We present an alternating direction method based on an augmented Lagrangian framework for solving semidefinite programming (SDP) problems in standard form. At each iteration, the algorithm, also known as a two-splitting scheme, minimizes the dual augmented Lagrangian function sequentially with respect to the Lagrange multipliers corresponding to the linear constraints, then the dual slack varia...

In this paper we study decomposition methods based on separable approximations for minimizing the augmented Lagrangian. In particular, we study and compare the Diagonal Quadratic Approximation Method (DQAM) of Mulvey and Ruszczyński [13] and the Parallel Coordinate Descent Method (PCDM) of Richtárik and Takáč [23]. We show that the two methods are equivalent for feasibility problems up to the s...

A generally nonconvex optimization problem with equality constraints is studied. The problem is introduced as an “inf sup” of a generalized augmented Lagrangian function. A dual problem is defined as the “sup inf’ of the same generalized augmented Lagrangian. Sufftcient conditions are derived for constructing the augmented Lagrangian function such that the extremal values of the primal and dual...

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 ...

Local Convergence Analysis of Augmented Lagrangian Methods for Piecewise Linear-Quadratic Composite Optimization Problems

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...

This is a documentation of a framework for robotmotion optimization that aims to draw on classical constrained optimization methods. With one exception the underlying algorithms are classical ones: Gauss-Newton (with adaptive stepsize and damping), Augmented Lagrangian, log-barrier, etc. The exception is a novel any-time version of the Augmented Lagrangian. The contribution of this framework is...

Recently, many variational models involving high order derivatives have been widely used in image processing, because they can reduce staircase effects during noise elimination. However, it is very challenging to construct efficient algorithms to obtain the minimizers of original high order functionals. In this paper, we propose a new linearized augmented Lagrangian method for Euler’s elastica ...

We consider the augmented Lagrangian dual for integer programming, and provide a primal characterization of the resulting bound. As a corollary, we obtain proof that the augmented Lagrangian is a strong dual for integer programming. We are able to show that the penalty parameter applied to the augmented Lagrangian term may be placed at a fixed, large value and still obtain strong duality for pu...