scholarly journals Convergence of a distributed method for minimizing sum of convex functions with fixed point constraints

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Nawarat Ekkarntrong ◽  
Tipsuda Arunrat ◽  
Nimit Nimana
Keyword(s):  
Fixed Point ◽  
Convergence Rate ◽  
Numerical Experiment ◽  
Theoretical Result ◽  
Convex Functions ◽  
Optimization Problem ◽  
Distributed Optimization ◽  
Distributed Method

AbstractIn this paper, we consider a distributed optimization problem of minimizing sum of convex functions over the intersection of fixed-point constraints. We propose a distributed method for solving the problem. We prove the convergence of the generated sequence to the solution of the problem under certain assumption. We further discuss the convergence rate with an appropriate positive stepsize. A numerical experiment is given to show the effectiveness of the obtained theoretical result.

DISTRIBUTED PROXIMAL-GRADIENT METHOD FOR CONVEX OPTIMIZATION WITH INEQUALITY CONSTRAINTS

2014 ◽  
Vol 56 (2) ◽  
pp. 160-178 ◽  
Author(s):  
JUEYOU LI ◽  
CHANGZHI WU ◽  
ZHIYOU WU ◽  
QIANG LONG ◽  
XIANGYU WANG
Keyword(s):  
Convergence Rate ◽  
Optimization Problem ◽  
Optimization Problems ◽  
Distributed Optimization ◽  
Inequality Constraints ◽  
Gradient Algorithm ◽  
Subgradient Methods ◽  
Penalty Function Method ◽  
Proximal Gradient Method ◽  
Primal Dual

AbstractWe consider a distributed optimization problem over a multi-agent network, in which the sum of several local convex objective functions is minimized subject to global convex inequality constraints. We first transform the constrained optimization problem to an unconstrained one, using the exact penalty function method. Our transformed problem has a smaller number of variables and a simpler structure than the existing distributed primal–dual subgradient methods for constrained distributed optimization problems. Using the special structure of this problem, we then propose a distributed proximal-gradient algorithm over a time-changing connectivity network, and establish a convergence rate depending on the number of iterations, the network topology and the number of agents. Although the transformed problem is nonsmooth by nature, our method can still achieve a convergence rate, ${\mathcal{O}}(1/k)$, after $k$ iterations, which is faster than the rate, ${\mathcal{O}}(1/\sqrt{k})$, of existing distributed subgradient-based methods. Simulation experiments on a distributed state estimation problem illustrate the excellent performance of our proposed method.

scholarly journals Asynchronous Gossip-Based Gradient-Free Method for Multiagent Optimization

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Deming Yuan
Keyword(s):  
Objective Function ◽  
Convex Functions ◽  
Optimization Problem ◽  
Objective Functions ◽  
Step Size ◽  
Optimal Point ◽  
Specific Agent ◽  
Gossip Algorithm ◽  
Distributed Method

This paper considers the constrained multiagent optimization problem. The objective function of the problem is a sum of convex functions, each of which is known by a specific agent only. For solving this problem, we propose an asynchronous distributed method that is based on gradient-free oracles and gossip algorithm. In contrast to the existing work, we do not require that agents be capable of computing the subgradients of their objective functions and coordinating their step size values as well. We prove that with probability 1 the iterates of all agents converge to the same optimal point of the problem, for a diminishing step size.

Solving Mono- and Multi-Objective Problems Using Hybrid Evolutionary Algorithms and Nelder-Mead Method

2021 ◽  
Vol 12 (4) ◽  
pp. 98-116
Author(s):  
Noureddine Boukhari ◽  
Fatima Debbat ◽  
Nicolas Monmarché ◽  
Mohamed Slimane
Keyword(s):  
Convergence Rate ◽  
Optimization Problem ◽  
Optimization Problems ◽  
State Of The Art ◽  
Hybrid Approach ◽  
Optimization Methods ◽  
Global Optimum ◽  
Stochastic Methods ◽  
Local Optima ◽  
Portfolio Optimization Problem

Evolution strategies (ES) are a family of strong stochastic methods for global optimization and have proved their capability in avoiding local optima more than other optimization methods. Many researchers have investigated different versions of the original evolution strategy with good results in a variety of optimization problems. However, the convergence rate of the algorithm to the global optimum stays asymptotic. In order to accelerate the convergence rate, a hybrid approach is proposed using the nonlinear simplex method (Nelder-Mead) and an adaptive scheme to control the local search application, and the authors demonstrate that such combination yields significantly better convergence. The new proposed method has been tested on 15 complex benchmark functions and applied to the bi-objective portfolio optimization problem and compared with other state-of-the-art techniques. Experimental results show that the performance is improved by this hybridization in terms of solution eminence and strong convergence.

scholarly journals Heliostat Field Layout Optimization with Evolutionary Algorithms

10.29007/7p6t ◽  
2018 ◽  
Author(s):  
Pascal Richter ◽  
David Laukamp ◽  
Levin Gerdes ◽  
Martin Frank ◽  
Erika Ábrahám
Keyword(s):  
Power Plant ◽  
Convex Optimization ◽  
Evolutionary Algorithms ◽  
Convergence Rate ◽  
Evolutionary Algorithm ◽  
Computer Science ◽  
Energy Supply ◽  
Optimization Problem ◽  
Convex Optimization Problem ◽  
New Genotype

The exploitation of solar power for energy supply is of increasing importance. While technical development mainly takes place in the engineering disciplines, computer science offers adequate techniques for optimization. This work addresses the problem of finding an optimal heliostat field arrangement for a solar tower power plant.We propose a solution to this global, non-convex optimization problem by using an evolutionary algorithm. We show that the convergence rate of a conventional evolutionary algorithm is too slow, such that modifications of the recombination and mutation need to be tailored to the problem. This is achieved with a new genotype representation of the individuals.Experimental results show the applicability of our approach.

A content-adaptive unstructured grid based regularized CT reconstruction method with a SART-type preconditioned fixed-point proximity algorithm

2022 ◽  
Author(s):  
Yun Chen ◽  
Yao Lu ◽  
Xiangyuan Ma ◽  
Yuesheng Xu
Keyword(s):  
Integral Equation ◽  
Fixed Point ◽  
Optimization Problem ◽  
Unstructured Grid ◽  
Reconstruction Technique ◽  
Reconstruction Method ◽  
Ct Reconstruction ◽  
Image Domain ◽  
Computational Costs ◽  
Content Adaptive

Abstract The goal of this study is to develop a new computed tomography (CT) image reconstruction method, aiming at improving the quality of the reconstructed images of existing methods while reducing computational costs. Existing CT reconstruction is modeled by pixel-based piecewise constant approximations of the integral equation that describes the CT projection data acquisition process. Using these approximations imposes a bottleneck model error and results in a discrete system of a large size. We propose to develop a content-adaptive unstructured grid (CAUG) based regularized CT reconstruction method to address these issues. Specifically, we design a CAUG of the image domain to sparsely represent the underlying image, and introduce a CAUG-based piecewise linear approximation of the integral equation by employing a collocation method. We further apply a regularization defined on the CAUG for the resulting illposed linear system, which may lead to a sparse linear representation for the underlying solution. The regularized CT reconstruction is formulated as a convex optimization problem, whose objective function consists of a weighted least square norm based fidelity term, a regularization term and a constraint term. Here, the corresponding weighted matrix is derived from the simultaneous algebraic reconstruction technique (SART). We then develop a SART-type preconditioned fixed-point proximity algorithm to solve the optimization problem. Convergence analysis is provided for the resulting iterative algorithm. Numerical experiments demonstrate the outperformance of the proposed method over several existing methods in terms of both suppressing noise and reducing computational costs. These methods include the SART without regularization and with quadratic regularization on the CAUG, the traditional total variation (TV) regularized reconstruction method and the TV superiorized conjugate gradient method on the pixel grid.

Eigenvalue Decomposition Based Modified Newton Algorithm

2013 ◽  
Vol 347-350 ◽  
pp. 2586-2589
Author(s):  
Wen Jun Wang ◽  
Ju Bo Zhu ◽  
Xiao Jun Duan
Keyword(s):  
Convergence Rate ◽  
Numerical Experiment ◽  
Hessian Matrix ◽  
New Method ◽  
Eigenvalue Decomposition ◽  
Newton Algorithm ◽  
Negative Eigenvalues ◽  
Classical Algorithm ◽  
Newton Direction ◽  
Modified Algorithm

When the Hessian matrix is not positive, the Newton direction maybe not the descending direction. A new method named eigenvalue decomposition based modified Newton algorithm is presented, which first takes eigenvalue decomposition on the Hessian matrix, then replaces the negative eigenvalues with their absolutely values, finally reconstruct Hessian matrix and modify searching direction. The new searching direction is always the descending direction, and the convergence of the algorithm is proved and conclusion on convergence rate is presented qualitatively. At last, a numerical experiment is given for comparing the convergence domains of modified algorithm and classical algorithm.

scholarly journals Composite Differential Search Algorithm

2014 ◽  
Vol 2014 ◽  
pp. 1-15 ◽  
Author(s):  
Bo Liu
Keyword(s):  
Convergence Rate ◽  
Optimization Problem ◽  
Search Algorithm ◽  
Search Algorithms ◽  
Control Parameters ◽  
Original Algorithm ◽  
New Space ◽  
Random Method ◽  
Better Than

Differential search algorithm (DS) is a relatively new evolutionary algorithm inspired by the Brownian-like random-walk movement which is used by an organism to migrate. It has been verified to be more effective than ABC, JDE, JADE, SADE, EPSDE, GSA, PSO2011, and CMA-ES. In this paper, we propose four improved solution search algorithms, namely “DS/rand/1,” “DS/rand/2,” “DS/current to rand/1,” and “DS/current to rand/2” to search the new space and enhance the convergence rate for the global optimization problem. In order to verify the performance of different solution search methods, 23 benchmark functions are employed. Experimental results indicate that the proposed algorithm performs better than, or at least comparable to, the original algorithm when considering the quality of the solution obtained. However, these schemes cannot still achieve the best solution for all functions. In order to further enhance the convergence rate and the diversity of the algorithm, a composite differential search algorithm (CDS) is proposed in this paper. This new algorithm combines three new proposed search schemes including “DS/rand/1,” “DS/rand/2,” and “DS/current to rand/1” with three control parameters using a random method to generate the offspring. Experiment results show that CDS has a faster convergence rate and better search ability based on the 23 benchmark functions.

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