An optimization problem involving symmetric PSD matrices $A,B,C \in \Re^{n \times n}$ is
$\min\limits_{A,B,C}\ Tr(AB) + ||A-C||^2_{F}$ , s.t. $A \succeq 0$.
An equivalent optimization problem holding matrices $B$ and $C$ constant is
$\min\limits_{A} \ ||A - C + B||^2_{F}$ , s.t. $A \succeq 0$.
I am trying to work out how the equivalence of these two optimization problems can be shown.
Note that $$ \|A - C + B\|_F^2 = \operatorname{Tr}((A - C + B)^2)\\ = \operatorname{Tr}[(A - C)^2] + 2 \operatorname{Tr}((A - C)B) + \operatorname{Tr}(B^2)\\ = \operatorname{Tr}[(A - C)^2] + 2 \operatorname{Tr}(AB) + [\operatorname{Tr}(B^2) - 2\operatorname{Tr}(BC)]\\ = \operatorname{Tr}[(A - C)^2] + \operatorname{Tr}(A[2B]) + \left[\frac 14 \operatorname{Tr}([2B]^2) - \operatorname{Tr}([2B]C)\right]. $$ In other words: if $f(A,B,C) = \operatorname{Tr}(AB) + \|A - C\|_F^2$ and $g(A,B,C) = \|A - C + B\|_F^2$, then there exists some "constant" $K$ (dependent on only $B$ and $C$) for which $g(A,B,C) = f(A,2B,C) + K$.
So, the $A$ that minimizes $f$ given matrices $B = B_0,C = C_0$ is the same $A$ that minimizes $g$ given matrices $B = B_0/2$, $C = C_0$.