I am studying Convex Optimization and my book says that if I have the function $y=x^TAx+2b^Tx$ and the solution $x^*=-A^\dagger b$, then $y$ can be reduced to $y^*=-b^TA^\dagger b$ $\quad$ ($\dagger$ is pseudo-inverse.)
But, I can't understand how $(x^*)^TAx^*$ can be equivalent to $b^TA^\dagger b$ in order to obtain $y^*$.
Thanks for your help.
That comes from the properties of Moore-Penrose inverse . $$A^\dagger AA^\dagger=A^\dagger$$ Combined this with the fact that A is symmetric(and if its convex positive semi-definite but that's not required) :
$(x^*)^TAx^*=b^T(A^\dagger)^TAA^\dagger b=b^T(A^T)^{\dagger}AA^\dagger b=b^TA^\dagger AA^\dagger b=b^TA^\dagger b$