论文标题
乘数的内部点 - 倍数方法，用于正面半准编程的乘数
An Interior Point-Proximal Method of Multipliers for Positive Semi-Definite Programming
论文作者
论文摘要
在本文中，我们概括了乘数（IP-PMM）的内部点 - 高方法（IP-PMM），该方法在[用于凸出的二次编程，计算优化和应用的内部点 - 乘法方法，78，307--351（2021）]，用于将Semai Realsive（SDP）的线性阳性阳性式溶液解决方案（SDP），允许使用新的问题。特别是，我们将一种不可行的内部方法（IPM）与乘数（PMM）的近端方法相结合，并将算法（IP-PMM）解释为原始双重正则化IPM，适用于解决SDP问题。我们将IPM的一些迭代应用于PMM的每个子问题，直到找到令人满意的解决方案。然后，我们更新PMM参数，形成一个新的IPM社区，然后重复此过程。鉴于此框架，我们证明了算法的多项式复杂性，在温和的假设下，而无需对牛顿方向进行精确计算。此外，我们为缺乏强双重性提供了必要的条件，这可以用作构建检测机制的基础，以识别IP-PMM中的病理病例。
In this paper we generalize the Interior Point-Proximal Method of Multipliers (IP-PMM) presented in [An Interior Point-Proximal Method of Multipliers for Convex Quadratic Programming, Computational Optimization and Applications, 78, 307--351 (2021)] for the solution of linear positive Semi-Definite Programming (SDP) problems, allowing inexactness in the solution of the associated Newton systems. In particular, we combine an infeasible Interior Point Method (IPM) with the Proximal Method of Multipliers (PMM) and interpret the algorithm (IP-PMM) as a primal-dual regularized IPM, suitable for solving SDP problems. We apply some iterations of an IPM to each sub-problem of the PMM until a satisfactory solution is found. We then update the PMM parameters, form a new IPM neighbourhood, and repeat this process. Given this framework, we prove polynomial complexity of the algorithm, under mild assumptions, and without requiring exact computations for the Newton directions. We furthermore provide a necessary condition for lack of strong duality, which can be used as a basis for constructing detection mechanisms for identifying pathological cases within IP-PMM.
