Question: 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????

Answer: 
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](https://www.csie.ntu.edu.tw/~cjlin/papers/guide/guide.pdf)).

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](https://en.wikipedia.org/wiki/Least_absolute_deviations#Contrasting_least_squares_with_least_absolute_deviations) 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.
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>