Optimal value of a semidefinite program Is a local optimum value of a SDP always the global one? 

If not, what are the conditions for that?
Here are a few more details for Suresh and Yoshio's answer. Following e.g. https://en.wikipedia.org/wiki/Semidefinite_programming, an SDP is of the form

$$\begin{array}{rl}
{\displaystyle\min_{X \in \mathbb{S}^n}} 
&amp; \langle C, X \rangle_{\mathbb{S}^n} \quad\text{subject to}\\
&amp; \langle A^{(k)}, X \rangle_{\mathbb{S}^n} \leq b_k, \quad (\forall k \in \{1,\ldots,m\}) \\
&amp; X \succeq 0
\end{array}$$
where $\mathbb{S}^n$ is the space of all $n\times n$ real symmetric matrices, $\langle C, X \rangle_{\mathbb{S}^n} = \sum_{ij} C_{ij} X_{ij}$, and $X \succeq 0$ constrains $X$ to be positive semi-definite.
The latter constraint is equivalent to (sometimes by definition)
$$y^T X y \ge 0~~~(\forall y\in \mathbb{R}^n).$$
(See e.g. here.)
Hence, the SDP is equivalent to
$$\begin{array}{rr@{}ll}
{\displaystyle\min_{X \in \mathbb{R}^{n\times n}}} 
&amp; \sum_{ij} C_{ij} X_{ij}  &amp;\text{subject to}\\
&amp; \sum_{ij} A^{(k)}_{ij} X_{ij} &amp;\leq b_k,
&amp; (\forall k \in \{1,\ldots,m\}) \\
&amp; \sum_{ij} y_i y_j X_{ij} &amp;\ge 0 
&amp; (\forall y\in \mathbb{R}^n)\\
&amp; X_{ij} &amp;= X_{ji} &amp; (\forall i,j\in\{1,\ldots,n\}).
\end{array}$$

So, we can think of the SDP as minimizing (or maximizing) a linear function of the vector $X$ subject to (infinitely many) linear constraints on $X$.  Hence, it is an optimization problem with a convex feasible region and a linear objective.  So any local optimum is a global optimum.