I was going through Stephen Boyd and lieven vandenberghe's Convex Optimization Book, In chapter 4 he explains Semidefinite programming (SDP) and it's the standard form.
min $c^{T}x$
subject to $x_{1}F_{1} + x_{2}F_{2} +... + x_{n}F_{n} + G <= 0$
I was wondering as SDP is the superset to Quadratic Programming having quadratic objectives should be allowed right?
In general, semidefinite programming uses dot product of vectors. The SDP can be written as follows:
$$ \begin{array}{rl} \min_{(x_i)_{i=1}^{n} \in \mathbb{R}^n} & \sum_{j=1}^{n}\sum_{i=1}^{n} c_{i,j} \langle x_i, x_j\rangle \\ \text{subject to} & \sum_{j=1}^{n}\sum_{i=1}^{n} a_{i,j,k} \langle x^i , x^j\rangle \leq b_k, \quad \forall k \\ \end{array} $$