论文标题
在Riemannian歧管上的非线性优化问题的顺序二次优化
Sequential Quadratic Optimization for Nonlinear Optimization Problems on Riemannian Manifolds
论文作者
论文摘要
我们考虑对平等和不平等约束的Riemannian歧管上的优化问题，我们称之为Riemannian非线性优化（RNLO）问题。尽管它们有许多应用，但对它们的现有研究尤其是在算法方面受到限制。在本文中，我们提出了Riemannian顺序二次优化（RSQO），该官方搜索技术具有ELL_1惩罚函数的线路搜索技术，作为在欧几里得空间中约束非线性优化问题的标准SQO算法扩展到Riemannianianianianianianianianianianianianianian的。我们通过并行运输和指数映射证明了它与RNLO问题的Karush-Kuhn-Tucker点的全球融合。此外，我们通过分析RSQO与Riemannian Newton方法产生的序列之间的关系来建立其局部二次收敛。我们的是第一种具有在Riemannian歧管上受约束的非线性优化的全局和局部收敛属性的算法。经验结果表明，与现有的Riemannian罚款和增强的Lagrangian方法相比，RSQO更稳定地发现解决方案，并且准确性更高。
We consider optimization problems on Riemannian manifolds with equality and inequality constraints, which we call Riemannian nonlinear optimization (RNLO) problems. Although they have numerous applications, the existing studies on them are limited especially in terms of algorithms. In this paper, we propose Riemannian sequential quadratic optimization (RSQO) that uses a line-search technique with an ell_1 penalty function as an extension of the standard SQO algorithm for constrained nonlinear optimization problems in Euclidean spaces to Riemannian manifolds. We prove its global convergence to a Karush-Kuhn-Tucker point of the RNLO problem by means of parallel transport and the exponential mapping. Furthermore, we establish its local quadratic convergence by analyzing the relationship between sequences generated by RSQO and the Riemannian Newton method. Ours is the first algorithm that has both global and local convergence properties for constrained nonlinear optimization on Riemannian manifolds. Empirical results show that RSQO finds solutions more stably and with higher accuracy compared with the existing Riemannian penalty and augmented Lagrangian methods.
