We combine the Lagrangian dual decomposition, barrier smoothing, path-following, and proximal Newton techniques to develop a new inexact interior-point Lagrangian decomposition method to solve a broad class of constrained composite convex optimization problems. Our method allows one to approximately solve the primal subproblems (called the slave problems ), which leads to inexact oracles (i.e., inexact function value, gradient, and Hessian) of the smoothed dual problem (called the master problem ). By appropriately controlling the inexact computation in both the slave and master problems, we can still establish a polynomial-time iteration complexity of our algorithm and recover primal solutions. We illustrate the performance of our method through two numerical examples and compare it with existing methods.

中文翻译：

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具有不精确预言机的不精确内点拉格朗日分解算法

我们结合了拉格朗日对偶分解、屏障平滑、路径跟踪和近端牛顿技术，开发了一种新的不精确内点拉格朗日分解方法来解决广泛的约束复合凸优化问题。我们的方法允许近似解决原始子问题（称为从问题），这导致平滑对偶问题（称为主问题）的不精确预言（即不精确的函数值、梯度和 Hessian）。通过适当控制从属和主控问题中的不精确计算，我们仍然可以建立算法的多项式时间迭代复杂度并恢复原始解。我们通过两个数值示例说明了我们方法的性能，并将其与现有方法进行了比较。