I have an interior point optimization question. The problem I am looking at it is a standard SDP:

**minimize trace(C’X)**

**subject to**

**trace(A_i X) = b_i, i = 1,2,…,N**

**X>=0**

X is a square matrix; X>=0 means positive definiteness. The interior point algorithm replaces the X>=0 condition with a barrier function so that the (relaxed) objective becomes **trace(C’X) - 1/t log det (X)** . t is a scalar controlling the severity of the barrier. As t->inf, the solution X*(t) to the relaxed objective approaches that of the true objective X*(inf). That is, lim t->inf X*(t) = X*(inf).

**The question:** Can it be proven (or contradicted) that trace(C’X*(t)) is a convex function of t? It seems logical but I have failed to find a proof.