Convexity of SDP solution with respect to barrier penalty

1

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.

2

Since it is CVX unrelated then this question belongs elsewhere e.g. stack exchange.

To me the answer seems no though.

3

Thanks, cross-posted to stack exchange. Any hints on why you think the answer might be no?

4

To me it seems you ask whether the central path is convex. The central path is the set of all X(t).

The central is definitely not convex. Why should it be that?

5

You are right and my mistake. I need to track the value of trace(C’X*(t)) and prove/disprove that it is a convex function of t.