JOURNAL OF INDUSTRIAL AND doi:10.3934/jimo.2018035 MANAGEMENT OPTIMIZATION Volume 15, Number 1, January 2019 pp. 113–130 A NOVEL MODELING AND SMOOTHING TECHNIQUE IN GLOBAL OPTIMIZATION Ahmet Sahiner ∗ , Nurullah Yilmaz and Gulden Kapusuz Suleyman Demirel University, Department of Mathematics Isparta, 32100, Turkey (Communicated by Gerhard-Wilhelm Weber) Abstract. In this paper, we introduce a new methodology for modeling of the given data and finding the global optimum value of the model function. First, a new surface blending technique is offered by using Bezier curves and a smooth objective function is obtained with the help of this technique. Second, a new global optimization method followed by an adapted algorithm is presented to reach the global minimizer of the objective function. As an application of this new methodology, we consider energy conformation problem in Physical Chemistry as a very important real-world problem. 1. Introduction. Surface modeling consists of finding the best continuous function for representing the scattered data points. It is an important issue in Computer- Aided Geometric Design (CAGD) and in many branches of engineering such as mechanic, geology, electric, industrial design and imaging sciences. Various methods for this aim have been investigated depending on the approximation or interpolation in decades [11, 17, 18, 21, 27]. Bezier surface method is one of the most important mathematical spline methods which is commonly used to model the data at hand smoothly. The Bezier surface is defined by its control points, as in the Bezier curve. Bezier curves, began to be developed in 1950. Paul de Casteljau, in fact, is the first scientist developed the Bezier curves in 1959. This approach is comprehensively handled in important books [6, 7]. Bezier surfaces have been used in geometric design for decades as well as being used in data modeling problems in recent years [20, 26]. The blending of surfaces, especially blending of Bezier surfaces is another im- portant issue for CAGD. The blending of surfaces includes two main steps: first, smoothing the transition of each intersecting surfaces; second, obtaining one smooth- ed surface representing each of the surfaces. The common way in literature to blend the surfaces is cutting the region near the intersection curve and filling the gap by a suitable surface patch which contact with the given surfaces at connection points at which the resulting surface is at least first order differentiable [13, 31]. There are many important studies to solve the blending problem parametrically such as [1, 3]. But from the optimization point of view, we need further information about the rectangular form of surfaces and their continuous derivatives. 2010 Mathematics Subject Classification. Primary: 90C26; Secondary: 65D05, 65D17, 97M10. Key words and phrases. Global optimization, blending surface, data modeling, smoothing tech- nique . * Corresponding author: ahmetsahiner@sdu.edu.tr. 113