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I have a problem where I need to find the optimal $X\in S_{++}^n$ (i.e. $X$ is positive definite) for a strictly convex function $f(X)$.

For what I understand, I need to assign a positive semidefinite Lagrange multiplier $U\in S_+^n$ for the constraint that $X$ is positive definite. The primal is then:

$L(X,U) = f(X)+tr( (X-X^T)U)=f(X)+tr(XU)-tr(X^TU)=f(X)+tr(XU)-tr(XU^T)=f(X)$

Is the expression above right? If so, what is the meaning of assigning a multiplier for the constraint?

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