锥规划中算子分裂方法的局部线性收敛性
On the Local Linear Convergence of Operator Splitting Methods for Conic Programming
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中文总结 AI 辅助
本文针对锥规划的算子分裂方法，通过统一原始对偶误差界框架，证明严格互补性等条件可使PDHG与ADMM局部线性收敛，并将框架推广至凸复合优化。
中文摘要 AI 辅助
原始对偶混合梯度方法（PDHG）、交替方向乘子法（ADMM）等算子分裂方法在锥规划问题上常表现出线性收敛性，尽管现有一般理论仅保证次线性收敛速率。本文确定了两种几何条件——严格互补性与二次面违反，以解释该局部行为：当以足够接近收敛的严格互补解初始化时，PDHG与ADMM会在这些条件下线性收敛至最优解。我们通过统一且可验证的原始对偶误差界框架建立该结果：首先，证明严格互补性结合相关互补面的二次面违反性质，意味着原始与对偶增广拉格朗日函数在严格互补解附近具有一致二次增长；其次，证明三类正则性条件的局部等价性：增广拉格朗日函数的一致二次增长、局部平滑原始对偶间隙的二次增长、鞍点映射的度量次正则性，该等价性阐明了先前提出的局部线性收敛相关条件间的关系；第三，利用统一公式进行简洁分析，表明这些等价条件可使PDHG与ADMM实现局部线性收敛。我们验证了标准多面体锥、对称锥，以及指数锥、幂锥的相关面的二次面违反性质，并证明该性质在笛卡尔积下保持；还通过重启Halpern方案获得了改进的局部速率；最后，将该框架推广至凸复合优化，引入二次次梯度违反条件，其是二次面违反的泛化。
英文摘要
Operator-splitting methods such as the primal-dual hybrid gradient method (PDHG) and the alternating direction method of multipliers (ADMM) often exhibit linear convergence on conic programs, although general theory guarantees only sublinear rates. We identify two geometric conditions -- strict complementarity and quadratic facial violation -- that explain this local behavior: under these conditions, PDHG and ADMM converge linearly to an optimal solution when initialized sufficiently close to the converging strictly complementary solution. We establish this result through a unified and verifiable primal-dual error-bound framework. First, we show that strict complementarity, together with a quadratic facial-violation property of the associated complementary faces, implies uniform quadratic growth of both the primal and dual augmented Lagrangians near a strictly complementary solution. Second, we prove the local equivalence of three regularity conditions: uniform quadratic growth of the augmented Lagrangians, quadratic growth of a localized smoothed primal-dual gap, and metric subregularity of the saddle-point mapping. This equivalence clarifies the relationship among previously proposed conditions for local linear convergence. Third, using a unified formulation, we give a concise analysis showing that these equivalent conditions yield local linear convergence of PDHG and ADMM. We verify the quadratic facial-violation property for standard polyhedral and symmetric cones, as well as relevant faces of exponential and power cones, and show that it is preserved under Cartesian products. We also obtain an improved local rate using a restarted Halpern scheme. Finally, we extend the framework to convex composite optimization through a quadratic subdifferential-violation condition, which generalizes the quadratic facial-violation.
