic algorithms are often helpless. This article takes you into the world of stochastic optimization, exploring how to transform the problem of finding minimum energy into finding maximum probability. We will delve into the physical intuition and mathematical principles of the Simulated Annealing algorithm, demonstrate its elegant mechanism of ‘high-temperature exploration, low-temperature locking’ through dynamic visualization, and derive the Pincus Theorem in detail, mathematically proving why the annealing algorithm can find the global optimal solution. [Read More]

Deterministic optimization is the cornerstone for understanding modern MCMC algorithms (like HMC, Langevin). This article delves into three classic deterministic optimization strategies: Newton’s Method (second-order perspective using curvature), Coordinate Descent (the divide-and-conquer predecessor to Gibbs), and Steepest Descent (greedy first-order exploration). Through mathematical derivation and Python visualization, we compare their behavioral patterns and convergence characteristics across different terrains (convex surfaces, narrow valleys, strong coupling). [Read More]

The Metropolis algorithm is the cornerstone of MCMC. We delve into its strategy for handling unnormalized densities, from the random walk mechanism to sampling 2D correlated Gaussians, complete with Python implementation and visual diagnostics. [Read More]