Kiyuob Jung, Jehan Oh
In this paper, we find the special case of the subgradient method minimizing a one-dimensional real-valued function, which we term the specular gradient method, that converges root-linearly without any additional assumptions except the convexity. Furthermore, we suggest a way to implement the specular gradient method without explicitly calculating specular derivatives.
Kiyuob Jung
This paper introduces specular differentiation, which generalizes Gâteaux and Fréchet differentiation in normed vector spaces. We investigate its fundamental theoretical properties and establish weak forms of the Mean Value Theorem and Fermat's Theorem in the specular sense. Finally, we identify a distinguished element of the Fréchet subdifferential of a convex function through specular differentiation.
Kiyuob Jung
This paper deals with nonsmooth convex optimization problems in Euclidean spaces. We identify special elements of the subdifferential of a convex function, called specular gradients. Based on this observation, we propose three numerical methods that use specular gradients in subgradient methods. We prove the convergence of the proposed methods under suitable step sizes. Numerical experiments demonstrate that the proposed methods are capable of minimizing non-differentiable functions that classical methods fail to minimize.
Kiyuob Jung
This paper proposes specular differentiation in one-dimensional Euclidean space and provides its fundamental analysis, including a quasi-Fermat theorem and a quasi-Mean Value Theorem. As an application, this paper develops several numerical schemes for solving initial value problems for first-order ordinary differential equations. Based on numerical simulations, we select one scheme and prove its second-order consistency and convergence. By modifying this scheme, we also obtain a numerical scheme with zero local truncation error for ODEs whose solution trajectories are ellipses.