How does barrier function method for semidefinite programming avoid the case when even eigenvalues are negative?

Question

Log-barrier function adds the expression $-\log(\det X)$ to the objective function to ensure that the matrix $X$ is positive definite.But when even eigenvalues of X is negative this expression is still finite.How does barrier function method avoid this?

Answer 1

I would say that you need the "correct" definition of log-det: $$ -\operatorname{log\,det}X := \begin{cases} -\log(\det(X)) & \text{if } X \succeq 0 \\ +\infty & \text{otherwise}. \end{cases} $$ Also note that the function defined via $$ f(X) := \begin{cases} -\log(\det(X)) & \text{if } \det(X) \ge 0 \\ +\infty & \text{otherwise} \end{cases} $$ fails to be convex (in dimensions at least $2$).