# Optimization with a symmetric matrix constraint

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?