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.

Clustering is one of the most fundamental and important tasks in data mining. Traditional clustering algorithms, such as K-means, assign every data point to exactly one cluster. However, in real-world datasets, the clusters may overlap with each other. Furthermore, often, there are outliers that should not belong to any cluster. We recently proposed the NEO-K-Means (Non-Exhaustive, Overlapping ...

where f : Rn → R and g : Rn → Rp are nonlinear continuous functions and Ω = {x ∈ Rn : −∞ < l ≤ x ≤ u < ∞}. Problems with equality constraints, h(x) = 0, can be reformulated into the above form by converting into a couple of inequality constraints h(x)− β ≤ 0 and −h(x)− β ≤ 0, where β is a small positive relaxation parameter. Since we do not assume that the objective function f is convex, the pr...

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...