# Use RBF as a Sampling Method in Multistart Global Optimization Method
## 1 Introduction  
A novel method to draw samples for global optimization methods is presented here. The process of locating the global minimum of a continuous and differentiable function \(f:S\to R,S\subset R^{n}\) is described as, determine  
\[x^{*}=\arg\min_{x\in S}f(x) \tag{1}\]  
with \(S\):  
\[S=[a_{1},b_{1}]\otimes[a_{2},b_{2}]\otimes\ldots[a_{n},b_{n}]\]  
The above problem is commonly used to describe problems in economics [1; 2; 3], physics [4; 5; 6], chemistry [7; 8; 9], medicine [10; 11] etc. The global optimization methods have two major categories: deterministic and stochastic methods. The most common methods of the first category are the so-called Interval methods [12; 13; 14], where the set \(S\) is divided iteratively in subregions and some subregions that do not contain the global solution are discarded using some pre-defined criteria. The majority of the methods belong to the second category where the reader can find Controlled Random Search methods [15; 16; 17], Simulated Annealing methods [18; 19], Differential Evolution methods [20; 21], Genetic algorithms [22; 23; 24], Particle Swarm optimization methods [25; 26], Ant Colony methods [27; 28] etc. Moreover, many hybrid stochastic methods have appeared recently in the relevant literature, such as methods that combine Particle Swarm Optimization and Simulated Annealing [29; 30], methods that combine Genetic Algorithms and Differential Evolution [31; 32], combinations of Genetic Algorithms and Particle Swarm Optimization [33] etc. Furthermore, due to the wide spread of parallel architectures in recent years as well as the widespread use of Graphics Processing Units (GPU), many methods have emerged that exploit such architectures [34; 35; 36]. This paper proposes an innovative sampling technique for the Multistart stochastic global sampling method. The Multistart technique is one of the simplest stochastic global optimization techniques and is the basis for many modern global optimization methods. Inthe Multistart method, a series of random samples are taken from the objective function and then a local optimization method is started from each sample. Regarding its simplicity, the method has been used with success in a wide area of practical applications, such as the Travelling Salesman Problem (TSP) [37, 38, 39], the vehicle routing problem [40, 41], the facility location problem [42], the maximum clique problem [43], the maximum fire risk insured capital problem [44], aerodynamic shape problems [45] etc. In addition, the Multistart method has been thoroughly studied by many researchers in recent years, and many works have been proposed on this method, such as methods for finding all local minima of a function [46, 47, 48], hybrid techniques [49, 50], GRASP methods [51], new termination rules [52, 53, 54], parallel techniques [55, 56]. Usually, in the Multistart method, samples are used from the objective function using some distribution such as the uniform distribution. In the present work, it is proposed that these samples are obtained from an RBF network [57], which has already been trained on a limited number of real samples from the objective function. RBF networks have been widely used in many real world problems, such as face recognition [58], function approximation [59, 60], image classification [61], water quality prediction [62] etc. The proposed sampling methodology generates an approximation of the objective function by first taking some samples from it and then trains a neural network to approximate the function. Once the neural network training process is completed, a bunch of points can be sampled from the neural network and those with the lowest functional value will be used as starting points for the Multistart method. This way, the actual function will not be sampled but the neural network approximating it, which should significantly reduce the required number of function calls. Furthermore, using points with a low function value as starting points is expected to speed up the location of the global minimum.