2012-08-31T17:02:03Z pmendes@uchc.edu Mendes Pedro University of Connecticut School of Medicine

Weierstrass function

Part of a set of benchmark functions for global optimisation

variables: 2
interval: [-0.5,0.5] (all variables)
global minimum: (0,0)
minimum value: 4

f(x₁,x₂)=- SUM(i=1,2)( SUM(k=0,20)(a^k cos( 2 π b^k (x₁ + 0.5) ) ) - 2 * SUM(k=0,20)( a^k cos( π b^k) ) )

with a=0.5 and b=3

The Weierstrass function is highly rugged and moderately repetitive. It has several global minima (but only one in the domain considered here). The function has a near-fractal structure. Weierstrass developed this function and it was the first function shown to be continuous everywhere and differentiable nowhere. This function is a good test for rugged and repetitive landscapes. Since it is non-differentiable, it cannot be used with gradient-based algorithms.

Defined in: Mishra, S.K. (2006) Some new test functions for global optimization and performance of repulsive particle swarm method. Munich Personal RePEc Archive paper 2718 (http://mpra.ub.uni-muenchen.de/2718/).

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2012-08-31T17:29:26Z 2012-08-31T17:02:24Z 2012-08-31T17:02:23Z cos(2*pi*(<CN=Root,Model=Weierstrass function,Vector=Values[x1],Reference=InitialValue>+0.5))+<CN=Root,Model=Weierstrass function,Vector=Values[a],Reference=InitialValue>*cos(2*pi*<CN=Root,Model=Weierstrass function,Vector=Values[b],Reference=InitialValue>*(<CN=Root,Model=Weierstrass function,Vector=Values[x1],Reference=InitialValue>+0.5))+<CN=Root,Model=Weierstrass function,Vector=Values[a],Reference=InitialValue>^2*cos(2*pi*<CN=Root,Model=Weierstrass function,Vector=Values[b],Reference=InitialValue>^2*(<CN=Root,Model=Weierstrass function,Vector=Values[x1],Reference=InitialValue>+0.5))+<CN=Root,Model=Weierstrass function,Vector=Values[a],Reference=InitialValue>^3*cos(2*pi*<CN=Root,Model=Weierstrass function,Vector=Values[b],Reference=InitialValue>^3*(<CN=Root,Model=Weierstrass function,Vector=Values[x1],Reference=InitialValue>+0.5))+<CN=Root,Model=Weierstrass function,Vector=Values[a],Reference=InitialValue>^4*cos(2*pi*<CN=Root,Model=Weierstrass function,Vector=Values[b],Reference=InitialValue>^4*(<CN=Root,Model=Weierstrass function,Vector=Values[x1],Reference=InitialValue>+0.5))+<CN=Root,Model=Weierstrass function,Vector=Values[a],Reference=InitialValue>^5*cos(2*pi*<CN=Root,Model=Weierstrass function,Vector=Values[b],Reference=InitialValue>^5*(<CN=Root,Model=Weierstrass function,Vector=Values[x1],Reference=InitialValue>+0.5))+<CN=Root,Model=Weierstrass function,Vector=Values[a],Reference=InitialValue>^6*cos(2*pi*<CN=Root,Model=Weierstrass function,Vector=Values[b],Reference=InitialValue>^6*(<CN=Root,Model=Weierstrass function,Vector=Values[x1],Reference=InitialValue>+0.5))+<CN=Root,Model=Weierstrass function,Vector=Values[a],Reference=InitialValue>^7*cos(2*pi*<CN=Root,Model=Weierstrass function,Vector=Values[b],Reference=InitialValue>^7*(<CN=Root,Model=Weierstrass function,Vector=Values[x1],Reference=InitialValue>+0.5))+<CN=Root,Model=Weierstrass function,Vector=Values[a],Reference=InitialValue>^8*cos(2*pi*<CN=Root,Model=Weierstrass function,Vector=Values[b],Reference=InitialValue>^8*(<CN=Root,Model=Weierstrass function,Vector=Values[x1],Reference=InitialValue>+0.5))+<CN=Root,Model=Weierstrass function,Vector=Values[a],Reference=InitialValue>^9*cos(2*pi*<CN=Root,Model=Weierstrass function,Vector=Values[b],Reference=InitialValue>^9*(<CN=Root,Model=Weierstrass function,Vector=Values[x1],Reference=InitialValue>+0.5))+<CN=Root,Model=Weierstrass function,Vector=Values[a],Reference=InitialValue>^10*cos(2*pi*<CN=Root,Model=Weierstrass function,Vector=Values[b],Reference=InitialValue>^10*(<CN=Root,Model=Weierstrass function,Vector=Values[x1],Reference=InitialValue>+0.5))+<CN=Root,Model=Weierstrass function,Vector=Values[a],Reference=InitialValue>^11*cos(2*pi*<CN=Root,Model=Weierstrass function,Vector=Values[b],Reference=InitialValue>^11*(<CN=Root,Model=Weierstrass function,Vector=Values[x1],Reference=InitialValue>+0.5))+<CN=Root,Model=Weierstrass function,Vector=Values[a],Reference=InitialValue>^12*cos(2*pi*<CN=Root,Model=Weierstrass function,Vector=Values[b],Reference=InitialValue>^12*(<CN=Root,Model=Weierstrass function,Vector=Values[x1],Reference=InitialValue>+0.5))+<CN=Root,Model=Weierstrass function,Vector=Values[a],Reference=InitialValue>^13*cos(2*pi*<CN=Root,Model=Weierstrass function,Vector=Values[b],Reference=InitialValue>^13*(<CN=Root,Model=Weierstrass function,Vector=Values[x1],Reference=InitialValue>+0.5))+<CN=Root,Model=Weierstrass function,Vector=Values[a],Reference=InitialValue>^14*cos(2*pi*<CN=Root,Model=Weierstrass function,Vector=Values[b],Reference=InitialValue>^14*(<CN=Root,Model=Weierstrass function,Vector=Values[x1],Reference=InitialValue>+0.5))+<CN=Root,Model=Weierstrass function,Vector=Values[a],Reference=InitialValue>^15*cos(2*pi*<CN=Root,Model=Weierstrass function,Vector=Values[b],Reference=InitialValue>^15*(<CN=Root,Model=Weierstrass function,Vector=Values[x1],Reference=InitialValue>+0.5))+<CN=Root,Model=Weierstrass function,Vector=Values[a],Reference=InitialValue>^16*cos(2*pi*<CN=Root,Model=Weierstrass function,Vector=Values[b],Reference=InitialValue>^16*(<CN=Root,Model=Weierstrass function,Vector=Values[x1],Reference=InitialValue>+0.5))+<CN=Root,Model=Weierstrass function,Vector=Values[a],Reference=InitialValue>^17*cos(2*pi*<CN=Root,Model=Weierstrass function,Vector=Values[b],Reference=InitialValue>^17*(<CN=Root,Model=Weierstrass function,Vector=Values[x1],Reference=InitialValue>+0.5))+<CN=Root,Model=Weierstrass function,Vector=Values[a],Reference=InitialValue>^18*cos(2*pi*<CN=Root,Model=Weierstrass function,Vector=Values[b],Reference=InitialValue>^18*(<CN=Root,Model=Weierstrass function,Vector=Values[x1],Reference=InitialValue>+0.5))+<CN=Root,Model=Weierstrass function,Vector=Values[a],Reference=InitialValue>^19*cos(2*pi*<CN=Root,Model=Weierstrass function,Vector=Values[b],Reference=InitialValue>^19*(<CN=Root,Model=Weierstrass function,Vector=Values[x1],Reference=InitialValue>+0.5))+<CN=Root,Model=Weierstrass function,Vector=Values[a],Reference=InitialValue>^20*cos(2*pi*<CN=Root,Model=Weierstrass function,Vector=Values[b],Reference=InitialValue>^20*(<CN=Root,Model=Weierstrass function,Vector=Values[x1],Reference=InitialValue>+0.5))-2*(cos(pi)+<CN=Root,Model=Weierstrass function,Vector=Values[a],Reference=InitialValue>*cos(pi*<CN=Root,Model=Weierstrass function,Vector=Values[b],Reference=InitialValue>)+<CN=Root,Model=Weierstrass function,Vector=Values[a],Reference=InitialValue>^2*cos(pi*<CN=Root,Model=Weierstrass function,Vector=Values[b],Reference=InitialValue>^2)+0.5^3*cos(pi*<CN=Root,Model=Weierstrass function,Vector=Values[b],Reference=InitialValue>^3)+0.5^4*cos(pi*<CN=Root,Model=Weierstrass function,Vector=Values[b],Reference=InitialValue>^4)+0.5^5*cos(pi*<CN=Root,Model=Weierstrass function,Vector=Values[b],Reference=InitialValue>^5)+0.5^6*cos(pi*<CN=Root,Model=Weierstrass function,Vector=Values[b],Reference=InitialValue>^6)+0.5^7*cos(pi*<CN=Root,Model=Weierstrass function,Vector=Values[b],Reference=InitialValue>^7)+0.5^8*cos(pi*<CN=Root,Model=Weierstrass function,Vector=Values[b],Reference=InitialValue>^8)+0.5^9*cos(pi*<CN=Root,Model=Weierstrass function,Vector=Values[b],Reference=InitialValue>^9)+0.5^10*cos(pi*<CN=Root,Model=Weierstrass function,Vector=Values[b],Reference=InitialValue>^10)+0.5^11*cos(pi*<CN=Root,Model=Weierstrass function,Vector=Values[b],Reference=InitialValue>^11)+0.5^12*cos(pi*<CN=Root,Model=Weierstrass function,Vector=Values[b],Reference=InitialValue>^12)+0.5^13*cos(pi*<CN=Root,Model=Weierstrass function,Vector=Values[b],Reference=InitialValue>^13)+0.5^14*cos(pi*<CN=Root,Model=Weierstrass function,Vector=Values[b],Reference=InitialValue>^14)+0.5^15*cos(pi*<CN=Root,Model=Weierstrass function,Vector=Values[b],Reference=InitialValue>^15)+0.5^16*cos(pi*<CN=Root,Model=Weierstrass function,Vector=Values[b],Reference=InitialValue>^16)+0.5^17*cos(pi*<CN=Root,Model=Weierstrass function,Vector=Values[b],Reference=InitialValue>^17)+0.5^18*cos(pi*<CN=Root,Model=Weierstrass function,Vector=Values[b],Reference=InitialValue>^18)+0.5^19*cos(pi*<CN=Root,Model=Weierstrass function,Vector=Values[b],Reference=InitialValue>^19)+0.5^20*cos(pi*<CN=Root,Model=Weierstrass function,Vector=Values[b],Reference=InitialValue>^20))+cos(2*pi*(<CN=Root,Model=Weierstrass function,Vector=Values[x2],Reference=InitialValue>+0.5))+<CN=Root,Model=Weierstrass function,Vector=Values[a],Reference=InitialValue>*cos(2*pi*<CN=Root,Model=Weierstrass function,Vector=Values[b],Reference=InitialValue>*(<CN=Root,Model=Weierstrass function,Vector=Values[x2],Reference=InitialValue>+0.5))+<CN=Root,Model=Weierstrass function,Vector=Values[a],Reference=InitialValue>^2*cos(2*pi*<CN=Root,Model=Weierstrass function,Vector=Values[b],Reference=InitialValue>^2*(<CN=Root,Model=Weierstrass function,Vector=Values[x2],Reference=InitialValue>+0.5))+<CN=Root,Model=Weierstrass function,Vector=Values[a],Reference=InitialValue>^3*cos(2*pi*<CN=Root,Model=Weierstrass function,Vector=Values[b],Reference=InitialValue>^3*(<CN=Root,Model=Weierstrass function,Vector=Values[x2],Reference=InitialValue>+0.5))+<CN=Root,Model=Weierstrass function,Vector=Values[a],Reference=InitialValue>^4*cos(2*pi*<CN=Root,Model=Weierstrass function,Vector=Values[b],Reference=InitialValue>^4*(<CN=Root,Model=Weierstrass function,Vector=Values[x2],Reference=InitialValue>+0.5))+<CN=Root,Model=Weierstrass function,Vector=Values[a],Reference=InitialValue>^5*cos(2*pi*<CN=Root,Model=Weierstrass function,Vector=Values[b],Reference=InitialValue>^5*(<CN=Root,Model=Weierstrass function,Vector=Values[x2],Reference=InitialValue>+0.5))+<CN=Root,Model=Weierstrass function,Vector=Values[a],Reference=InitialValue>^6*cos(2*pi*<CN=Root,Model=Weierstrass function,Vector=Values[b],Reference=InitialValue>^6*(<CN=Root,Model=Weierstrass function,Vector=Values[x2],Reference=InitialValue>+0.5))+<CN=Root,Model=Weierstrass function,Vector=Values[a],Reference=InitialValue>^7*cos(2*pi*<CN=Root,Model=Weierstrass function,Vector=Values[b],Reference=InitialValue>^7*(<CN=Root,Model=Weierstrass function,Vector=Values[x2],Reference=InitialValue>+0.5))+<CN=Root,Model=Weierstrass function,Vector=Values[a],Reference=InitialValue>^8*cos(2*pi*<CN=Root,Model=Weierstrass function,Vector=Values[b],Reference=InitialValue>^8*(<CN=Root,Model=Weierstrass function,Vector=Values[x2],Reference=InitialValue>+0.5))+<CN=Root,Model=Weierstrass function,Vector=Values[a],Reference=InitialValue>^9*cos(2*pi*<CN=Root,Model=Weierstrass function,Vector=Values[b],Reference=InitialValue>^9*(<CN=Root,Model=Weierstrass function,Vector=Values[x2],Reference=InitialValue>+0.5))+<CN=Root,Model=Weierstrass function,Vector=Values[a],Reference=InitialValue>^10*cos(2*pi*<CN=Root,Model=Weierstrass function,Vector=Values[b],Reference=InitialValue>^10*(<CN=Root,Model=Weierstrass function,Vector=Values[x2],Reference=InitialValue>+0.5))+<CN=Root,Model=Weierstrass function,Vector=Values[a],Reference=InitialValue>^11*cos(2*pi*<CN=Root,Model=Weierstrass function,Vector=Values[b],Reference=InitialValue>^11*(<CN=Root,Model=Weierstrass function,Vector=Values[x2],Reference=InitialValue>+0.5))+<CN=Root,Model=Weierstrass function,Vector=Values[a],Reference=InitialValue>^12*cos(2*pi*<CN=Root,Model=Weierstrass function,Vector=Values[b],Reference=InitialValue>^12*(<CN=Root,Model=Weierstrass function,Vector=Values[x2],Reference=InitialValue>+0.5))+<CN=Root,Model=Weierstrass function,Vector=Values[a],Reference=InitialValue>^13*cos(2*pi*<CN=Root,Model=Weierstrass function,Vector=Values[b],Reference=InitialValue>^13*(<CN=Root,Model=Weierstrass function,Vector=Values[x2],Reference=InitialValue>+0.5))+<CN=Root,Model=Weierstrass function,Vector=Values[a],Reference=InitialValue>^14*cos(2*pi*<CN=Root,Model=Weierstrass function,Vector=Values[b],Reference=InitialValue>^14*(<CN=Root,Model=Weierstrass function,Vector=Values[x2],Reference=InitialValue>+0.5))+<CN=Root,Model=Weierstrass function,Vector=Values[a],Reference=InitialValue>^15*cos(2*pi*<CN=Root,Model=Weierstrass function,Vector=Values[b],Reference=InitialValue>^15*(<CN=Root,Model=Weierstrass function,Vector=Values[x2],Reference=InitialValue>+0.5))+<CN=Root,Model=Weierstrass function,Vector=Values[a],Reference=InitialValue>^16*cos(2*pi*<CN=Root,Model=Weierstrass function,Vector=Values[b],Reference=InitialValue>^16*(<CN=Root,Model=Weierstrass function,Vector=Values[x2],Reference=InitialValue>+0.5))+<CN=Root,Model=Weierstrass function,Vector=Values[a],Reference=InitialValue>^17*cos(2*pi*<CN=Root,Model=Weierstrass function,Vector=Values[b],Reference=InitialValue>^17*(<CN=Root,Model=Weierstrass function,Vector=Values[x2],Reference=InitialValue>+0.5))+<CN=Root,Model=Weierstrass function,Vector=Values[a],Reference=InitialValue>^18*cos(2*pi*<CN=Root,Model=Weierstrass function,Vector=Values[b],Reference=InitialValue>^18*(<CN=Root,Model=Weierstrass function,Vector=Values[x2],Reference=InitialValue>+0.5))+<CN=Root,Model=Weierstrass function,Vector=Values[a],Reference=InitialValue>^19*cos(2*pi*<CN=Root,Model=Weierstrass function,Vector=Values[b],Reference=InitialValue>^19*(<CN=Root,Model=Weierstrass function,Vector=Values[x2],Reference=InitialValue>+0.5))+<CN=Root,Model=Weierstrass function,Vector=Values[a],Reference=InitialValue>^20*cos(2*pi*<CN=Root,Model=Weierstrass function,Vector=Values[b],Reference=InitialValue>^20*(<CN=Root,Model=Weierstrass function,Vector=Values[x1],Reference=InitialValue>+0.5))-2*(cos(pi)+<CN=Root,Model=Weierstrass function,Vector=Values[a],Reference=InitialValue>*cos(pi*<CN=Root,Model=Weierstrass function,Vector=Values[b],Reference=InitialValue>)+<CN=Root,Model=Weierstrass function,Vector=Values[a],Reference=InitialValue>^2*cos(pi*<CN=Root,Model=Weierstrass function,Vector=Values[b],Reference=InitialValue>^2)+0.5^3*cos(pi*<CN=Root,Model=Weierstrass function,Vector=Values[b],Reference=InitialValue>^3)+0.5^4*cos(pi*<CN=Root,Model=Weierstrass function,Vector=Values[b],Reference=InitialValue>^4)+0.5^5*cos(pi*<CN=Root,Model=Weierstrass function,Vector=Values[b],Reference=InitialValue>^5)+0.5^6*cos(pi*<CN=Root,Model=Weierstrass function,Vector=Values[b],Reference=InitialValue>^6)+0.5^7*cos(pi*<CN=Root,Model=Weierstrass function,Vector=Values[b],Reference=InitialValue>^7)+0.5^8*cos(pi*<CN=Root,Model=Weierstrass function,Vector=Values[b],Reference=InitialValue>^8)+0.5^9*cos(pi*<CN=Root,Model=Weierstrass function,Vector=Values[b],Reference=InitialValue>^9)+0.5^10*cos(pi*<CN=Root,Model=Weierstrass function,Vector=Values[b],Reference=InitialValue>^10)+0.5^11*cos(pi*<CN=Root,Model=Weierstrass function,Vector=Values[b],Reference=InitialValue>^11)+0.5^12*cos(pi*<CN=Root,Model=Weierstrass function,Vector=Values[b],Reference=InitialValue>^12)+0.5^13*cos(pi*<CN=Root,Model=Weierstrass function,Vector=Values[b],Reference=InitialValue>^13)+0.5^14*cos(pi*<CN=Root,Model=Weierstrass function,Vector=Values[b],Reference=InitialValue>^14)+0.5^15*cos(pi*<CN=Root,Model=Weierstrass function,Vector=Values[b],Reference=InitialValue>^15)+0.5^16*cos(pi*<CN=Root,Model=Weierstrass function,Vector=Values[b],Reference=InitialValue>^16)+0.5^17*cos(pi*<CN=Root,Model=Weierstrass function,Vector=Values[b],Reference=InitialValue>^17)+0.5^18*cos(pi*<CN=Root,Model=Weierstrass function,Vector=Values[b],Reference=InitialValue>^18)+0.5^19*cos(pi*<CN=Root,Model=Weierstrass function,Vector=Values[b],Reference=InitialValue>^19)+0.5^20*cos(pi*<CN=Root,Model=Weierstrass function,Vector=Values[b],Reference=InitialValue>^20)) 0 3.9999980926513672 0 0 0.5 3 <CN=Root,Model=Weierstrass function,Vector=Values[Weierstrass function],Reference=InitialValue> This table outputs time and the target function of the optimization.