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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Test functions for optimization ： ウィキペディア英語版 Test functions for optimization In applied mathematics, test functions, known as artificial landscapes, are useful to evaluate characteristics of optimization algorithms, such as: * Velocity of convergence. * Precision. * Robustness. * General performance. Here some test functions are presented with the aim of giving an idea about the different situations that optimization algorithms have to face when coping with these kinds of problems. In the first part, some objective functions for single-objective optimization cases are presented. In the second part, test functions with their respective Pareto fronts for multi-objective optimization problems (MOP) are given. The artificial landscapes presented herein for single-objective optimization problems are taken from Bäck, Haupt et. al. and from Rody Oldenhuis software. Given the amount of problems (55 in total), just a few are presented here. The complete list of test functions is found on the Mathworks website. The test functions used to evaluate the algorithms for MOP were taken from Deb,〔Deb, Kalyanmoy (2002) Multiobjective optimization using evolutionary algorithms (Repr. ed.). Chichester (): Wiley. ISBN 0-471-87339-X.〕 Binh et. al.〔Binh T. and Korn U. (1997) MOBES: A Multiobjective Evolution Strategy for Constrained Optimization Problems. In: Proceedings of the Third International Conference on Genetic Algorithms. Czech Republic. pp. 176-182〕 and Binh.〔Binh T. (1999) A multiobjective evolutionary algorithm. The study cases. Technical report. Institute for Automation and Communication. Barleben, Germany〕 You can download the software developed by Deb,〔Deb K. (2011) Software for multi-objective NSGA-II code in C. Available at URL:http://www.iitk.ac.in/kangal/codes.shtml. Revision 1.1.6〕 which implements the NSGA-II procedure with GAs, or the program posted on Internet, which implements the NSGA-II procedure with ES. Just a general form of the equation, a plot of the objective function, boundaries of the object variables and the coordinates of global minima are given herein. ==Test functions for single-objective optimization problems== \right)}\right) $-\backslash exp\backslash left(0.5\backslash left(\backslash cos\backslash left(2\backslash pi\; x\backslash right)+\backslash cos\backslash left(2\backslash pi\; y\backslash right)\backslash right)\backslash right)\; +\; e\; +\; 20$ ||$f(0,0)\; =\; 0$ ||$-5\backslash le\; x,y\; \backslash le\; 5$ |- | Sphere function || || $f(\backslash boldsymbol)\; =\; \backslash sum\_^\; x\_^$ || $f(x\_,\; \backslash dots,\; x\_)\; =\; f(0,\; \backslash dots,\; 0)\; =\; 0$ || $-\backslash infty\; \backslash le\; x\_\; \backslash le\; \backslash infty$, $1\; \backslash le\; i\; \backslash le\; n$ |- | Rosenbrock function || || $f(\backslash boldsymbol)\; =\; \backslash sum\_^\; \backslash left(100\; \backslash left(x\_\; -\; x\_^\backslash right)^\; +\; \backslash left(x\_\; -\; 1\backslash right)^\backslash right)$ || $\backslash text\; =$ \begin n=2 & \rightarrow \quad f(1,1) = 0, \\ n=3 & \rightarrow \quad f(1,1,1) = 0, \\ n>3 & \rightarrow \quad f\left(\underbrace_ || $-\backslash infty\; \backslash le\; x\_\; \backslash le\; \backslash infty$, $1\; \backslash le\; i\; \backslash le\; n$ |- | Beale's function || || $f(x,y)\; =\; \backslash left(\; 1.5\; -\; x\; +\; xy\; \backslash right)^\; +\; \backslash left(\; 2.25\; -\; x\; +\; xy^\backslash right)^$ $+\; \backslash left(2.625\; -\; x+\; xy^\backslash right)^$ || $f(3,\; 0.5)\; =\; 0$ || $-4.5\; \backslash le\; x,y\; \backslash le\; 4.5$ |- | Goldstein–Price function: || || $f(x,y)\; =\; \backslash left(1+\backslash left(x+y+1\backslash right)^\backslash left(19-14x+3x^-14y+6xy+3y^\backslash right)\backslash right)$ $\backslash left(30+\backslash left(2x-3y\backslash right)^\backslash left(18-32x+12x^+48y-36xy+27y^\backslash right)\backslash right)$ || $f(0,\; -1)\; =\; 3$ || $-2\; \backslash le\; x,y\; \backslash le\; 2$ |- | Booth's function: || ||$f(x,y)\; =\; \backslash left(\; x\; +\; 2y\; -7\backslash right)^\; +\; \backslash left(2x\; +y\; -\; 5\backslash right)^$ ||$f(1,3)\; =\; 0$ ||$-10\; \backslash le\; x,y\; \backslash le\; 10$ |- | Bukin function N.6: || || $f(x,y)\; =\; 100\backslash sqrt\; +\; 0.01\; \backslash left|x+10\; \backslash right|.\backslash quad$ || $f(-10,1)\; =\; 0$ || $-15\backslash le\; x\; \backslash le\; -5$, $-3\backslash le\; y\; \backslash le\; 3$ |- | Matyas function: || || $f(x,y)\; =\; 0.26\; \backslash left(\; x^\; +\; y^\backslash right)\; -\; 0.48\; xy$ || $f(0,0)\; =\; 0$ || $-10\backslash le\; x,y\; \backslash le\; 10$ |- | Lévi function N.13: || || $f(x,y)\; =\; \backslash sin^\backslash left(3\backslash pi\; x\backslash right)+\backslash left(x-1\backslash right)^\backslash left(1+\backslash sin^\backslash left(3\backslash pi\; y\backslash right)\backslash right)$ $+\backslash left(y-1\backslash right)^\backslash left(1+\backslash sin^\backslash left(2\backslash pi\; y\backslash right)\backslash right)$ || $f(1,1)\; =\; 0$ || $-10\backslash le\; x,y\; \backslash le\; 10$ |- | Three-hump camel function: || || $f(x,y)\; =\; 2x^\; -\; 1.05x^\; +\; \backslash frac\; +\; xy\; +\; y^$ || $f(0,0)\; =\; 0$ || $-5\backslash le\; x,y\; \backslash le\; 5$ |- | Easom function: || || $f(x,y)\; =\; -\backslash cos\; \backslash left(x\backslash right)\backslash cos\; \backslash left(y\backslash right)\; \backslash exp\backslash left(-\backslash left(\backslash left(x-\backslash pi\backslash right)^\; +\; \backslash left(y-\backslash pi\backslash right)^\backslash right)\backslash right)$ || $f(\backslash pi\; ,\; \backslash pi)\; =\; -1$ || $-100\backslash le\; x,y\; \backslash le\; 100$ |- | Cross-in-tray function: || || $f(x,y)\; =\; -0.0001\; \backslash left(\; \backslash left|\; \backslash sin\; \backslash left(x\backslash right)\; \backslash sin\; \backslash left(y\backslash right)\; \backslash exp\; \backslash left(\; \backslash left|100\; -\; \backslash frac\}\}\; \backslash right|\backslash right)\backslash right|\; +\; 1\; \backslash right)^$ || $\backslash text\; =$ \begin f\left(1.34941, -1.34941\right) & = -2.06261 \\ f\left(1.34941, 1.34941\right) & = -2.06261 \\ f\left(-1.34941, 1.34941\right) & = -2.06261 \\ f\left(-1.34941,-1.34941\right) & = -2.06261 \\ \end || $-10\backslash le\; x,y\; \backslash le\; 10$ |- | Eggholder function: || || $f(x,y)\; =\; -\; \backslash left(y+47\backslash right)\; \backslash sin\; \backslash left(\backslash sqrt+47\backslash right|\}\backslash right)\; -\; x\; \backslash sin\; \backslash left(\backslash sqrt\backslash right)$ || $f(512,\; 404.2319)\; =\; -959.6407$ || $-512\backslash le\; x,y\; \backslash le\; 512$ |- | Hölder table function: || || $f(x,y)\; =\; -\; \backslash left|\backslash sin\; \backslash left(x\backslash right)\; \backslash cos\; \backslash left(y\backslash right)\; \backslash exp\; \backslash left(\backslash left|1\; -\; \backslash frac\}\}\; \backslash right|\backslash right)\backslash right|$ || $\backslash text\; =$ \begin f\left(8.05502, 9.66459\right) & = -19.2085 \\ f\left(-8.05502, 9.66459\right) & = -19.2085 \\ f\left(8.05502,-9.66459\right) & = -19.2085 \\ f\left(-8.05502,-9.66459\right) & = -19.2085 \end || $-10\backslash le\; x,y\; \backslash le\; 10$ |- | McCormick function: || || $f(x,y)\; =\; \backslash sin\; \backslash left(x+y\backslash right)\; +\; \backslash left(x-y\backslash right)^\; -\; 1.5x\; +\; 2.5y\; +\; 1$ || $f(-0.54719,-1.54719)\; =\; -1.9133$ || $-1.5\backslash le\; x\; \backslash le\; 4$, $-3\backslash le\; y\; \backslash le\; 4$ |- | Schaffer function N. 2: || || $f(x,y)\; =\; 0.5\; +\; \backslash frac\; -\; y^\backslash right)\; -\; 0.5\}\backslash right)\; \backslash right)^\}$ || $f(0,\; 0)\; =\; 0$ || $-100\backslash le\; x,y\; \backslash le\; 100$ |- | Schaffer function N. 4: || || $f(x,y)\; =\; 0.5\; +\; \backslash frac\; -\; y^\backslash right|\backslash right)\backslash right)\; -\; 0.5\}\backslash right)\; \backslash right)^\}$ || $f(0,1.25313)\; =\; 0.292579$ || $-100\backslash le\; x,y\; \backslash le\; 100$ |- | Styblinski–Tang function: || || $f(\backslash boldsymbol)\; =\; \backslash frac\; x\_^\; -\; 16x\_^\; +\; 5x\_\}$ || , $1\backslash le\; i\; \backslash le\; n$. |- | Simionescu function: || || $f(x,y)\; =\; 0.1xy$, $\backslash text\; x^2+y^2\backslash le\backslash left(r\_+r\_\backslash cos\backslash left(n\; \backslash arctan\; \backslash frac\; \backslash right)\backslash right)^2$ $\backslash text\; r\_=1,\; r\_=0.2\; \backslash text\; n\; =\; 8$ || $f(\backslash pm\; 0.85586214,\backslash mp\; 0.85586214)\; =\; -0.072625$ || $-1.25\backslash le\; x,y\; \backslash le\; 1.25$ |} 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Test functions for optimization」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. 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