# Closing Duality Gaps of SDPs through Perturbation



Let $$({\bf P},{\bf D})$$ be a primal-dual pair of SDPs with a nonzero finite duality gap. Under such circumstances, $${\bf P}$$ and $${\bf D}$$ are weakly feasible and if we perturb the problem data to recover strong feasibility, the (common) optimal value function $$v$$ as a function of the perturbation is not well-defined at zero (unperturbed data) since there are two different optimal values'' $$v({\bf P})$$ and $$v({\bf D})$$, where $$v({\bf P})$$ and $$v({\bf D})$$ are the optimal values of $${\bf P}$$ and $${\bf D}$$ respectively. Thus, continuity of $$v$$ is lost at zero though $$v$$ is continuous elsewhere. Nevertheless, we show that a limiting version $${v_a}$$ of $$v$$ is a well-defined monotone decreasing continuous bijective function connecting $$v({\bf P})$$ and $$v({\bf D})$$ with domain $$[0, \pi/2]$$ under the assumption that both $${\bf P}$$ and $${\bf D}$$ have singularity degree one. The domain $$[0, \pi/2]$$ corresponds to directions of perturbation defined in a certain manner. Thus, $${v_a}$$ completely fills'' the nonzero duality gap under a mild regularity condition. Our result is tight in that there exists an instance with singularity degree two for which $${v_a}$$ is not continuous.