An International Journal of Optimization and Control: Theories & Applications ISSN:2146-0957 eISSN:2146-5703 Vol.9, No.3, pp.31-38 (2019) http://doi.org/10.11121/ijocta.01.2019.00671 RESEARCH ARTICLE A new auxiliary function approach for inequality constrained global optimization problems Nurullah Yilmaz and Ahmet Sahiner Department of Mathematics, Suleyman Demirel University, Isparta, Turkey nurullahyilmaz@sdu.edu.tr, ahmetsahiner@sdu.edu.tr ARTICLE INFO ABSTRACT Article History: Received 14 August 2018 Accepted 26 January 2019 Available 15 April 2019 In this study, we deal with the nonlinear constrained global optimization prob- lems. First, we introduce a new smooth exact penalty function for solving constrained optimization problems. We combine the exact penalty function with the auxiliary function in regard to constrained global optimization. We present a new auxiliary function approach and the adapted algorithm in order to solve non-linear inequality constrained global optimization problems. Fi- nally, we illustrate the efficiency of the algorithm on some numerical examples. Keywords: Constrained optimization Global optimization Smoothing approach Penalty function AMS Classification 2010: 90C26; 90C30; 65D10; 65K10 1. Introduction We consider the following continuous constrained optimization problem (P ) min x∈R n f (x) s.t. g j (x) ≤ 0, j =1, 2, ..., m, where f : R n → R and g j (x) : R n → R, j ∈ J = {1, 2, ..., m} are continuously differen- tiable functions. The problem (P ) is considered in many problems of engineering and natural sci- ences [1–4] and it is studied in many papers [6,7]. There exists a very rich theory for the solu- tion of the problem (P ) [5]. One of the tra- ditional but effective method to solve the prob- lem (P ) is the penalty function method [8]. The penalty function method has been proposed in order to transform a constrained optimization problem to an unconstrained optimization prob- lem. The method offers constructing a barrier on the boundary of the set of feasible solutions which is defined as D 0 := {x ∈ R n : g j (x) ≤ 0,j =1, 2,...,m} and it is assumed that D 0 is not empty. In order to construct a barrier the “b(t)= − log(−t)”, “b(t) = max(t, 0)” functions are used. The penalized objective function is de- fined as F (x, ρ)= f (x)+ ρ m  j =1 b(g j (x)), (1) and problem (P) re-stated as (P ρ ) min x∈R n F (x, ρ), where ρ> 0 is a penalty parameter. If b(t)= max(t, 0) is in the formula (1), the penalty func- tion is called as exact penalty function according to Zangwill [9]. It can be observed that the ex- act penalty function may be non-smooth. When the penalty function approach is non-smooth, one of the conventional approaches is constructing a smoothing approach. The smoothing approach is based on modifying the objective function or approximating the objective function by smooth functions [10]. In order to improve the smooth- ing approaches, different types of valuable tech- niques and algorithms are developed [11–14]. In recent years, the smoothing approaches have been used for many non-smooth problems such as min- max [15, 16], exact penalty [17–20] and etc. [21]. 31 Special issue of the International Conference on Applied Mathematics in Engineering (ICAME'18)