Convergence Analysis of L-ADMM for Multi-block Linear-Constrained Separable Convex Minimization Problem

We focus on the convergence analysis of the extended linearized alternating direction method of multipliers(L-ADMM)for solving convex minimization problems with three or more separable blocks in the objective functions.Previous convergence analysis of the L-ADMM needs to reduce the multi-block convex minimization problems to two blocks by grouping the variables.Moreover,there has been no rate of convergence analysis for the L-ADMM.In this paper,we construct a counter example to show the failure of convergence of the extended L-ADMM.We prove the convergence and establish the sublinear convergence rate of the extended L-ADMM under the assumptions that the proximal gradient step sizes are smaller than certain values,and any two coefficient matrices in linear constraints are orthogonal.

DOUBLE INERTIAL PROXIMAL GRADIENT ALGORITHMS FOR CONVEX OPTIMIZATION PROBLEMS AND APPLICATIONS

In this paper, we propose double inertial forward-backward algorithms for solving unconstrained minimization problems and projected double inertial forward-backward algorithms for solving constrained minimization problems. We then prove convergence theorems under mild conditions. Finally, we provide numerical experiments on image restoration problem and image inpainting problem. The numerical results show that the proposed algorithms have more efficient than known algorithms introduced in the literature.

An accelerated augmented Lagrangian method for linearly constrained convex programming with the rate of convergence O(1/k^2)

In this paper, we propose and analyze an accelerated augmented Lagrangian method（denoted by AALM） for solving the linearly constrained convex programming. We show that the convergence rate of AALM is O（1/k^2） while the convergence rate of the classical augmented Lagrangian method（ALM） is O1 k. Numerical experiments on the linearly constrained 1-2minimization problem are presented to demonstrate the effectiveness of AALM.

ITERATIVE l1 MINIMIZATION FOR NON-CONVEX COMPRESSED SENSING

An algorithmic framework, based on the difference of convex functions algorithm （D- CA）, is proposed for minimizing a class of concave sparse metrics for compressed sensing problems. The resulting algorithm iterates a sequence ofl1 minimization problems. An exact sparse recovery theory is established to show that the proposed framework always improves on the basis pursuit （l1 minimization） and inherits robustness from it. Numerical examples on success rates of sparse solution recovery illustrate further that, unlike most existing non-convex compressed sensing solvers in the literature, our method always out- performs basis pursuit, no matter how ill-conditioned the measurement matrix is. Moreover, the iterative l1 （ILl） algorithm lead by a wide margin the state-of-the-art algorithms on l1/2 and logarithimic minimizations in the strongly coherent （highly ill-conditioned） regime, despite the same objective functions. Last but not least, in the application of magnetic resonance imaging （MRI）, IL1 algorithm easily recovers the phantom image with just 7 line projections.

AN INDEFINITE-PROXIMAL-BASED STRICTLY CONTRACTIVE PEACEMAN-RACHFORD SPLITTING METHOD

The Peaceman-Rachford splitting method is efficient for minimizing a convex optimization problem with a separable objective function and linear constraints.However,its convergence was not guaranteed without extra requirements.He et al.(SIAM J.Optim.24:1011-1040,2014)proved the convergence of a strictly contractive Peaceman-Rachford splitting method by employing a suitable underdetermined relaxation factor.In this paper,we further extend the so-called strictly contractive Peaceman-Rachford splitting method by using two different relaxation factors.Besides,motivated by the recent advances on the ADMM type method with indefinite proximal terms,we employ the indefinite proximal term in the strictly contractive Peaceman-Rachford splitting method.We show that the proposed indefinite-proximal strictly contractive Peaceman-Rachford splitting method is convergent and also prove the o(1/t)convergence rate in the nonergodic sense.The numerical tests on the l 1 regularized least square problem demonstrate the efficiency of the proposed method.

Sparse Estimation of High-Dimensional Inverse Covariance Matrices with Explicit Eigenvalue Constraints

Firstly,this paper proposes a generalized log-determinant optimization model with the purpose of estimating the high-dimensional sparse inverse covariance matrices.Under the normality assumption,the zero components in the inverse covariance matrices represent the conditional independence between pairs of variables given all the other variables.The generalized model considered in this study,because of the setting of the eigenvalue bounded constraints,covers a large number of existing estimators as special cases.Secondly,rather than directly tracking the challenging optimization problem,this paper uses a couple of alternating direction methods of multipliers(ADMM)to solve its dual model where 5 separable structures are contained.The first implemented algorithm is based on a single Gauss–Seidel iteration,but it does not necessarily converge theoretically.In contrast,the second algorithm employs the symmetric Gauss–Seidel(sGS)based ADMM which is equivalent to the 2-block iterative scheme from the latest sGS decomposition theorem.Finally,we do numerical simulations using the synthetic data and the real data set which show that both algorithms are very effective in estimating high-dimensional sparse inverse covariance matrix.

CT local reconstruction of solid rocket motor based on PI-POCS-TVM

In order to test the defects of solid rocket motor(SRM)accurately and efficiently,the computed tomography(CT)inspection scheme for SRM's defects was investigated and a CT local reconstruction algorithm called prior information-projection onto convex sets-total variation minimization(PI-POCS-TVM)was developed.The SRM was first inspected by industrial CT(ICT)to generate a low-resolution SRM image used as the prior information,then high-resolution local inspection was carried out for SRM's defects.To validate the effectiveness of the CT inspection scheme for SRM's defects,one SRM was inspected by a narrow fan beam ICT system.Filtered back projection(FBP)algorithm,POCS,POCS-TVM and PI-POCS-TVM algorithms were applied to reconstruct CT images.The performance of these algorithms was also compared.Results demonstrated the effectiveness of the CT inspection scheme for SRM's defects.Moreover,the PI-POCS-TVM algorithm has better local reconstruction image quality than the other three algorithms,showing great significance for accurate measurement of SRM's defects.

A STOCHASTIC MOVING BALLS APPROXIMATION METHOD OVER A SMOOTH INEQUALITY CONSTRAINT

We consider the problem of minimizing the average of a large number of smooth component functions over one smooth inequality constraint.We propose and analyze a stochastic Moving Balls Approximation(SMBA)method.Like stochastic gradient(SG)met hods,the SMBA method's iteration cost is independent of the number of component functions and by exploiting the smoothness of the constraint function,our method can be easily implemented.Theoretical and computational properties of SMBA are studied,and convergence results are established.Numerical experiments indicate that our algorithm dramatically outperforms the existing Moving Balls Approximation algorithm(MBA)for the structure of our problem.