scholarly journals On the asymptotic behavior of the Douglas–Rachford and proximal-point algorithms for convex optimization

2021 ◽  
Author(s):  
Goran Banjac ◽  
John Lygeros
Keyword(s):  
Convex Optimization ◽  
Optimization Problems ◽  
Proximal Point ◽  
Finite Dimensional ◽  
Constraint Set ◽  
Optim Theory ◽  
The Difference ◽  
Primal And Dual ◽  
Closed Convex Set ◽  
Euclidean Setting

AbstractBanjac et al. (J Optim Theory Appl 183(2):490–519, 2019) recently showed that the Douglas–Rachford algorithm provides certificates of infeasibility for a class of convex optimization problems. In particular, they showed that the difference between consecutive iterates generated by the algorithm converges to certificates of primal and dual strong infeasibility. Their result was shown in a finite-dimensional Euclidean setting and for a particular structure of the constraint set. In this paper, we extend the result to real Hilbert spaces and a general nonempty closed convex set. Moreover, we show that the proximal-point algorithm applied to the set of optimality conditions of the problem generates similar infeasibility certificates.

scholarly journals Decentralized and parallel primal and dual accelerated methods for stochastic convex programming problems

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Darina Dvinskikh ◽  
Alexander Gasnikov
Keyword(s):  
Convex Optimization ◽  
Inverse Problems ◽  
Convex Programming ◽  
Data Science ◽  
Optimization Problems ◽  
Stochastic Gradient ◽  
Logarithmic Factor ◽  
Accelerated Methods ◽  
Convex Optimization Problems ◽  
Primal And Dual

Abstract We introduce primal and dual stochastic gradient oracle methods for decentralized convex optimization problems. Both for primal and dual oracles, the proposed methods are optimal in terms of the number of communication steps. However, for all classes of the objective, the optimality in terms of the number of oracle calls per node takes place only up to a logarithmic factor and the notion of smoothness. By using mini-batching technique, we show that the proposed methods with stochastic oracle can be additionally parallelized at each node. The considered algorithms can be applied to many data science problems and inverse problems.

scholarly journals A proximal point method for nonsmooth convex optimization problems in Banach spaces

1997 ◽  
Vol 2 (1-2) ◽  
pp. 97-120 ◽  
Author(s):  
Y. I. Alber ◽  
R. S. Burachik ◽  
A. N. Iusem
Keyword(s):  
Banach Space ◽  
Convex Optimization ◽  
Banach Spaces ◽  
Optimization Problems ◽  
Geometric Characteristic ◽  
Point Method ◽  
Proximal Point Method ◽  
Projection Operators ◽  
Proximal Point ◽  
Generalized Projection Operators

In this paper we show the weak convergence and stability of the proximal point method when applied to the constrained convex optimization problem in uniformly convex and uniformly smooth Banach spaces. In addition, we establish a nonasymptotic estimate of convergence rate of the sequence of functional values for the unconstrained case. This estimate depends on a geometric characteristic of the dual Banach space, namely its modulus of convexity. We apply a new technique which includes Banach space geometry, estimates of duality mappings, nonstandard Lyapunov functionals and generalized projection operators in Banach spaces.

scholarly journals A Line-Search-Based Partial Proximal Alternating Directions Method for Separable Convex Optimization

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Yu-hua Zeng ◽  
Yu-fei Yang ◽  
Zheng Peng
Keyword(s):  
Convex Optimization ◽  
Line Search ◽  
Optimization Problems ◽  
Proximal Point Method ◽  
Search Technique ◽  
Proximal Point ◽  
Separable Convex Optimization ◽  
Line Search Technique ◽  
Alternating Directions Method ◽  
Proximal Term

We propose an appealing line-search-based partial proximal alternating directions (LSPPAD) method for solving a class of separable convex optimization problems. These problems under consideration are common in practice. The proposed method solves two subproblems at each iteration: one is solved by a proximal point method, while the proximal term is absent from the other. Both subproblems admit inexact solutions. A line search technique is used to guarantee the convergence. The convergence of the LSPPAD method is established under some suitable conditions. The advantage of the proposed method is that it provides the tractability of the subproblem in which the proximal term is absent. Numerical tests show that the LSPPAD method has better performance compared with the existing alternating projection based prediction-correction (APBPC) method if both are employed to solve the described problem.

scholarly journals New inertial proximal gradient methods for unconstrained convex optimization problems

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Peichao Duan ◽  
Yiqun Zhang ◽  
Qinxiong Bu
Keyword(s):  
Convex Optimization ◽  
Optimization Problems ◽  
Gradient Methods ◽  
Gradient Algorithm ◽  
Proximal Gradient Method ◽  
Proximal Point ◽  
Convex Optimization Problem ◽  
Convex Optimization Problems ◽  
Acceleration Methods ◽  
Weak Convergence Theorem

AbstractThe proximal gradient method is a highly powerful tool for solving the composite convex optimization problem. In this paper, firstly, we propose inexact inertial acceleration methods based on the viscosity approximation and proximal scaled gradient algorithm to accelerate the convergence of the algorithm. Under reasonable parameters, we prove that our algorithms strongly converge to some solution of the problem, which is the unique solution of a variational inequality problem. Secondly, we propose an inexact alternated inertial proximal point algorithm. Under suitable conditions, the weak convergence theorem is proved. Finally, numerical results illustrate the performances of our algorithms and present a comparison with related algorithms. Our results improve and extend the corresponding results reported by many authors recently.

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