Convergence of symmetric rank-one method based on modified Quasi-Newton equation

In this paper we investigate on convergence rate of a modified symmetric rank-one (SR1) method for unconstrained optimization problems. In general, the modified SR1 method incorporates a modified secant equation into the standard SR1 method. Also a restart procedure is applied to avoid the loss of positive definiteness and zero denominator. A remarkable feature of the modified SR1 method is that it possesses at most $n+1$-step $q$-superlinearly convergent and $2n$-step quadratic convergent without uniformly independent assumptions of steps

PROVIDING A COMPREHENSIVE MODEL TO MEASURE THE PERFORMANCE DIMENSIONS OF INDUSTRIAL CLUSTERS USING THE HYBRID APPROACH OF Q-FACTOR ANALYSIS AND CLUSTER ANALYSIS

One of the most important development strategies with the emphasis on small and medium industries is the geographical concentration of production units and the formation of cluster. The industrial cluster is a globally economic phenomenon that has been proposed as a modern model for economic development. Theoretically, an industrial cluster can strengthen specialized sectors and facilitate industrial cooperation. The aim of this study was to provide a comprehensive model for measuring the performance dimensions of industrial clusters in Markazi province using a hybrid approach of Q-factor analysis and cluster analysis. For this purpose, at first and in the first phase of the research, 41 effective factors in the clustering of the performance dimensions of the statistical population were identified with the study of previous research and the use of Q-factor analysis, and in the second phase, a model for comprehensive performance measurement of industrial clusters was presented using cluster analysis in R software. The results of the study indicated that industrial clusters in Markazi province have four financial, competitive, economic and environmental performance dimensions

A whale optimization algorithm (WOA) approach for clustering

Clustering is a powerful technique in data-mining, which involves identifing homogeneous groups of objects based on the values of attributes. Meta-heuristic algorithms such as particle swarm optimization, artificial bee colony, genetic algorithm and differential evolution are now becoming powerful methods for clustering. In this paper, we propose a new meta-heuristic clustering method, the Whale Clustering Optimization Algorithm, based on the swarm foraging behavior of humpback whales. After a detailed formulation and explanation of its implementation, we will then compare the proposed algorithm with other existing well-known algorithms in clustering, including PSO, ABC, GA, DE and k-means. Proposed algorithm was tested using one artificial and seven real benchmark data sets from the UCI machine learning repository. Simulations show that the proposed algorithm can successfully be used for data clustering

Quasi-Newton methods based on ordinary differential equation approach for unconstrained nonlinear optimization

In this paper, we propose a hybrid ODE-based quasi-Newton (QN) method for unconstrained optimization problems, which combines the idea of low-order implicit Runge–Kutta (RK) techniques for gradient systems with the QN type updates of the Jacobian matrix such as the symmetric rank-one (SR1) update. The main idea of this approach is to associate a QN matrix to approximate numerically the Jacobian matrix in the gradient system. Fundamentally this is a gradient system based on the first order optimality conditions of the optimization problem. To further extend the methods for solving large-scale problems, a feature incorporated to the proposed methods is that a limited memory setting for matrix–vector multiplications is used, thus avoiding the computational and storage issues when computing Jacobian information. Under suitable assumptions, global convergence of the proposed method is proved. Practical insights on the effectiveness of these approaches on a set of test functions are given by a numerical comparison with that of the limited memory BFGS algorithm (L-BFGS) and conjugate gradient algorithm (CG)

Limited memory methods with improved symmetric rank-one updates and its applications on nonlinear image restoration

The iterative solution of unconstrained optimization problems has been found in a variety of significant applications of research areas, such as image restoration. In this paper, we present an efficient limited memory quasi-Newton technique based on symmetric rank-one updating formula to compute meaningful solutions for large-scale problems arising in some image restoration problems. Numerical experiments and comparisons on various well-known methods in the literature are presented to illustrate the effectiveness of the proposed method particularly for images of large size