## 3 Nelder-Mead Algorithm for the Optimal Completion of Incomplete PCMs
### Nelder-Mead Algorithm  
The Nelder-Mead algorithm (also known as the _simplex search algorithm_) is one of the most popular direct search methods for unconstrained optimization problems since it originally appeared in 1965 [33], and it is suitable for the minimization of functions of several variables. It does not require derivative information, and it is more appropriate for the function minimization where its derivative is unknown or discontinuous [39]. The Nelder-Mead algorithm uses a simplex with \(k+1\) vertices (points) for \(k-\)dimensional vectors (for a function of \(k\) variables) in finding the optimum point. In each iteration of the algorithm, the \(k+1\) vertices are updated and ordered according to the increasing values of the objective function on the \(k+1\) vertices (see, e.g., [39; 40]). The algorithm has four possible steps in a single iteration: _reflection, expansion, contraction_ and _shrink_. The associated scalar parameters are: coefficients of reflection (\(\rho\)), expansion (\(\chi\)), contraction (\(\gamma\)) and shrink (\(\sigma\)). They must meet the following constraints' criteria: \(\rho>0\), \(\chi>1\), \(\chi>\rho\), \(0<\gamma<1\), \(0<\sigma<1\). In most circumstances, the Nelder-Mead algorithm achieves substantial improvements and produces very pleasant results in the first few iterations. In addition, except for shrink transformation, which is exceedingly unusual in practice, the algorithm normally only requires one or two function evaluations per iteration. This is very useful in real applications where the function evaluation takes a lengthy amount of time or the evaluation is costly. For such situations, the method is frequently faster than other existing derivative-free optimization methods [41; 42]. The Nelder-Mead algorithm is popular and widely used in practice. The fundamental reason for its popularity in practice, aside from its simplicity to be understood and coded, is its ability to achieve a significant reduction in function value with a little number of function evaluations [41]. The method has been vastly used in practical applications, especially in chemistry, medicine and chemical engineering [42]. It has also been implemented and included in different libraries and software packages. For instance, numerical recipes in C [43]; MATLAB's fminsearch [44]; Python [45]; and MATHEMATICA [46]. The practical implementation of the Nelder-Mead algorithm is often reasonable, although it may, in some rare cases, get stuck in a non-stationary point. Restarting the algorithm several times can be used as a heuristic approach when stagnation occurs [47]. The convergence properties of the algorithm for low and high dimensions were studied by [39; 48; 49; 50]. Furthermore, the complexity analysis for a single iteration of the Nelder-Mead algorithm has been reported in the literature [51; 52]. Another version of the standard Nelder-Mead algorithm [48] has been implemented using adaptive parameters and the authors found that the algorithm performs better than MATLAB's fminsearch by utilizing several benchmark testing functions for higher dimensional problems, but they have not clearly stated the convergence properties of the method. A modified Nelder-Mead algorithm [35] has also been proposed for solving a general nonlinear constrained optimization problem that handles linear and nonlinear (in)equality constraints. Several benchmark problems have been examined and compared with various methods (the \(\alpha\) constrained method with mutation [53]; the genetic algorithm [54]; and the bees algorithm [55]--to mention a few) to evaluate the performance of their algorithm. Regarding the effectiveness and efficiency, the authors discovered that it is competitive to such algorithms. Nonetheless, our approach to handle the interval constraints is different.