Table 1 summarizes the test functions used in the following experiments together with the initial $m(0)$ and $σ(0)$. The initial covariance matrix $C(0)$ and the diagonal decoding matrix $D(0)$ are always set to the identity matrix. The random vectors and the random matrix appearing in Table 1 are initialized randomly for each problem instance, but the same values are used for different algorithms for fair comparison. A trial is considered as success if the target function value of $10-8$ is reached before the algorithm spends $5×104n$ function evaluations, otherwise regarded as failure. For $n⩽40$ we conducted 20 independent trials for each setting, 10 for $80⩽n⩽320$, and 3 for $n⩾640$. When the computational effort of evaluating the test function scales worse than linear with the dimension (i.e., on rotated functions) we may omit dimensions larger than 320.

Table 1:
Test function definitions and initial conditions. The unit vector $u$ is either $e1=(1,0,…,0)$ for separable scenarios, or random vectors drawn uniformly on the unit sphere for nonseparable scenarios or for Ell-Cig and Ell-Dis. A vector $z$ represents an orthogonal transformation $Rx$ of the input vector $x$, where the orthogonal matrix $R$ is the identity matrix for separable scenarios, or is constructed by generating normal random elements and applying the Gram-Schmidt procedure. The diagonal matrix $Dell=diag(1,…,10i-1n-1,…,10)$ represents a coordinate-wise scaling and the vector $y$ is a coordinate-wisely transformed input vector $y=Dell2x$.
$f(x$$m(0)$$σ(0)$
Sphere $∥x∥2$ $3·1$
Cigar $〈u,x〉2+106(∥x∥2-〈u,x〉2)$ $3·1$
Discus $106〈u,x〉2+(∥x∥2-〈u,x〉2)$ $3·1$
Ellipsoid $∥Dell3z∥2$ $3·1$
TwoAxes $106∑i=1n/2[z]i2+∑i=n/2+1n[z]i2$ $3·1$
Ell-Cig $10-4〈mu,y〉2+(∥y∥2-〈u,y〉2)$ $3·1$
Ell-Dis $104〈u,y〉2+(∥y∥2-〈u,y〉2)$ $3·1$
Rosenbrock $∑i=1n-1100([z]i2-[z]i+1)2+([z]i-1)2$ $0$ 0.1
Bohachevsky $∑i=1n-1([z]i2+2[z]i+12-0.3cos(3π[z]i)⋯$ $N(0,82I)$
$-0.4cos(4π[z]i+1)+0.7)$
Rastrigin $∑i=1n[z]i2+10(1-cos(2π[z]i))$ $N(0,32I)$
$f(x$$m(0)$$σ(0)$
Sphere $∥x∥2$ $3·1$
Cigar $〈u,x〉2+106(∥x∥2-〈u,x〉2)$ $3·1$
Discus $106〈u,x〉2+(∥x∥2-〈u,x〉2)$ $3·1$
Ellipsoid $∥Dell3z∥2$ $3·1$
TwoAxes $106∑i=1n/2[z]i2+∑i=n/2+1n[z]i2$ $3·1$
Ell-Cig $10-4〈mu,y〉2+(∥y∥2-〈u,y〉2)$ $3·1$
Ell-Dis $104〈u,y〉2+(∥y∥2-〈u,y〉2)$ $3·1$
Rosenbrock $∑i=1n-1100([z]i2-[z]i+1)2+([z]i-1)2$ $0$ 0.1
Bohachevsky $∑i=1n-1([z]i2+2[z]i+12-0.3cos(3π[z]i)⋯$ $N(0,82I)$
$-0.4cos(4π[z]i+1)+0.7)$
Rastrigin $∑i=1n[z]i2+10(1-cos(2π[z]i))$ $N(0,32I)$

Close Modal