# An Adaptive Procedure for the Global Minimization of a Class of Polynomial Functions
## 5 Experimentation  
The experimentation is performed with a 3.2 GHz 8-core Intel Xeon W processor machine using IEEE 754-64 bit double precision floating point format (with \(eps=2.22\times 10^{-16}\)) and is carried out on the three nonnegatively constrained test problems described in Sections 5.1-5.3 and on the unconstrained test problem described in Section 5.4. The following methods are applied to each test problem. * Class 1 methods: library versions of Nelder-Mead method (denoted in the tables by the name NM), Differential Evolution (denoted DE) and Simulated Annealing (denoted SA). These methods are run for at most 200 iterations and are applied for comparison purpose. * Class 2 methods: the Multistart procedure described in Section 3 (denoted MS) and the adaptive procedure described in Section 4 (denoted AD). Two values for the tolerance \(\epsilon\) in (4) are used by MS and AD, namely \(\epsilon_{1}=10^{-8}\sim eps^{1/2}\) and \(\epsilon_{2}=10^{-12}\sim eps^{3/4}\). The names MS1 and AD1 refer to \(\epsilon_{1}\) and the names MS2 and AD2 refer to \(\epsilon_{2}\). The number of randomly generated initial points is \(q=10\). The bound to the number of iterations for each run of MS is \(d_{\max}=1000\), while the overall number of iterations of AD is bounded by \(d_{\max}=5000\). With Class 2 methods the procedure \(\mathcal{L}\) used as local minimizer differs according to the presence of constraints or not. In the test problems two different situations occur:  
* The problem has a region \(D\) related to nonnegativity constraints. In this case, a block nonlinear Gauss-Seidel scheme [10] is implemented as \(\mathcal{L}\), in order to find constrained local minima of \(f(\mathbf{x})\). The vector \(\mathbf{x}\) is decomposed into \(\mu\) disjoint subvectors, cyclically updated until some convergent criterium is satisfied. The convergence of this scheme, more specifically called in our case Alternating Nonnegative Least Squares (ANLS), is analyzed in [10]. We apply it for the particular case of Problem (1) with \(f(\mathbf{x})\) as in (2) in the two-dimensional case (\(\mu=2\) in Sections 5.1 and 5.2) and in the three-dimensional case (\(\mu=3\) in Section 5.3). * The problem is unconstrained (Section 5.4). In this case the local minimizer \(\mathcal{L}\) is a library procedure based on Levenberg-Marquardt method. The aim of the experimentation is:
* To analyze, in terms of both execution time and accuracy, the effects on the performance of Class 2 methods of the two chosen values for the tolerance \(\epsilon\) used in the stopping condition (4). Of course, we expect that a weaker request for the tolerance (\(\epsilon_{1}=10^{-8}\)) would result in lower accuracy and time saving than a stronger request (\(\epsilon_{2}=10^{-12}\)). This issue is experimentally analyzed by comparing the performances of method MS1 with those of method MS2 and the performances of method AD1 with those of method AD2. * To compare the performances of the adaptive and non adaptive procedures of Class 2. We wonder whether the adaptive procedure, with its policy of discarding the less promising items, can overlook the item which eventually would turn out to be the best one. This issue is experimentally analyzed by comparing the performances of method MS1 with those of method AD1 and the performances of method MS2 with those of method AD2. * To compare the performances of Class 1 and of Class 2 methods. Naturally, we expect that in the case of constrained problems the execution times of Class 1 methods, which implement general purpose procedures, would be larger than those of Class 2, specifically tailored to functions of type (2). Beforehand, it is not clear whether the latter ones would pay the time saving with a lower accuracy. For each problem we assume that an accurate approximation of the solution \(f^{*}\) of (1) is known. In the tables two performance indices are listed:  
* the execution time (denoted _time_) in seconds.