发表机构
School of Mathematics and Physics, The University of Queensland（昆士兰大学数理学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对带等式约束的非凸优化，提出一种基于共轭残差法的非精确原始-对偶牛顿方法，直接处理子问题非正定性，避免有害正则化，并具有全局收敛保证和良好实证表现。
AI 中文摘要
具有非线性等式约束的优化问题出现在科学、工程以及日益增长的机器学习领域中。解决此类问题的著名方法包括序列二次规划，以及更广泛的原始-对偶牛顿方法。这些方法的经典分析通常依赖于强假设，其中最显著的可能是拉格朗日 Hessian 在约束雅可比矩阵零空间上的正定性。在实践中，这一假设往往需要强正则化或使用正定 Hessian 替代矩阵。此外，在大规模场景下，精确求解原始-对偶牛顿子问题通常在计算上不可行。为了解决这些问题，我们提出了一种非精确原始-对偶牛顿方法，其内部求解器基于共轭残差（CR）方法。利用 CR 方法近期建立的性质，包括负曲率检测、迭代单调性和下降保证，我们的方法直接处理子问题中出现的非正定性。因此，我们的方法避免了拉格朗日 Hessian 的有害正则化，同时自然地适应非精确求解。我们建立了最坏情况下的全局收敛保证，并在大规模非凸问题上展示了强大的实证性能。
英文摘要
Optimization problems with nonlinear equality constraints arise throughout science, engineering, and increasingly in machine learning. Prominent methods for solving such problems include sequential quadratic programming and, more broadly, primal-dual Newton methods. Classical analyses of these methods typically rely on strong assumptions, perhaps most notably positive definiteness of the Lagrangian Hessian on the null space of the constraint Jacobian. In practice, this assumption often necessitates strong regularization or the use of a positive definite Hessian surrogate. Moreover, in large-scale settings, solving the primal-dual Newton subproblem exactly is often computationally infeasible. To address these issues, we propose an inexact primal-dual Newton method with an inner solver based on the conjugate residual (CR) method. Exploiting recently established properties of CR, including negative-curvature detection, iterate monotonicity, and descent guarantees, our method handles indefiniteness in the subproblem directly as it arises. Our method thereby avoids detrimental regularization of the Lagrangian Hessian while naturally accommodating inexact solves. We establish worst-case global convergence guarantees and demonstrate strong empirical performance on large-scale, nonconvex problems.
