multivariable quadratic form

Question

I see the multivariable quadratic form given by 2 different expressions:

$$f(X)=X^{T}AX$$ versus $$f(X)=\frac{1}{2}X^{T}AX +B^{T}X+C$$

Which is right? and crucially why the difference? What affect does it have?

Answer 1

$X^{T}AX$ is a quadratic form.

$\frac{1}{2}X^{T}AX +B^{T}X+C$ is a quadratic polynomial, of which the quadratic term, $\frac{1}{2}X^{T}AX$, is a quadratic form.

The factor of $\frac{1}{2}$ in the quadratic polynomial is there "for convenience" because it makes the Hessian of the quadratic polynomial equal to $A$ (presuming $A$ is symmetric, which it always can be made to be). However, the factor of $\frac{1}{2}$ could instead be absorbed into (included in) $A$, in which case the Hessian would be $2A$ (again assuming $A$ is symmetric).

In the case of optimization with quadratic objective, the version $\frac{1}{2}X^{T}AX +B^{T}X+C$ can serve as a general objective (although the presence or absence of the constant $C$ has no effect on the optimal argument values (argmin or argmax). $X^{T}AX$ could be used a a quadratic objective, but if so, it would be for the special case in which there is no linear term.