TL;DR
This paper introduces an approach to study what is called the canonical conditioner μf, the continuity, Lipschitz continuity, differentiability, and subdifferentiability properties of μf\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb}
We propose an adaptive accelerated gradient method for solving smooth convex optimization problems. The method incorporates a scheme to determine the step size adaptively, by means of a local estimation of the smoothness constant, which is assumed unknown, without resorting to line search procedures. The sequence generated by this method converges weakly to a minimizer of the objective function, and the function values converge at a fast rate of O1k2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {O}\left( \frac{1}{k^2} \right) $$\end{document}. Moreover, if the objective function is strongly convex, the function values converge at a linear rate O1k2(1-ρ)k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {O}\left( \frac{1}{k^2}(1-\rho )^k \right) $$\end{document}, with ρ=OμL\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho =\mathcal {O}\left( \frac{\mu }{L} \right) $$\end{document}, without knowledge of the strong convexity parameter.
We characterize the well-posedness of the higher order regularity problem in the upper half-space with data in Sobolev Banach function spaces by proving its equivalence to natural weighted estimates for the Hardy–Littlewood maximal operator. The generality of our framework allows for applications to Lebesgue spaces, rearrangement-invariant spaces such as Orlicz spaces, and variable exponent Lebesgue spaces, as well as their weighted counterparts, among others. This is established for the family of second-order, homogeneous, elliptic, constant complex coefficient systems in Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}^n$$\end{document} that admit a distinguished coefficient tensor, a natural condition that always holds in the scalar case and for the Lamé system of elasticity.
We use cookies to run the site and, with your consent, for analytics and to show ads. See our Cookie Policy.
