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Table 9 Minimax optimization test problems

From: A hybrid cuckoo search algorithm with Nelder Mead method for solving global optimization problems

Test problem

Problem defination

Problem 1 (Yang 2010b)

\(FM _{1}(x)=\text {max}\;{f_i(x)},\; i=1,2,3,\)

\(f_1(x)=x_1^2+x_2^4,\)

\(f_2(x)=(2-x1)^2+(2-x_2)^2,\)

\(f_3(x)=2exp(-x_1+x_2)\)

Problem 2 (Yang 2010b)

\(FM _{2}(x)=\text {max}\;{f_i(x)}, \;i=1,2,3,\)

\(f_1(x)=x_1^4+x_2^2\)

\(f_2(x)=(2-x1)^2+(2-x_2)^2,\)

\(f_3(x)=2exp(-x_1+x_2)\)

Problem 3 (Yang 2010B)

\(FM _{3}(x)=x_1^2+x_2^2+2x_3^2+x_4^2-5x_1-5x_2-21x_3+7x_4,\)

\(g_2(x)=-x_1^2-x_2^2-x_3^3-x_4^2-x_1+x_2-x_3+x_4+8,\)

\(g_3(x)=-x_1^2-2x_2^2-x_3^2-2x_4+x_1+x_4+10,\)

\(g_4(x)=-x_1^2-x_2^2-x_3^2-2x_1+x_2+x_4+5\)

Problem 4 (Yang 2010B)

\(FM _{4}(x)=\text {max}\;{f_i(x)}\;i=1,\ldots ,5\)

\(f_{1}(x)=(x_1-10)^2+5(x_2-12)^2+x_3^4+3(x_4-11)^2 +10x_5^6+7x_6^2+x_7^4-4x_6x_7-10x_6-8x_7,\)

\(f_2(x)=f_1(x)+10(2x_1^2+3x_2^4+x_3+4x_4^2+5x_5-127),\)

\(f_3(x)=f_1(x)+10(7x_1+3x_2+10x_3^2+x_4-x_5-282),\)

\(f_4(x)=f_1(x)+10(23x_1+x_2^2+6x_6^2-8x_7-196),\)

\(f_5(x)=f_1(x)+10(4x_1^2+x_2^2-3x_1x_2+2x_3^2+5x_6-11x_7\)

Problem 5 (Schwefel 1995)

\(FM _{5}(x)=\text {max}\;{f_i(x)},\;i=1,2,\)

\(f_1(x)=|x_1+2x_2-7|\),

\(f_2(x)=|2x_1+x_2-5|\)

Problem 6 (Schwefel 1995)

\(FM _{6}(x)=\text {max}\;{f_i(x)},\)

\(f_i(x)=|x_i|,\;i= 1,\ldots ,10\)

Problem 7 (Lukšan and Vlcek 2000)

\(FM _{7}(x)= \text {max}\;{f_i(x)},\;i=1,2,\)

\(f_1(x)=(x_1-\sqrt{(x_1^2+x_2^2)}cos\sqrt{x_1^2+x_2^2})^2+0.005(x_1^2+x_2^2)^2,\)

\(f_2(x)=(x_2-\sqrt{(x_1^2+x_2^2)}sin\sqrt{x_1^2+x_2^2})^2+0.005(x_1^2+x_2^2)^2\)

Problem 8 (Lukšan and Vlcek 2000)

\(FM _{8}(x)= \text {max}\;{f_i(x)},\;i=1,\ldots ,4,\)

\(f_1(x)=(x_1-(x_4+1)^4)^2+(x_2-(x_1-(x_4+1)^4)^4)^2 +2x_3^2+x_4^2-5(x_1-(x_4+1)^4)-5(x_2-(x1-(x_4+1)^4)^4)-21x_3+7x_4,\)

\(f_2(x)=f_1(x)+10[(x_1-(x_4+1)^4)^2+(x_2-(x_1-(x_4+1)^4)^4)^2 +x_3^2+x_4^2+(x_1-(x_4+1)^4)-(x_2-(x_1-(x_4+1)^4)^4)+x_3-x_4-8],\)

\(f_3(x)=f_1(x)+10[(x_1-(x_4+1)^4)^2+2(x_2-(x_1 -(x_4+1)^4)^4)^2+x_3^2+2x_4^2-(x_1-(x_4+1)^4)-x_4-10]\)

\(f_4(x)=f_1(x)+10[(x_1-(x_4+1)^4)^2+(x_2-(x_1 -(x_4+1)^4)^4)^2+x_3^2+2(x_1-(x_4+1)^4)-(x_2-(x_1-(x_4+1)^4)^4)-x_4-5]\)

Problem 9 (Lukšan and Vlcek 2000)

\(FM _{9}(x)= \text {max}\;{f_i(x)},\;i=1,\ldots ,5,\)

\(f_1(x)=(x_1-10)^2+5(x_2-12)^2+x_3^4+3(x_4-11)^2+10x_5^6+7x_6^2+x_7^4-4x_6x_7-10x_6-8x_7\),

\(f_2(x)=-2x_1^2-2x_3^4-x_3-4x_4^2-5x_5+127\),

\(f_3(x)=-7x_1-3x_2-10x_3^2-x_4+x_5+282\),

\(f_4(x)=-23x_1-x_2^2-6x_6^2+8x_7+196\),

\(f_5(x)=-4x_1^2-x_2^2+3x_1x_2-2x_3^2-5x_6+11x_7\)

Problem 10 (Lukšan and Vlcek 2000)

\(FM _{10}(x)=\text {max}\;{|f_i(x)|}, \;i=1,\ldots ,21,\)

\(f_i(x)=x_1exp(x_3t_i)+x_2exp(x_4t_i)-\frac{1}{1+t_i},\)

\(t_i=-0.5+\frac{i-1}{20}\)