The objective function is $$ f(X)=\min_X trace(B^TX^TCXBD) $$ we know the following derivatives from Matrix Cookbook, $$ \frac{\delta{trace(B^TX^TCXB)}}{\delta X}=C^TXBB^T+CXBB^T \\ \frac{\delta trace(XD)}{\delta X}=D^T $$ then, is it reasonable for the following rule? $$ \frac{\delta f(X)}{\delta X}=\frac{\delta f(X)}{\delta \{B^TX^TCXB\}}\cdot \frac{\delta{trace(B^TX^TCXB)}}{\delta X}=D^T(C^TXBB^T+CXBB^T) $$
2026-03-28 00:26:28.1774657588
Is it right for chain rule in trace function?
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Sorry,I've found the following relations $$ tr(ABC)=tr(BCA)=tr(CAB) $$ then, we have $$ tr(B^TX^TCXBD)=tr(CXBDB^TX^T) $$ then, from the Matrix Cookbook, we have $$ \frac{\delta{tr(EXFX^TG)}}{\delta X}=E^TG^TXF^T+GEXF $$ then, let $$ E=C,F=BD,G=I $$ we could get the answer.