具有部分可分结构的分布式非线性锥优化
Distributed Nonlinear Conic Optimisation with partially separable Structure
AI总结：
本文将原始对偶乘子法推广至锥约束场景，提出可分布式求解部分可分半定规划等问题的算法，给出两类更新方案的收敛性，并验证了其在分布式最大割问题近似求解中的应用。
摘要：
本文研究图上带锥约束的可分凸代价函数分布式非线性优化问题。我们借助凸分析、单调算子理论与不动点理论，将最初为等式约束优化设计、近期被扩展至包含线性不等式约束的原始对偶乘子法（PDMM）推广至可处理锥约束。所得算法能以完全分布式的方式求解各类优化问题，包括具有部分可分结构的重要半定规划类别。我们通过将Peaceman-Rachford分裂算法应用于与提升对偶问题相关的单调包含问题，推导得到更新方程。锥约束通过提升对偶域中的反射法实现，其中辅助变量相对于极锥与连接对偶域和提升对偶域的子空间的交进行反射。本文给出了同步更新方案和随机更新方案的收敛性结果，并演示了该算法的一项应用：以完全分布式方式实现基于半定规划的最大割问题近似算法。
英文摘要：
In this paper we consider the problem of distributed nonlinear optimisation of a separable convex cost function over a graph subject to cone constraints. We show how to generalise, using convex analysis, monotone operator theory and fixed-point theory, the primal-dual method of multipliers (PDMM), originally designed for equality constraint optimisation and recently extended to include linear inequality constraints, to accommodate for cone constraints. The resulting algorithm can be used to implement a variety of optimisation problems, including the important class of semidefinite programs with partially separable structure, in a fully distributed fashion. We derive update equations by applying the Peaceman-Rachford splitting algorithm to the monotonic inclusion related to the lifted dual problem. The cone constraints are implemented by a reflection method in the lifted dual domain where auxiliary variables are reflected with respect to the intersection of the polar cone and a subspace relating the dual and lifted dual domain. Convergence results for both synchronous and stochastic update schemes are provided and an application of the proposed algorithm is demonstrated to implement an approximate algorithm for maximum cut problems based on semidefinite programming in a fully distributed fashion.
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