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We introduce a novel primal-dual flow for affine constrained convex optimization problem. As a modification of the standard saddle-point system, our primal-dual flow is proved to possesses the exponential decay property, in terms of a tailored Lyapunov function. Then a class of primal-dual methods for the original optimization problem are obtained from numerical discretizations of the continuous flow, and with a unified discrete Lyapunov function, nonergodic convergence rates are established. Among those algorithms, we can recover the (linearized) augmented Lagrangian method and the quadratic penalty method with continuation technique. Also, new methods with a special inner problem, that is a linear symmetric positive definite system or a nonlinear equation which may be solved efficiently via the semi-smooth Newton method, have been proposed as well. Especially, numerical tests on the linearly constrained $l_1$-$l_2$ minimization show that our method outperforms the accelerated linearized Bregman method.
This paper considers a general convex constrained problem setting where functions are not assumed to be differentiable nor Lipschitz continuous. Our motivation is in finding a simple first-order method for solving a wide range of convex optimization
This paper investigates accelerating the convergence of distributed optimization algorithms on non-convex problems. We propose a distributed primal-dual stochastic gradient descent~(SGD) equipped with powerball method to accelerate. We show that the
In this work, we introduce ADAPD, $textbf{A}$ $textbf{D}$ecentr$textbf{A}$lized $textbf{P}$rimal-$textbf{D}$ual algorithmic framework for solving non-convex and smooth consensus optimization problems over a network of distributed agents. ADAPD makes
The augmented Lagrangian method (ALM) is a fundamental tool for solving the canonical convex minimization problem with linear constraints, and efficiently and easily how to implement the original ALM is affirmatively significant. Recently, He and Yua
We study constrained stochastic programs where the decision vector at each time slot cannot be chosen freely but is tied to the realization of an underlying random state vector. The goal is to minimize a general objective function subject to linear c