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 continous and the domain is compact, everything is fine, right?
Any help or hint to literature would be appreciated.
If the primal problem is feasible and its dual problem possesses a strictly feasible point, i.e. a point belonging to the (relative) interior of the feasible region, then the primal problem
(note however that under these hypotheses the dual itself is not guaranteed to attain its optimal value).
A classical fact that is often used is that a continuous function on a compact set attains its minimum AND maximum
More flexible is, however, the following Theorem (see e.g., Thm 1.9 in Rockafellar & Wets's Variational Analysis)
Defn (level bounded) A function $f : R^n \to [-\infty,+\infty]$ is level-bounded if for every $\alpha \in R$, the set $\{x \in R^n | f(x) \le \alpha\}$ is bounded.
Theorem Let $f: R^n \to [-\infty,+\infty]$ be a lower semi-continuous, level-bounded and proper function. Then the value of $\inf f$ is finite and the set $\text{argmin} f$ is nonempty and compact.
