Abstract —This paper proposes a new Particle Swarm Optimization (PSO) paradigm based on Nelder-Mead simplex search method for solving optimization problems. Nelder-Mead is a classical derivative-free optimization technique, which gives it the advantage over the other methods in its class. On the other hand, PSO is a recently introduced metaheuristic that also has the merit of its simplicity of modeling. Merging the two methods in a new introduced way improved the solution speed and lead to results that are better and competent to the results reported in literature. Keywords— Heuristics, Metaheuristics, Nelder-Mead, Optimization problems, Particle Swarm Optimization. I. INTRODUCTION HE complexity inherent with models and equations representing engineering problems, business or economic aspects, and industrial and planning concerns is raising the attention towards the utilization of efficient solution methodologies to tackle any class of these problems. This paper proposes a new solution methodology to solve unconstrained optimization problems. This method is based on Nelder-Mead simplex search algorithm and the nature inspired intelligence metaheuristic; Particle Swarm Optimization (PSO). Generally, the solution of optimization problems can be divided majorly into two categories, the classical local search methods such as the Newton’s method, the Gradient method, the Nelder Mead and many others. Alternatively, there are the heuristics and/or the metaheuristic techniques; such as the PSO, Genetic Algorithm, Simulated Annealing and many others. For each category there are merits and shortcomings. For the classical methods it is well known that it is a fast way of search, but with the problem of being trapped in local optima due to the weak ability of exploration search. Conversely, the metaheuristic approaches are well known with their powerful exploration capabilities, however, with slow convergence rates. Combining the two approaches can help in overcoming or at least rectifying the shortcomings in both classes. Presently, the use of the Nelder Mead integrated with PSO came out of two main reasons. First, the complexity of the A. Abdelhalim is with the Egypt Japan University of Science & Technology, New Borg El-Arab City, 21934,Alexandria, Egypt (phone: 002-03-459 9520; fax: 002-03-459 9520; e-mail: alyaa.adel@ejust.edu.eg ). A. Eltawil is with the Egypt Japan University of Science & Technology, New Borg El-Arab City, 21934,Alexandria, Egypt (phone: 002-03-459 9520; fax: 002-03-459 9520; e-mail: eltawil@ejust.edu.eg ). K. Nakata is with Department of Industrial Engineering and Management, Tokyo Institute of Technology, 2-12-1-W9-60, Ohokayama, Meguro-ku, Tokyo 152-8552, Japan (e-mail: nakata.k.ac@m.titech.ac.jp ). M. El-Alem is with Mathematics and Computer Science Department, Faculty of Science, Alexandria University, Egypt, (e-mail: mmelalem@hotmail.com ). engineering problems and applications we tend to face in the real world that makes it difficult and sometimes impossible, to differentiate the equations and models representing these problems while Nelder-Mead is derivative free. Second, the weak point in PSO methodology of the local search ability that makes hybrid methods perform better than using standard PSO solution methodology alone. Both Nelder Mead and the PSO were used to solve plethora of problems. PSO is used to solve power systems[1], feature selection for structure-activity correlations in medical applications [2], biological applications [3], size and shape optimization [4]; [5], environmental applications [6], analysis in chemical processes [7], bioinformatics [8], task assignment problems [9], industrial control [10] and numerical optimization [11], and [12]. Moreover, PSO is used as a recommended solution methodology for even more complicated problems as reviewed in [13] and [14]. Similarly, the Nelder–Mead simplex method has been applied in physics [15], crystallography [16], biology [17], chemistry [18] and health care [19]. The rest of the paper is organized as follows; section II is a description of the Nelder-Mead simplex search method. Section III is a description of the PSO metaheuristic approach. Section IV will give detailed explanation of the integrated model of Nelder-Mead and PSO. Section V illustrates the results obtained from our developed model and discusses the comparison with other models found in literature. Finally, section VI concludes the presented work in this paper and gives some ideas for future work. II. NELDER MEAD SIMPLEX MEHTOD Nelder-Mead simplex search method was first introduced by Nelder and Mead in 1965 [20]. It worth mentioning that the simplex term used here is not related to the simplex term used in linear programming. For the n-dimensional space, a convex hull with n+1 vertices are forming the simplex for that given problem. The method works in a way that it rescales the simplex in each step according to the resulted local information fed in each step and iteration. Four steps the method goes through for examining the next step for the simplex modification. These steps are named; reflection, expansion, contraction, and shrinkage or reduction through which the simplex successively improve itself until it zero in on the optimum. Figure 1 shows the pseudocode that describes Nelder-Mead algorithm. The detailed steps of the Nelder-Mead algorithm is described below: 1. Initialization: as mentioned earlier, for n-dimensional space, a simplex with  vertices is initialized by random generation of the point x i . Herein comes the first A. Abdelhalim, A. Eltawil, K. Nakata, M. El-Alem A Guided Particle Swarm Optimization Technique to Solve Optimization Problems T