# When can the optimal value of a SDP be achieved?

Looking at semidefinite programs, are there any sufficient conditions for the solvability (i.e. the optimal value can be achieved, that is infimum=minimum)?

Obviously if the problem is unbounded, the optimal value cannot be attained. Also, if my objective function is continuous and the domain is compact, everything is fine, right?

Any help or hint to literature would be appreciated.

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Hint to literature: Stephen Boyd's Convex Optimization text has at least one section on this exact problem (semi-definite programming). The text is located here: stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf and check on page 168 of the text for the specific section. –  jrand Jun 13 '11 at 16:36
@vger Yes if the domain is compact, you're minimizing a continuous function over a closed bounded set, so the inf is attained. For more general solvability, check duality theory. Typically, there must exist a strictly feasible point, i.e., a positive definite matrix $X$ satisfying the linear constraints and a positive definite matrix $Z$ satisfying the linear constraints of the dual. –  Dominique Oct 28 '11 at 18:11