通过扰动消除半定规划的对偶间隙
Closing Duality Gaps of SDPs through Perturbation
AI总结：
本文研究了具有非零有限对偶间隙的半定规划（SDP）原始-对偶对，通过引入扰动恢复强可行性，证明了在奇异性度为一的假设下，最优值函数的极限版本 ${v_a}$ 是一个定义良好的单调递减连续双射函数，能够完全填补对偶间隙。
摘要：
设 $({\bf P},{\bf D})$ 为一个具有非零有限对偶间隙的半定规划（SDP）原始-对偶对。在这种情况下，${\bf P}$ 和 ${\bf D}$ 是弱可行的，如果我们扰动问题数据以恢复强可行性，作为扰动函数的（公共）最优值函数 $v$ 在零点（未扰动数据）处定义不明确，因为存在“两个不同的最优值” $v({\bf P})$ 和 $v({\bf D})$，其中 $v({\bf P})$ 和 $v({\bf D})$ 分别是 ${\bf P}$ 和 ${\bf D}$ 的最优值。因此，尽管 $v$ 在其他地方是连续的，但其在零点处失去了连续性。尽管如此，我们证明了在 ${\bf P}$ 和 ${\bf D}$ 的奇异性度均为一的假设下，$v$ 的极限版本 ${v_a}$ 是一个定义良好的单调递减连续双射函数，它将 $v({\bf P})$ 和 $v({\bf D})$ 连接起来，定义域为 $[0, π/2]$。定义域 $[0,π/2]$ 对应于以特定方式定义的扰动方向。因此，在温和的正则性条件下，${v_a}$ “完全填补”了非零对偶间隙。我们的结果是紧致的，因为存在一个奇异性度为二的实例，使得 ${v_a}$ 不连续。
英文摘要：
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 $[0, π/2]$ under the assumption that both ${\bf P}$ and ${\bf D}$ have singularity degree one. The domain $[0,π/2]$ 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.
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