分布光滑Optimization在限制割线不等式条件下的指数收敛性
Exponential Convergence for Distributed Smooth Optimization Under the Restricted Secant Inequality Condition
摘要：
本文研究了分布式光滑最优化问题，其目标是利用局部信息交换，使由局部光滑费用函数之和构成的全局费用函数最小。证明一阶方法的指数/线性收敛的标准假设是费用函数的强凸性，这在许多实际应用中并不成立。本文首先证明了在全局费用函数满足限制性割线不等式条件下，连续时间分布式原始-对偶梯度算法指数收敛于一个全局极小点.这个条件比强凸性条件弱，因为它不需要凸性，全局极小点也不需要唯一。然后，我们证明了离散时间分布式原始对偶法构造使用欧拉近似方法收敛到一个全局极小在相同的条件下线性。通过数值模拟说明了理论结果。
英文摘要：
This paper considers the distributed smooth optimization problem in which the objective is to minimize a global cost function formed by a sum of local smooth cost functions, by using local information exchange. The standard assumption for proving exponential/linear convergence of first-order methods is the strong convexity of the cost functions, which does not hold for many practical applications. In this paper, we first show that the continuous-time distributed primal-dual gradient algorithm converges to one global minimizer exponentially under the assumption that the global cost function satisfies the restricted secant inequality condition. This condition is weaker than the strong convexity condition since it does not require convexity and the global minimizers are not necessary to be unique. We then show that the discrete-time distributed primal-dual algorithm constructed by using the Euler's approximation method converges to one global minimizer linearly under the same condition. The theoretical results are illustrated by numerical simulations.
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