SVM - why quadratic programming problem? In Support Vector Machine, why is it a quadratic programming problem instead of a linear programming problem to obtain the optimal separating hyperplane. I only find, in book references, that the author choose the quadratic, my question is why????
 A: In order to find an optimal separating hyperplane, the norm of the weight vector $||\overline{w}||$ should be minimized, subject to constraints $y_i(\overline{w} \cdot \varphi(x_i) + b) ≥ 1 − \xi_i$, $\xi_i \geqslant 0, i=1,\dots, l$ (see here).
While it's technically possible to minimize $\mathcal{l}^1$-norm $||\overline{w}|| = \sum_i^n |w_i|$ (i.e. to solve linear programming problem) instead of $\mathcal{l}^2$ norm (quadratic problem), the $l^1$-approach has a number of disadvantages over the $l^2$: 
(a) the solutions for $l^1$-norm minimization problem lack stability, 
(b) the solution isn't unique,
(c) it's harder to provide computationally efficient method for $l^1$-minimization, as compared to $l^2$-minimization. 
On the other hand, while the solution of $l^1$-minimization problem is more robust to outliers than of the corresponding $l^2$ problem, this  doesn't play a great role specifically for SVMs, since there's a very small chance for an outlier to become a support vector.
A: Because the optimal separating hyperplane between classes of data
\begin{equation}
D(\mathbf{x})=\mathbf{w}^T\mathbf{x} + b = c   \quad \quad -1 < c < 1,
\end{equation}
is found by minimizing the objective function
\begin{equation}
\begin{split}
Q(\mathbf{w})&=\frac{1}{2} ||\mathbf{w}||^2\\
\mathrm{w.r.t } & \quad y_i(\mathbf{w}^T\mathbf{x}_i + b) \geq 1,\\
\end{split}
\end{equation}
which is linear in the inequality contraint but is a quadratic objective function due to the squared term.  The square of the Euclidean norm $||\mathbf{w}||$ makes the optimization problem "quadratic programming."  The quadratic objective function with inequality constraints results in a function value that is unique, but the solutions are nonunique.   
There's also a definition in optimization theory:
Definition: An optimization problem for which the objective function, inequality, and equality constraints are linear is said to be a linear program.  However, if the objective function is quadratic while the constraints are all linear, then the optimization problem is called a quadratic program. 
A: This has nothing to do with the L1 norm. Both are the Euclidean norm. Conversion to QP is due to practical reasons so that the gradient is continuous at the origin.
See https://math.stackexchange.com/a/439168/532462
