Algorithm 1: Piecewise-linear Algorithm Ø The number of ${a}_{i}\ge 0\left(i=1,2,\cdots ,m\right)$ is C, let t = C/m. Ø If $t\ge 0.5$ , the minimax point is searched from the left $x=-1$ , otherwise, from $x=1$ . We only describe the search process of start from the left, and the right is the same. Solving the intersection coordinates $\left({X}_{i},{Y}_{i}\right)$ of the $x=-1$ and ${y}_{i}\left(i=1,\cdots ,m\right)$ , let $Y=\left({Y}_{1},\cdots ,{Y}_{m}\right)$ and ${Y}_{p}=\mathrm{max}\left(Y\right)$ , if the corresponding ${a}_{p}\ge 0$ , then terminate the search and $x=-1$ is the solution of the subproblem. Ø If ${a}_{p}\le 0$ , then solving the intersection coordinates $\left({\stackrel{˜}{X}}_{j},{\stackrel{˜}{Y}}_{j}\right)$ of the straight line with a slope bigger than ${a}_{p}$ and the current line ${y}_{p}$ . Ø Here, the number of lines where the slope is larger than ${a}_{p}$ is recorded as w, remember $\stackrel{˜}{Y}={\stackrel{˜}{Y}}_{1},\cdots ,{\stackrel{˜}{Y}}_{W}$ , let ${\stackrel{˜}{Y}}_{h}=\mathrm{max}\left(\stackrel{˜}{Y}\right)$ , and judge whether the abscissa ${\stackrel{˜}{X}}_{h}$ is out of bounds, if ${\stackrel{˜}{X}}_{h}>1$ , then x = 1 is the solution of the subproblem, otherwise judge whether or not ${a}_{h}\ge 0$ , if ${a}_{h}\ge 0$ , then stop to research, and ${\stackrel{˜}{X}}_{h}$ is the solution of the subproblem, otherwise repeat the step 3, until the minimax point is found.