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Register a new userA First-Order Primal-Dual Method for Nonconvex Constrained Optimization Based On the Augmented Lagrangian
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Nonlinearly constrained nonconvex and nonsmooth optimization models play an increasingly important role in machine learning, statistics and data analytics. In this paper, based on the augmented Lagrangian function we introduce a flexible first-order primal-dual method, to be called nonconvex auxiliary problem principle of augmented Lagrangian (NAPP-AL), for solving a class of nonlinearly constrained nonconvex and nonsmooth optimization problems. We demonstrate that NAPP-AL converges to a stationary solution at the rate of o(1/sqrt{k}), where k is the number of iterations. Moreover, under an additional error bound condition (to be called VP-EB in the paper), we further show that the convergence rate is in fact linear. Finally, we show that the famous Kurdyka- Lojasiewicz property and the metric subregularity imply the afore-mentioned VP-EB condition.
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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 Yuan have proposed a balanced version of ALM [B.S. He and X.M. Yuan, arXiv:2108.08554, 2021], which reshapes the original ALM by balancing its subproblems and makes the benchmark ALM easier to implement without any additional condition. In practice, the balanced ALM updates the new iterate by a primal-dual order. In this note, exploiting the variational inequality structure of the most recent balanced ALM, we propose a dual-primal version of the balanced ALM for linearly constrained convex minimization problems. The novel proposed method generates the new iterate by a dual-primal order and enjoys the same computational difficulty with the original primal-dual balanced ALM. Furthermore, under the lens of the proximal point algorithm, we conduct the convergence analysis of the novel introduced method in the context of variational inequalities. Numerical tests on the basic pursuit problem demonstrate that the introduced method enjoys the same high efficiency with the prototype balanced ALM.
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 problems with minimal requirements. We study the method of weighted dual averages (Nesterov, 2009) in this setting and prove that it is an optimal method.
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.
First-order methods (FOMs) have been widely used for solving large-scale problems. A majority of existing works focus on problems without constraint or with simple constraints. Several recent works have studied FOMs for problems with complicated functional constraints. In this paper, we design a novel augmented Lagrangian (AL) based FOM for solving problems with non-convex objective and convex constraint functions. The new method follows the framework of the proximal point (PP) method. On approximately solving PP subproblems, it mixes the usage of the inexact AL method (iALM) and the quadratic penalty method, while the latter is always fed with estimated multipliers by the iALM. We show a complexity result of $O(varepsilon^{-frac{5}{2}}|logvarepsilon|)$ for the proposed method to achieve an $varepsilon$-KKT point. This is the best known result. Theoretically, the hybrid method has lower iteration-complexity requirement than its counterpart that only uses iALM to solve PP subproblems, and numerically, it can perform significantly better than a pure-penalty-based method. Numerical experiments are conducted on nonconvex linearly constrained quadratic programs and nonconvex QCQP. The numerical results demonstrate the efficiency of the proposed methods over existing ones.
This paper is devoted to studying an inexact augmented Lagrangian method for solving a class of manifold optimization problems, which have non-smooth objective functions and non-negative constraints. Under the constant positive linear dependence condition on manifold, we show that the proposed method converges to a stationary point of the non-smooth manifold optimization problem. Moreover, we propose a globalized semi-smooth Newton method to solve the augmented Lagrangian subproblem on manifolds efficiently. The local superlinear convergence of the manifold semi-smooth Newton method is also established under some suitable conditions. Finally, numerical experiments on compressed modes and (constrained) sparse PCA illustrate the advantages of the proposed method in terms of accuracy and computational efficiency.
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