scholarly journals A Preconditioning Technique for First-Order Primal-Dual Splitting Method in Convex Optimization

2017 ◽  
Vol 2017 ◽  
pp. 1-11 ◽  
Author(s):  
Meng Wen ◽  
Jigen Peng ◽  
Yuchao Tang ◽  
Chuanxi Zhu ◽  
Shigang Yue
Keyword(s):  
Iterative Algorithm ◽  
General Framework ◽  
Optimization Problems ◽  
Splitting Method ◽  
Iteration Process ◽  
First Order ◽  
Preconditioning Technique ◽  
Primal Dual ◽  
Preconditioned Iterative ◽  
Better Than

We introduce a preconditioning technique for the first-order primal-dual splitting method. The primal-dual splitting method offers a very general framework for solving a large class of optimization problems arising in image processing. The key idea of the preconditioning technique is that the constant iterative parameters are updated self-adaptively in the iteration process. We also give a simple and easy way to choose the diagonal preconditioners while the convergence of the iterative algorithm is maintained. The efficiency of the proposed method is demonstrated on an image denoising problem. Numerical results show that the preconditioned iterative algorithm performs better than the original one.

scholarly journals Sparse Constrained Reconstruction for Accelerating Parallel Imaging Based on Variable Splitting Method

2013 ◽  
Vol 2013 ◽  
pp. 1-10
Author(s):  
Wenlong Xu ◽  
Xiaofang Liu ◽  
Xia Li
Keyword(s):  
Magnetic Resonance ◽  
Parallel Imaging ◽  
Optimization Problems ◽  
Splitting Method ◽  
Second Order ◽  
Constrained Reconstruction ◽  
Reconstruction Process ◽  
First Order ◽  
Variable Splitting ◽  
Reconstruction Methods

Parallel imaging is a rapid magnetic resonance imaging technique. For the ill-conditioned problem, noise and aliasing artifacts are amplified during the reconstruction process and are serious especially for high accelerating imaging. In this paper, a sparse constrained reconstruction problem is proposed for parallel imaging, and an effective solution based on the variable splitting method is contrived. First-order and second-order norm optimization problems are first split, and then they are transferred to unconstrained minimization problem by the augmented Lagrangian method. At last, first-order norm and second-order norm optimization problems are alternatively resolved by different methods. With a discrepancy principle as the stopping criterion, analysis of simulated and actual parallel magnetic resonance image reconstruction is presented and discussed. Compared with the routine parallel imaging reconstruction methods, the results show that the noise and aliasing artifacts in the reconstructed image are evidently reduced at large acceleration factors.

Numerical Techniques for Nonlinear Lagrangian Dynamical Systems and Their Application

2012 ◽  
Vol 476-478 ◽  
pp. 1513-1516
Author(s):  
Jin Li
Keyword(s):  
Dynamical Systems ◽  
General Framework ◽  
Optimization Problems ◽  
Lagrangian Function ◽  
Lagrangian System ◽  
Numerical Techniques ◽  
Constrained Optimization Problems ◽  
First Order ◽  
Nonlinear Lagrangian ◽  
Special Case

We provide a general framework for solving constrained optimization problems, this framework relies on dynamical systems using a class of nonlinear Lagrangian function, we construct a first order derivatives based and a second order derivatives based differential systems. Under this framework, We show that the exponential Lagrangian system as the special case is discussed.

scholarly journals Accelerated Stochastic Algorithms for Convex-Concave Saddle-Point Problems

2021 ◽  
Author(s):  
Renbo Zhao
Keyword(s):  
Saddle Point ◽  
State Of The Art ◽  
Stochastic Algorithms ◽  
Stochastic Algorithm ◽  
Saddle Point Problems ◽  
First Order ◽  
Strongly Convex ◽  
Primal Dual ◽  
Oracle Complexity ◽  
Better Than

We develop stochastic first-order primal-dual algorithms to solve a class of convex-concave saddle-point problems. When the saddle function is strongly convex in the primal variable, we develop the first stochastic restart scheme for this problem. When the gradient noises obey sub-Gaussian distributions, the oracle complexity of our restart scheme is strictly better than any of the existing methods, even in the deterministic case. Furthermore, for each problem parameter of interest, whenever the lower bound exists, the oracle complexity of our restart scheme is either optimal or nearly optimal (up to a log factor). The subroutine used in this scheme is itself a new stochastic algorithm developed for the problem where the saddle function is nonstrongly convex in the primal variable. This new algorithm, which is based on the primal-dual hybrid gradient framework, achieves the state-of-the-art oracle complexity and may be of independent interest.

scholarly journals A First-Order Explicit-Implicit Splitting Method for a Convection-Diffusion Problem

2020 ◽  
Vol 20 (4) ◽  
pp. 769-782
Author(s):  
Amiya K. Pani ◽  
Vidar Thomée ◽  
A. S. Vasudeva Murthy
Keyword(s):  
Splitting Method ◽  
Diffusion Problem ◽  
Euler Method ◽  
Second Order ◽  
Time Step ◽  
Convection Diffusion ◽  
First Order ◽  
Backward Euler ◽  
Strang Splitting ◽  
Spatially Periodic

AbstractWe analyze a second-order in space, first-order in time accurate finite difference method for a spatially periodic convection-diffusion problem. This method is a time stepping method based on the first-order Lie splitting of the spatially semidiscrete solution. In each time step, on an interval of length k, of this solution, the method uses the backward Euler method for the diffusion part, and then applies a stabilized explicit forward Euler approximation on {m\geq 1} intervals of length {\frac{k}{m}} for the convection part. With h the mesh width in space, this results in an error bound of the form {C_{0}h^{2}+C_{m}k} for appropriately smooth solutions, where {C_{m}\leq C^{\prime}+\frac{C^{\prime\prime}}{m}}. This work complements the earlier study [V. Thomée and A. S. Vasudeva Murthy, An explicit-implicit splitting method for a convection-diffusion problem, Comput. Methods Appl. Math. 19 2019, 2, 283–293] based on the second-order Strang splitting.

scholarly journals A General Framework for Decentralized Optimization With First-Order Methods

2020 ◽  
Vol 108 (11) ◽  
pp. 1869-1889
Author(s):  
Ran Xin ◽  
Shi Pu ◽  
Angelia Nedic ◽  
Usman A. Khan
Keyword(s):  
General Framework ◽  
Decentralized Optimization ◽  
First Order ◽  
First Order Methods

scholarly journals Municipal Leachate Treatment by Fenton Process: Effect of Some Variable and Kinetics

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Mohammad Ahmadian ◽  
Sohyla Reshadat ◽  
Nader Yousefi ◽  
Seyed Hamed Mirhossieni ◽  
Mohammad Reza Zare ◽  
...  
Keyword(s):  
Reaction Time ◽  
Color Removal ◽  
Molar Ratio ◽  
Order Kinetic ◽  
Fenton Process ◽  
Solution Ph ◽  
Leachate Treatment ◽  
First Order ◽  
Order Kinetic Model ◽  
Better Than

Due to complex composition of leachate, the comprehensive leachate treatment methods have been not demonstrated. Moreover, the improper management of leachate can lead to many environmental problems. The aim of this study was application of Fenton process for decreasing the major pollutants of landfill leachate on Kermanshah city. The leachate was collected from Kermanshah landfill site and treated by Fenton process. The effect of various parameters including solution pH, Fe2+and H2O2dosage, Fe2+/H2O2molar ratio, and reaction time was investigated. The result showed that with increasing Fe2+and H2O2dosage, Fe2+/H2O2molar ratio, and reaction time, the COD, TOC, TSS, and color removal increased. The maximum COD, TOC, TSS, and color removal were obtained at low pH (pH: 3). The kinetic data were analyzed in term of zero-order, first-order, and second-order expressions. First-order kinetic model described the removal of COD, TOC, TSS, and color from leachate better than two other kinetic models. In spite of extremely difficulty of leachate treatment, the previous results seem rather encouraging on the application of Fenton’s oxidation.

Nonlinear Optimal Design of Dynamic Mechanical Systems

1992 ◽  
Author(s):  
T. E. Potter ◽  
K. D. Willmert ◽  
M. Sathyamoorthy
Keyword(s):  
Objective Function ◽  
Optimization Problems ◽  
Iteration Process ◽  
Optimization Method ◽  
Linear Constraints ◽  
Sum Of Squares ◽  
Function Evaluation ◽  
Deformation Analysis ◽  
Nonlinear Constraints ◽  
Quadratic Constraints

Abstract Mechanism path generation problems which use link deformations to improve the design lead to optimization problems involving a nonlinear sum-of-squares objective function subjected to a set of linear and nonlinear constraints. Inclusion of the deformation analysis causes the objective function evaluation to be computationally expensive. An optimization method is presented which requires relatively few objective function evaluations. The algorithm, based on the Gauss method for unconstrained problems, is developed as an extension of the Gauss constrained technique for linear constraints and revises the Gauss nonlinearly constrained method for quadratic constraints. The derivation of the algorithm, using a Lagrange multiplier approach, is based on the Kuhn-Tucker conditions so that when the iteration process terminates, these conditions are automatically satisfied. Although the technique was developed for mechanism problems, it is applicable to any optimization problem having the form of a sum of squares objective function subjected to nonlinear constraints.

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