$7.(15$ points$)$ The soft margin SVM classifier optimizes the following primal problem: N N N min max L$($w$,$ wo$,$ a$,$ b$,4)=$ kww$+$c$]$ En ww $\backslash$Sigma $\backslash$alpha $,$ an$($n$($w$+$an TT $+$ up$)-1+$n$)-$Suns. n$=1$ n$=1$ n$=1$ s$.$t$.$ an $>0$ Min $>0$ where w is the weight, wo is the bias, N is the number of data points. For n$-$th data point, En E & is its slack variable, an Ea and pin Eu are its Lagrange multipliers, In and yn are its input feature and target respectively. $($a$)(5$ points$)$ Please derive the formula of af aen $($b$)(5$ points$)$ Please show an $<$ C when &n is optimal. $($c$)(5$ points$)$ Discuss the value range of an in the following cases: $1$ when the n$-$th data point is classified correctly, when the n$-$th data point is misclassified.