scholarly journals Relaxed augmented Lagrangian-based proximal point algorithms for convex optimization with linear constraints

2014 ◽  
Vol 10 (3) ◽  
pp. 743-759 ◽  
Author(s):  
Yuan Shen ◽  
◽  
Wenxing Zhang ◽  
Bingsheng He ◽  
◽  
...  
Keyword(s):  
Convex Optimization ◽  
Augmented Lagrangian ◽  
Linear Constraints ◽  
Proximal Point ◽  
Proximal Point Algorithms

scholarly journals Proximal point algorithms involving fixed point iteration for nonexpansive mappings in CAT(κ) spaces

2018 ◽  
Vol 34 (2) ◽  
pp. 229-237
Author(s):  
NUTTAPOL PAKKARANANG ◽  
◽  
POOM KUMAM ◽  
PRASIT CHOLAMJIAK ◽  
RAWEEROTE SUPARATULATORN ◽  
...  
Keyword(s):  
Fixed Point ◽  
Convex Optimization ◽  
Iterative Algorithm ◽  
Proximal Point Algorithm ◽  
Nonexpansive Mappings ◽  
Proximal Point ◽  
Fixed Point Iteration ◽  
Proximal Point Algorithms ◽  
Common Solution ◽  
Mild Conditions

In this paper, we propose a new modified proximal point algorithm involving fixed point iteration for nonexpansive mappings in CAT(1) spaces. Under some mild conditions, we prove that the sequence generated by our iterative algorithm ∆-converges to a common solution between certain convex optimization and fixed point problems.

Modified proximal symmetric ADMMs for multi-block separable convex optimization with linear constraints

2021 ◽  
pp. 1-28
Author(s):  
Yuan Shen ◽  
Yannian Zuo ◽  
Liming Sun ◽  
Xiayang Zhang
Keyword(s):  
Convex Optimization ◽  
Splitting Method ◽  
Principal Component ◽  
Convergence Result ◽  
Linear Constraints ◽  
Convex Optimization Problem ◽  
Separable Convex Programming ◽  
Separable Convex Optimization ◽  
Linearly Constrained ◽  
Correction Step

We consider the linearly constrained separable convex optimization problem whose objective function is separable with respect to [Formula: see text] blocks of variables. A bunch of methods have been proposed and extensively studied in the past decade. Specifically, a modified strictly contractive Peaceman–Rachford splitting method (SC-PRCM) [S. H. Jiang and M. Li, A modified strictly contractive Peaceman–Rachford splitting method for multi-block separable convex programming, J. Ind. Manag. Optim. 14(1) (2018) 397-412] has been well studied in the literature for the special case of [Formula: see text]. Based on the modified SC-PRCM, we present modified proximal symmetric ADMMs (MPSADMMs) to solve the multi-block problem. In MPSADMMs, all subproblems but the first one are attached with a simple proximal term, and the multipliers are updated twice. At the end of each iteration, the output is corrected via a simple correction step. Without stringent assumptions, we establish the global convergence result and the [Formula: see text] convergence rate in the ergodic sense for the new algorithms. Preliminary numerical results show that our proposed algorithms are effective for solving the linearly constrained quadratic programming and the robust principal component analysis problems.

scholarly journals A Splitting Augmented Lagrangian Method for Low Multilinear-Rank Tensor Recovery

2015 ◽  
Vol 32 (01) ◽  
pp. 1540008 ◽  
Author(s):  
Lei Yang ◽  
Zheng-Hai Huang ◽  
Yu-Fan Li
Keyword(s):  
Augmented Lagrangian ◽  
Augmented Lagrangian Method ◽  
Convex Relaxation ◽  
Linear Constraints ◽  
Lagrangian Method ◽  
Splitting Technique ◽  
Completion Problem ◽  
Rank Tensor ◽  
Tensor Recovery ◽  
Multilinear Rank

This paper studies a recovery task of finding a low multilinear-rank tensor that fulfills some linear constraints in the general settings, which has many applications in computer vision and graphics. This problem is named as the low multilinear-rank tensor recovery problem. The variable splitting technique and convex relaxation technique are used to transform this problem into a tractable constrained optimization problem. Considering the favorable structure of the problem, we develop a splitting augmented Lagrangian method (SALM) to solve the resulting problem. The proposed algorithm is easily implemented and its convergence can be proved under some conditions. Some preliminary numerical results on randomly generated and real completion problems show that the proposed algorithm is very effective and robust for tackling the low multilinear-rank tensor completion problem.

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