The Augmented Lagrangian Method (ALM) is an iterative method for the solution of equality-constrained non-linear programming problems. In contrast to the quadratic penalty method, the ALM can satisfy equality constraints in an exact way. Further, ALM is claimed to converge in less iterations, indicating that it is superior in approach to a quadratic penalty method.
It is referred to as an advantage that the ALM solves equality constraints in an exact way, meaning that the penalty parameter does not need to go to infinity in order to yield accurate feasibility for the constraints. However, we sometimes actually want to minimize an unconstrained problem that has large quadratic penalty terms. In these situations it is unclear how the ALM could be utilized in the correct way. This paper presents the answer: We derive a modified version of the ALM that is also suitable for solving functions with large quadratic penalty terms.
In this paper, we consider the minimization of a Tikhonov functional with an $\ell_1$ penalty for solving linear inverse problems with sparsity constraints. One of the many approaches used to solve this problem uses the Nemskii operator to transform the Tikhonov functional into one with an $\ell_2$ penalty term but a nonlinear operator. The transformed problem can then be analyzed and minimized using standard methods. However, by the nature of this transform, the resulting functional is only once continuously differentiable, which prohibits the use of second order methods. Hence, in this paper…
In this work, we shed light on the so-called Kaczmarz method for solving Linear System (LS) and Linear Feasibility (LF) problems from a optimization point of view. We introduce well-known optimization approaches such as Lagrangian penalty and Augmented Lagrangian in the Randomized Kaczmarz (RK) method. In doing so, we propose two variants of the RK method namely the Randomized Penalty Kacmarz (RPK) method and Randomized Augmented Kacmarz (RAK) method. We carry out convergence analysis of the proposed methods and obtain linear convergence results.
We develop two fundamental stochastic sketching techniques; Penalty Sketching (PS) and Augmented Lagrangian Sketching (ALS) for solving consistent linear systems. The proposed PS and ALS techniques extend and generalize the scope of Sketch & Project (SP) method by introducing Lagrangian penalty sketches. In doing so, we recover SP methods as special cases and furthermore develop a family of new stochastic iterative methods. By varying sketch parameters in the proposed PS method, we recover novel stochastic methods such as Penalty Newton Descent, Penalty Kaczmarz, Penalty Stochastic Descent, Pe…
We propose two variants of Newton method for solving unconstrained minimization problem. Our method leverages optimization techniques such as penalty and augmented Lagrangian method to generate novel variants of the Newton method namely the Penalty Newton method and the Augmented Newton method. In doing so, we recover several well-known existing Newton method variants such as Damped Newton, Levenberg, and Levenberg-Marquardt methods as special cases. Moreover, the proposed Augmented Newton method can be interpreted as Newton method with adaptive heavy ball momentum. We provide global convergen…
On the convergence of iterated penalty methods for structure-preserving discretizations of saddle point problems
University of Oxford（牛津大学）；Charles University（查理大学）；University of Pittsburgh（匹兹堡大学）；U.S. Naval Research Laboratory（美国海军研究实验室）；University of Chicago（芝加哥大学）
A crucial issue in two-dimensional Nuclear Magnetic Resonance (NMR) is the speed and accuracy of the data inversion. This paper proposes a multi-penalty method with locally adapted regularization parameters for fast and accurate inversion of 2DNMR data. The method solves an unconstrained optimization problem whose objective contains a data-fitting term, a single $L1$ penalty parameter and a multiple parameter $L2$ penalty. We propose an adaptation of the Fast Iterative Shrinkage and Thresholding (FISTA) method to solve the multi-penalty minimization problem, and an automatic procedure to compu…
A Convergent ADMM Algorithm for Grain Boundary Energy Minimization
The Hong Kong University of Science and Technology（香港科技大学）；Shenzhen University（深圳大学）；HKUST Shenzhen-Hong Kong Collaborative Innovation Research Institute（港深创新研究院）
This paper is concerned with a novel deep learning method for variational problems with essential boundary conditions. To this end, we first reformulate the original problem into a minimax problem corresponding to a feasible augmented Lagrangian, which can be solved by the augmented Lagrangian method in an infinite dimensional setting. Based on this, by expressing the primal and dual variables with two individual deep neural network functions, we present an augmented Lagrangian deep learning method for which the parameters are trained by the stochastic optimization method together with a proje…
In this paper, we propose a novel overlapping domain decomposition method that can be applied to various problems in variational imaging such as total variation minimization. Most of recent domain decomposition methods for total variation minimization adopt the Fenchel--Rockafellar duality, whereas the proposed method is based on the primal formulation. Thus, the proposed method can be applied not only to total variation minimization but also to those with complex dual problems such as higher order models. In the proposed method, an equivalent formulation of the model problem with parallel str…
University of Innsbruck（因斯布鲁克大学）；University of Idaho（爱达荷大学）
We propose aNETT (augmented NETwork Tikhonov) regularization as a novel data-driven reconstruction framework for solving inverse problems. An encoder-decoder type network defines a regularizer consisting of a penalty term that enforces regularity in the encoder domain, augmented by a penalty that penalizes the distance to the data manifold. We present a rigorous convergence analysis including stability estimates and convergence rates. For that purpose, we prove the coercivity of the regularizer used without requiring explicit coercivity assumptions for the networks involved. We propose a possi…
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…
Total Variation, Adaptive Total Variation and Nonconvex Smoothly Clipped Absolute Deviation Penalty for Denoising Blocky Images
School of Computing Sciences, VIT University, Vellore, TN, India（计算机科学学院，VIT大学，Vellore，印度）；Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore, 637371（数学科学系，南洋理工大学，新加坡）
In this work we reformulate the method presented in App. Opt. 53:2297 (2014) as a constrained minimization problem using the augmented Lagrangian method. First we introduce the new method and then describe the numerical solution, which results in a simple algorithm. Numerical experiments with both synthetic and real fringe patterns show the accuracy and simplicity of the resulting algorithm.
In this paper, we introduce a class of nonsmooth nonconvex least square optimization problem using convex analysis tools and we propose to use the iterative minimization-majorization (MM) algorithm on a convex set with initializer away from the origin to find an optimal point for the optimization problem. For this, first we use an approach to construct a class of convex majorizers which approximate the value of non-convex cost function on a convex set. The convergence of the iterative algorithm is guaranteed when the initial point $x^{(0)}$ is away from the origin and the iterative points $x^{…
