Explore a lecture on optimization techniques focusing on dealing with linear constraints through random permutation. Delve into advanced concepts like multi-block ADMM variants, randomization tricks, and their applications in solving large-scale problems. Learn about the divergence of cyclic ADMM, spectral analysis of switched linear systems, and novel randomization rules. Examine the comparison of various algorithms, including cyclic coordinate descent, and their convergence rates. Gain insights into the relationship between different optimization methods and discover a new variant of the matrix AM-GM inequality.