Explore a groundbreaking framework for accelerated methods in Convex Programming through this 55-minute talk from the Society for Industrial and Applied Mathematics. Delve into the Bi-Level Unconstrained Minimization (BLUM) approach, which utilizes approximations of high-order proximal points to surpass traditional limits in Complexity Theory. Discover a novel second-order method achieving a convergence rate of O(k^(-4)), outperforming existing methods for functions with Lipschitz continuous Hessian. Examine new techniques with exact auxiliary search procedures, boasting impressive convergence rates of O(k^(-(3p+1)/2)) for proximal operators of order p≥1. Gain insights from Yurii Nesterov of Université Catholique de Louvain, Belgium, as he presents these innovative concepts in convex optimization.