The soft margin SVM optimisation problem is defined as
$$
\begin{aligned}
\text{minimise}_{\xi, w, b}& \frac{1}{2}||w||^2 + C\sum_{i=1}^n\xi_i \\
\text{s.t}\;\;&y^{(i)}(w^Tx^{(i)} + b) \geq 1-\xi_i,\;\;i =1,…n \\
&\xi_i \geq 0
\end{aligned}
$$
I know that $\frac{1}{2}||w||^2$ is a convex problem. Are the objective and the constraint functions convex as well? I know that additions of two convex functions $f(x) + g(x)$ is a convex function. However, the objective function involves the addition of multiple variable function $w$ and $\xi$. Will they still be convex?
Best Answer
Yes, it is convex.
Each of the constraint corresponds to a half-space. The feasible set is a polyhedral.