# Application of the Parabola Method in Nonconvex Optimization
## 4 Numerical Comparison of Zero-Order Methods: Multivariate Optimization Problems
_Treccani function_ \[f(x)=x_{1}^{4}+4x_{1}^{3}+4x_{1}^{2}+x_{2}^{2},\] \[X=\Big{\{}x\in R^{2}:\;-3\leq x_{1}\leq 3,\;-3\leq x_{2}\leq 3,\Big{\}}.\] Global optimum: \(x^{\star}=(0,0)\), \(x^{\star\star}=(2,0)\), \(f^{\star}=0\). 3. _Slubert function_ \[f(x)=\left(\sum_{i=1}^{5}i\cos((i+1)x_{1}+i)\right)\cdot\left(\sum_{i=1}^{5}i \cos((i+1)x_{2}+i)\right).\] \[X=\Big{\{}x\in R^{2}:\;-10\leq x_{1}\leq 10,\;-10\leq x_{2}\leq 10,\Big{\}}.\] The function has 760 local minimum points for \(x\in X\), 18 of them are global optimum points, and \(f^{\star}=-186.730909\). 4. _3-hump camel function_ \[f(x)=2x_{1}^{2}-1.05x_{1}^{4}+\frac{x_{1}^{6}}{6}-x_{1}x_{2}+x_{2}^{2},\] \[X=\Big{\{}x\in R^{2}:\;-3\leq x_{1}\leq 3,\;-3\leq x_{2}\leq 3,\Big{\}}.\] This function has three minimums for \(x\in X\), one of them is global: \(x^{\star}=(0,0)\), and \(f^{\star}=0\). 5. _6-hump camel function_ \[f(x)=4x_{1}^{2}-2.1x_{1}^{4}+\frac{x_{1}^{6}}{3}+x_{1}x_{2}-4x_{2}^{2}+4x_{2}^ {4},\] \[X=\Big{\{}x\in R^{2}:\;-3\leq x_{1}\leq 3,\;-1.5\leq x_{2}\leq 1.5,\Big{\}}.\] The 6-hump camel function has six minimum points for \(x\in X\), two of them are global optima: \(x^{\star}=(-0.089842,0.712656)\), \(x^{\star\star}=(0.089842,-0.712656)\), and \(f^{\star}=-1.031628\). 6. _Rosenbrock function_ \[X=\Big{\{}x\in R^{2}:\;-5\leq x_{1}\leq 10,\;0\leq x_{2}\leq 15,\Big{\}}.\]0 \[X=\Big{\{}x\in R^{2}:\;-5\leq x_{1}\leq 10,\;0\leq x_{2}\leq 15,\Big{\}}.\]1 The only global minimum of the function is \(x^{\star}=(1,1)\), and \(f^{\star}=0\). 7. _Levy-1 function_ \[X=\Big{\{}x\in R^{2}:\;-5\leq x_{1}\leq 10,\;0\leq x_{2}\leq 15,\Big{\}}.\]2 \[X=\Big{\{}x\in R^{2}:\;-5\leq x_{1}\leq 10,\;0\leq x_{2}\leq 15,\Big{\}}.\]3The function has approximately \(5^{n}\) local minima for \(x\in X\), the global minimum is \(x_{i}^{*}=1\), \(i=1,\ldots,n\), and \(f^{*}=0\). 8. _Levy-2 function_ \[X=\Big{\{}x\in R^{2}:\;-5\leq x_{1}\leq 10,\;0\leq x_{2}\leq 15,\Big{\}}.\]4 \[X=\Big{\{}x\in R^{2}:\;-5\leq x_{1}\leq 10,\;0\leq x_{2}\leq 15,\Big{\}}.\]5 \[X=\Big{\{}x\in R^{2}:\;-5\leq x_{1}\leq 10,\;0\leq x_{2}\leq 15,\Big{\}}.\]6 As with _Levy-2 function_, the function has approximately \(5^{n}\) local minima for \(x\in X\), the global minimum is \(x_{i}^{*}=1\), \(i=1,\ldots,n\), and \(f^{*}=0\). 9. _Levy-3 function_ \[X=\Big{\{}x\in R^{2}:\;-5\leq x_{1}\leq 10,\;0\leq x_{2}\leq 15,\Big{\}}.\]7 \[X=\Big{\{}x\in R^{2}:\;-5\leq x_{1}\leq 10,\;0\leq x_{2}\leq 15,\Big{\}}.\]8 \[X=\Big{\{}x\in R^{2}:\;-5\leq x_{1}\leq 10,\;0\leq x_{2}\leq 15,\Big{\}}.\]9 This function has approximately \(30^{n}\) local minima for \(x\in X\), the only global minimum is \(x_{i}^{*}=1\), \(i=1,\ldots,n\), and \(f^{*}=0\). The results of the numerical experiments are presented in Tables 8 and 9. The following notations are used: \(n\) is the number of variables; \(f_{best}\) is the best value of the objective function found during the executing of the algorithm; \(f^{*}\) is the optimal value of the objective function; \(k_{f}\) is the average number of function calculations during the execution of the algorithm considering all launches of the current problem; \(m\) is the number of problems solved successfully using the multistart procedure; \(M\) is the total number of randomly generated points. The multistart procedure launches the algorithm from one of the generated points. The designation \(m/M\) means that \(m\) launches of the algorithm from \(M\) starting points resulted in a successful problem solution (the global minimum point was found). Coordinate descent-par is the coordinate descent method in combination with the ATSA; Hooke-Jeeves-par is the Hooke and Jeeves method combined with the ATSA; Rosenbrock-dis is the Rosenbrock method with a discrete step in the search procedure; Rosenbrock-par is the Rosenbrock method in combination with the ATSA.