$\backslash$section$*\{$Solution$\}$

The constraints enforce an increasing order:

$\backslash [$

$0\backslash$leq x$\_1\backslash$leq x$\_2\backslash$leq $\backslash$dots $\backslash$leq x$\_$n $\backslash$leq $1.$

$\backslash ]$

To minimize $\backslash ($ c$^$T x $\backslash ),$ we should assign the smallest possible values to $\backslash ($ x$\_$j $\backslash ),$ following the constraint order.

$-$ If $\backslash ($ c$\_1\backslash$leq c$\_2\backslash$leq $\backslash$dots $\backslash$leq c$\_$n $\backslash ),$ the best choice is $\backslash ($ x$\_1=0\backslash )$ and $\backslash ($ x$\_$n $=1\backslash ),$ with intermediate values adjusting based on feasibility.

$-$ If $\backslash ($ c$\_1\backslash$geq c$\_2\backslash$geq $\backslash$dots $\backslash$geq c$\_$n $\backslash ),$ then the best choice is setting all $\backslash ($ x$\_$j $=0\backslash ).$

$-$ Otherwise, the best strategy is setting $\backslash ($ x$\_1=0\backslash )$ and then increasing values gradually, ensuring the constraint holds.

A guaranteed optimal solution is:

$\backslash [$

x$\_$j$^*=$

$\backslash$begin$\{$cases$\}$

$0,$ & $\backslash$text$\{$if $\}$ c$\_$j $\backslash$text$\{$ is among the largest values$\},\backslash \backslash$

$1,$ & $\backslash$text$\{$if $\}$ c$\_$j $\backslash$text$\{$ is among the smallest values$\},\backslash \backslash$

$\backslash$text$\{$any increasing sequence$\},$ & $\backslash$text$\{$if there are ties$\}.$

$\backslash$end$\{$cases$\}$

$\backslash ]$

The optimal value is:

$\backslash [$

c$^$T x$^*=\backslash$sum$\_\{$j$=1\}^\{$n$\}$ c$\_$j x$\_$j$^*.$

$\backslash ]$