A variational perspective on accelerated methods in optimization

Accelerated gradient methods play a central role in optimization, achieving optimal rates in many settings. Although many generalizations and extensions of Nesterov’s original acceleration method have been proposed, it is not yet clear what is the natural scope of the acceleration concept. In this paper, we study accelerated methods from a continuous-time perspective. We show that there is a Lagrangian functional that we call the Bregman Lagrangian, which generates a large class of accelerated methods in continuous time, including (but not limited to) accelerated gradient descent, its non-Euclidean extension, and accelerated higher-order gradient methods. We show that the continuous-time limit of all of these methods corresponds to traveling the same curve in spacetime at different speeds. From this perspective, Nesterov’s technique and many of its generalizations can be viewed as a systematic way to go from the continuous-time curves generated by the Bregman Lagrangian to a family of discrete-time accelerated algorithms.

Correct Formulation of the Stability Problem for Timoshenko Beam

This article is about the nonlinear problems of the theory of elastic Cosserat – Timoshenko’s rods in the material (Lagrangian) description with energy conjugated vectors of forces, moments and strains. The variational formulations of static problems was given. The differential equations of the plane stability problems were obtained from the second variation of the Lagrangian functional. The exact solutions of the stability problems for basic types of the end fixities of the rod were obtained for the Timoshenko’s rod (taking into account only bending and shear stiffness). It appears that classical well-known equilibrium stability functional and stability equations for the Timoshenko’s rod are incorrect. Also well-known Engesser formula (with bending and shear stiffness) is incorrect. The numerical solution of the stability problems for hinged Timoshenko’s rod with rigid support was obtained. Also, simplified formula for this problem was derived using asymptotic analysis.