AI 中文总结
本研究将无二次优化（QO-free）方法推广至带约束的黎曼流形优化，提出含主搜索方向、高阶校正方向及黎曼弧搜索的算法，证明其全局与超线性收敛性，数值实验显示其具竞争力。
AI 中文摘要
无二次优化（QO-free）方法是一类用于求解欧氏空间中带非线性约束优化问题的强大且有效的算法。本研究旨在将该方法推广至求解带额外等式和不等式约束的流形上的优化问题。我们首先在流形场景中提出一种特定算法：每次迭代时，求解三个共享同一线性算子的线性系统以确定主搜索方向；此外，通过求解一个降维线性最小二乘子问题得到高阶校正方向，以规避现有相关文献中假设不会出现的Maratos效应。随后，在当前迭代点的切空间内执行黎曼弧搜索以生成新的迭代点。在适当假设下，我们建立了所提方法的全局及强收敛性；进一步证明，弧搜索最终将接受单位步长，由此确立算法的超线性收敛性。最后，数值结果表明，与其他现有方法相比，所提方法极具竞争力。
英文摘要
The quadratic optimization-free (QO-free) method is a class of powerful and effective algorithms for solving nonlinearly constrained optimization problems in Euclidean spaces. The aim of the present work is to extend this method to solve optimization problems on manifolds with additional equality and inequality constraints. We first present a specific algorithm in the manifold setting. At each iteration, three linear systems sharing a common linear operator are solved to determine the master search direction. In addition, a higher-order correction direction is obtained by solving a reduced linear least squares subproblem to circumvent the Maratos effect which is assumed not to arise in existing related literature. A Riemannian arc search is then performed within the tangent space of the current iterate to generate the new iterate. Under appropriate assumptions, we establish the global and strong convergence of the proposed method. Moreover, we prove that the unit step size will eventually be accepted by the arc search, upon which the superlinear convergence of the algorithm is established. Finally, numerical results demonstrate that the proposed method is very competitive compared with other existing approaches.
Comments48 pages, 2 figures
