黎曼流形上退化非线性优化问题的稳定序列二次规划方法
A stabilized sequential quadratic programming method for degenerate nonlinear optimization problems on Riemannian manifolds
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中文总结 AI 辅助
本研究将稳定SQP框架扩展至黎曼流形，提出无约束规范假设下收敛的稳定SQP方法，经数值实验验证其对退化非线性优化问题的有效性。
中文摘要 AI 辅助
我们提出了一种用于求解黎曼流形上退化约束优化问题的稳定序列二次规划（SQP）方法。本研究考虑的问题是具有等式和不等式约束的黎曼非线性规划问题（RNLP），其中经典的约束规范可能失效。现有黎曼SQP方法仅在满足约束规范时才能保证全局收敛，而对于退化问题，其收敛性无法得到保证。为解决这一局限，我们将稳定SQP框架从欧几里得空间扩展到黎曼流形。在不假设任何约束规范的情况下，我们证明生成的序列存在一个聚点，该聚点是卡罗需-库恩-塔克（KKT）点、近似KKT（AKKT）点或相关可行性问题的平稳点。最后，我们开展数值实验以验证所提方法对退化问题的有效性。
英文摘要
We propose a stabilized sequential quadratic programming (SQP) method for degenerate constrained optimization problems on Riemannian manifolds. The problem considered in this study is a Riemannian nonlinear programming problem (RNLP) with equality and inequality constraints, where classical constraint qualifications may fail. While existing Riemannian SQP methods guarantee global convergence only under constraint qualifications, their convergence behavior is not ensured for degenerate problems. To address this limitation, we extend the stabilized SQP framework from Euclidean spaces to Riemannian manifolds. Without assuming any constraint qualification, we prove that the generated sequence has an accumulation point that is a Karush--Kuhn--Tucker (KKT) point, an approximate KKT (AKKT) point, or a stationary point of an associated feasibility problem. Finally, we conduct numerical experiments to confirm the effectiveness of the proposed method for degenerate problems.
