Metropolis Algorithm: Optimization and Probabilistic Approach

This lecture introduces the Metropolis algorithm, aiming to provide an approximate and probabilistic solution to discrete optimization problems by seeking the minimum of a function. The algorithm involves evolving a Markov chain with a state space and a limiting distribution. Different ideas are explored, such as defining a distribution on the state space and making the chain lazy. The instructor explains the process of creating a chain from a base state, letting it evolve to approximate a target distribution. Various strategies are discussed to make the chain lazy and explore the state space efficiently at different temperatures, leading to both local and global minima.