Ex$.12\mathrm{.}1$ Show that the criteria $(12\mathrm{.}25)$ and $(12\mathrm{.}8)$ are equivalent.

With $f(x)=h(x{)}^T+{}_0,$ consider the optimization problem

$mi{n}_{{}_0,}{\displaystyle \underset{i=1}{\overset{N}{}}}[1-y_{i}f(x_i){]}_{+}+\frac{}{2}|||{|}^2$

where the subscript $"+"$ indicates positive part. This has the form loss $+$

penalty, which is a familiar paradigm in function estimation. It is easy to

show $($Exercise $12\mathrm{.}1)$ that the solution to $(12\mathrm{.}25),$ with $=\frac{1}{C},$ is the

same as that for $(12\mathrm{.}8).$

$mi{n}_{,{}_0}\frac{1}{2}|||{|}^2+C{\displaystyle \underset{i=1}{\overset{N}{}}}{}_i$

subject $to{}_{i}0,y_i(x_i^T+{}_0)1-{}_i$AAi,

where the "cost" parameter $C$ replaces the constant in $(12\mathrm{.}7)$; the separable

case corresponds to $C=.$