A Survey on Some Recent Developments of Alternating Direction Method of Multipliers

Recently, alternating direction method of multipliers (ADMM) attracts much attentions from various fields and there are many variant versions tailored for different models. Moreover, its theoretical studies such as rate of convergence and extensions to nonconvex problems also achieve much progress. In this paper, we give a survey on some recent developments of ADMM and its variants.

Fast Global Minimization of the Chan–Vese Model for Image Segmentation Problem

The segmentation of weak boundary is still a difficult problem, especially sensitive to noise, which leads to the failure of segmentation. Based on the previous works, by adding the boundary indicator function with L 2,1 norm, a new convergent variational model is proposed. A novel strategy for the weak boundary image is presented. The existence of the minimizer for our model is given, by using the alternating direction method of multipliers (ADMMs) to solve the model. The experiments show that our new method is robust in segmentation of objects in a range of images with noise, low contrast, and direction.

Efficient Proximal Gradient Algorithms for Joint Graphical Lasso

We consider learning as an undirected graphical model from sparse data. While several efficient algorithms have been proposed for graphical lasso (GL), the alternating direction method of multipliers (ADMM) is the main approach taken concerning joint graphical lasso (JGL). We propose proximal gradient procedures with and without a backtracking option for the JGL. These procedures are first-order methods and relatively simple, and the subproblems are solved efficiently in closed form. We further show the boundedness for the solution of the JGL problem and the iterates in the algorithms. The numerical results indicate that the proposed algorithms can achieve high accuracy and precision, and their efficiency is competitive with state-of-the-art algorithms.

Hybrid FSK-PSK Waveform Optimization for Radar Based on Alternating Direction Method of Multiplier (ADMM)

In this paper, a new radar signal modulated with a hybrid of the frequency shift keying (FSK) and the phase shift keying (PSK) signal—i.e., the FSK-PSK signal—is studied. Different phase encoding sequences are used to modulate the sub-pulses to obtain lower sidelobe levels and ensure signal orthogonality. In addition, to counter intra-pulse slice repeater jamming of specific length generated by the enemy jammer, an orthogonal waveform made of sub-pulses of equal length based on the FSK-PSK modulation scheme is designed. The simulation results show that the optimized discrete phase encoding sequence can significantly enhance the orthogonality of the sub-pulse in the FSK-PSK signal and effectively suppress the slice repeater jamming. Two algorithms are proposed: (1) the low sidelobe waveform optimization algorithm based on ADMM (LSW-ADMM); and (2) the anti-slice-repeater-jamming algorithm based on ADMM (ASRJ-ADMM). Both algorithms exhibit fast convergence speed and low computational complexity.

Fast Algorithms for LS and LAD-Collaborative Regression

With the development of modern science and technology, it is easy to obtain a large number of high-dimensional datasets, which are related but different. Classical unimodel analysis is less likely to capture potential links between the different datasets. Recently, a collaborative regression model based on least square (LS) method for this problem has been proposed. In this paper, we propose a robust collaborative regression based on the least absolute deviation (LAD). We give the statistical interpretation of the LS-collaborative regression and LAD-collaborative regression. Then we design an efficient symmetric Gauss–Seidel-based alternating direction method of multipliers algorithm to solve the two models, which has the global convergence and the Q-linear rate of convergence. Finally we report numerical experiments to illustrate the efficiency of the proposed methods.