Function

Definition

Property

SCH1

$\min f\left(x\right)=\left(f_1\left(x\right),f_2\left(x\right)\right)\text{s}\text{.t}\text{.}x\in \left[-5,7\right]$

$f_1\left(x\right)=x^2$

$f_2\left(x\right)={\left(x-2\right)}^2$

Convex

SCH2

$\min f\left(x\right)=\left(f_1\left(x\right),f_2\left(x\right)\right)\text{s}\text{.t}.x\in \left[-5,10\right]$

$f_1\left(x\right)=\{\begin{array}{l}-x,\left(x\le 1\right)\\ -2+x,\left(1<x\le 3\right)\\ 4-x,\left(3<x\le 4\right)\\ -4+x,\left(x>4\right)\end{array}$

$f_2\left(x\right)={\left(x-5\right)}^2$

Discontinuous

ZDT2

$\min f\left(x\right)=\left(f_1\left(x\right),f_2\left(x\right)\right)\text{s}\text{.t}.x\in \left[-5,10\right]$

$f_1\left(x\right)=x_1$

$f_2\left(x\right)=g\left(x\right)\left[1-{\left(x_1/g\left(x\right)\right)}^2\right]$

$g\left(x\right)=1+9\left({\displaystyle \sum _{i=2}^n{x}_i}\right)/\left(n-1\right)$

Concave

ZDT3

$\min f\left(x\right)=\left(f_1\left(x\right),f_2\left(x\right)\right)\text{s}\text{.t}.x_i\in \left[0,1\right]$

$f_1\left(x\right)=x_1$

$f_2\left(x\right)=g\left(x\right)\left[1-\sqrt{x_1/g\left(x\right)}-\frac{x_1}{g\left(x\right)}\sin \left(10\text{π}{x}_1\right)\right]$

$g\left(x\right)=1+9\left({\displaystyle \sum _{i=2}^n{x}_i}\right)/\left(n-1\right)$

Discrete