This User’s Guide describes the functionality and basic usage of the Matlab package TVAL3 for total variation minimization. The main algorithm used in TVAL3 is briefly introduced in the appendix.

We are interested in solving the system [ A LT L 0 ][ c λ ] = [ F G ] , (1) by a variant of the augmented Lagrangian algorithm. This type of problem with nonsymmetric A typically arises in certain discretizations of the Navier–Stokes equations. Here A is a (n,n) matrix, c, F ∈ R, L is a (m,n) matrix, and λ,G ∈ R. We assume that A is invertible on the kernel of L. Convergence rates of the augmen...

This paper develops a new error criterion for the approximate minimization of augmented Lagrangian subproblems. This criterion is practical in the sense that it requires only information that is ordinarily readily available, such as the gradient (or a subgradient) of the augmented Lagrangian. It is also “relative” in the sense of relative error criteria for proximal point algorithms, in that it...

At each outer iteration of standard Augmented Lagrangian methods one tries to solve a box-constrained optimization problem with some prescribed tolerance. In the continuous world, using exact arithmetic, this subproblem is always solvable. Therefore, the possibility of finishing the subproblem resolution without satisfying the theoretical stopping conditions is not contemplated in usual converg...

This paper demonstrates that for generalized methods of multipliers for convex programming based on Bregman distance kernels | including the classical quadratic method of multipliers | the minimization of the augmented Lagrangian can be truncated using a simple, generally implementable stopping criterion based only on the norms of the primal iterate and the gradient (or a subgradient) of the au...

Many applications arising in a variety of fields can be well illustrated by the task of recovering the low-rank and sparse components of a given matrix. Recently, it is discovered that this NP-hard task can be well accomplished, both theoretically and numerically, via heuristically solving a convex relaxation problem where the widely-acknowledged nuclear norm and l1 norm are utilized to induce ...

The alternating direction of multipliers (ADMM) is a form of augmented Lagrangian algorithm that has experienced a renaissance in recent years due to its applicability to optimization problems arising from “big data” and image processing applications, and the relative ease with which it may be implemented in parallel and distributed computational environments. This chapter aims to provide an ac...

We present a new derivative-free trust-region (DFTR) algorithm to solve general nonlinear constrained problems with the use of an augmented Lagrangian method. No derivatives are used, neither for the objective function nor for the constraints. An augmented Lagrangian method, known as an effective tool to solve equality and inequality constrained optimization problems with derivatives, is exploi...

Abstract In this paper, we present a majorized semismooth Newton-CG augmented Lagrangian method, called SDPNAL+, for semidefinite programming (SDP) with partial or full nonnegative constraints on the matrix variable. SDPNAL+ is a much enhanced version of SDPNAL introduced by Zhao et al. (SIAM J Optim 20:1737–1765, 2010) for solving generic SDPs. SDPNAL works very efficiently for nondegenerate S...