Shokri Z. Selim

A simulated annealing algorithm for the clustering problem

Abstract In this paper we discuss the solution of the clustering problem usually solved by the K -means algorithm. The problem is known to have local minimum solutions which are usually what the K -means algorithm obtains. The simulated annealing approach for solving optimization problems is described and is proposed for solving the clustering problem. The parameters of the algorithm are discussed in detail and it is shown that the algorithm converges to a global solution of the clustering problem. We also find optimal parameters values for a specific class of data sets and give recommendations on the choice of parameters for general data sets. Finally, advantages and disadvantages of the approach are presented.

A simulated annealing algorithm for unit commitment

This paper presents a simulated annealing algorithm (SAA) to solve the unit commitment problem (UCP). New rules for randomly generating feasible solutions are introduced. The problem has two subproblems: a combinatorial optimization problem; and a nonlinear programming problem. The former is solved using the SAA while the latter problem is solved via a quadratic programming routine. Numerical results showed an improvement in the solutions costs compared to previously obtained results.

Integrating genetic algorithms, tabu search, and simulated annealing for the unit commitment problem

This paper presents a new algorithm based on integrating genetic algorithms, tabu search and simulated annealing methods to solve the unit commitment problem. The core of the proposed algorithm is based on genetic algorithms. Tabu search is used to generate new population members in the reproduction phase of the genetic algorithm. A simulated annealing method is used to accelerate the convergence of the genetic algorithm by applying the simulated annealing test for all the population members. A new implementation of the genetic algorithm is introduced. The genetic algorithm solution is coded as a mix between binary and decimal representation. The fitness function is constructed from the total operating cost of the generating units without penalty terms. In the tabu search part of the proposed algorithm, a simple short-term memory procedure is used to counter the danger of entrapment at a local optimum, and the premature convergence of the genetic algorithm. A simple cooling schedule has been implemented to apply the simulated annealing test in the algorithm. Numerical results showed the superiority of the solutions obtained compared to genetic algorithms, tabu search and simulated annealing methods, and to two exact algorithms.

New algorithms for solving the fuzzy clustering problem

Abstract Two new algorithms for fuzzy clustering are presented. Convergence of the proposed algorithms is proved. An empirical study of their convergence behavior is discussed. The performance of the new algorithms is compared with the fuzzy c-means algorithm by testing them on four published data sets. Experimental results show that the new algorithms are faster and lead to computational savings.

Soft clustering of multidimensional data: a semi-fuzzy approach

Abstract This paper discusses new approaches to unsupervised fuzzy classification of multidimensional data. In the developed clustering models, patterns are considered to belong to some but not necessarily all clusters. Accordingly, such algorithms are called ‘semi-fuzzy’ or ‘soft’ clustering techniques. Several models to achieve this goal are investigated and corresponding implementation algorithms are developed. Experimental results are reported.

Fuzzy c-means: Optimality of solutions and effective termination of the algorithm

Abstract In this paper, the solutions produced by the fuzzy c -means algorithm for a general class of problems are examined and a method to test for the local optimality of such solutions is established. An equivalent mathematical program is defined for the c -means problem utilizing a generalized norm, then the properties of the resulting optimization problem are investigated. It is shown that the gradient of the resulting objective function at the solution produced by the c -means algorithm in this case takes a special structure which can be used in terminating the algorithm. Moreover, the local optimality of the solution obtained is checked utilizing the Hessian of the criterion function. The solution is a local minimum point if the Hessian matrix at this point is positive semidefinite. Simple rules are proposed to help in checking the definiteness of the matrix.

A new genetic-based tabu search algorithm for unit commitment problem

This paper presents a new algorithm based on integrating the use of genetic algorithms and tabu search methods to solve the unit commitment problem. The proposed algorithm, which is mainly based on genetic algorithms incorporates tabu search method to generate new population members in the reproduction phase of the genetic algorithm. In the proposed algorithm, genetic algorithm solution is coded as a mix between binary and decimal representation. A fitness function is constructed from the total operating cost of the generating units without penalty terms. In the tabu search part of the algorithm, a simple short term memory procedure is used to counter the danger of entrapment at a local optimum by preventing cycling of solutions, and the premature convergence of the genetic algorithm. A significant improvement of the proposed algorithm results, over those obtained by either genetic algorithm or tabu search, has been achieved. Numerical examples also showed the superiority of the proposed algorithm compared with two classical methods in the literature.

A global algorithm for the fuzzy clustering problem

Abstract The Fuzzy clustering (FC) problem is a non-convex mathematical program which usually possesses several local minima. The global minimum solution of the problem is found using a simulated annealing-based algorithm. Some preliminary computational experiments are reported and the solution is compared with that generated by the Fuzzy C-means algorithm.

A thresholded fuzzy c -means algorithm for semi-fuzzy clustering

Abstract In this paper, the problem of achieving “semi-fuzzy” or “soft” clustering of multidimensional data is discussed. A technique based on thresholding the results of the fuzzy c -means algorithm is introduced. The proposed approach is analysed and contrasted with the soft clustering method (see S. Z. Selim and M. A. Ismail, Pattern Recognition 17 , 559–568) showing the merits of the new method. Separation of clusters in the semi-fuzzy clustering context is introduced and the use of the proposed technique to measure the degree of separation is explained.

A relaxation approach to the fuzzy clustering problem

Abstract In this paper a new algorithm for fuzzy clustering is presented. The proposed algorithm utilizes the idea of relaxation. Convergence of the proposed algorithm is proved and limits on the relaxation parameter are derived. Stopping criteria and resulting convergence behaviour of the algorithms are discussed. The performance of the new algorithm is compared to the fuzzy c -means algorithm by testing both on three published data sets. Theoretical and empirical results reported in this paper show that the new algorithm is more efficient and leads to significant computational savings.