AI 中文总结
研究退化多面体投影问题的对偶半光滑牛顿法，利用对偶表示非唯一性，引入原对偶提升投影等价集，建立位移界，开发不精确对偶SSN方法及全局化版本，具全局收敛和局部超线性收敛性，数值实验证其在高度退化设置下的鲁棒性。
AI 中文摘要
本文研究退化多面体投影问题的对偶半光滑牛顿（SSN）方法，其中对偶残差的广义雅可比矩阵即使在任意接近解集时也可能保持奇异。我们利用对偶表示的非唯一性，而非对这些奇异系统进行正则化。引入了一个原对偶提升投影等价集，其总是具有极点，且表明其极点几何确定了可构造对偶残差非奇异广义雅可比矩阵的对偶代表。此几何与满列秩条件和广义弱严格罗宾逊约束限定相关，说明牛顿步所需的正则性可恢复而非先验施加。还建立了位移界，将算法中代表选择与局部牛顿机制联系起来。在此变分框架基础上，开发了具有局部超线性收敛的不精确对偶SSN方法以及结合单调代表选择与沃尔夫线搜索的全局化版本。所得方法全局收敛并最终恢复快速局部收敛率。在正则化最优传输、电池调度可行性恢复和占用测度投影上的数值实验证明了其在高度退化设置下的鲁棒性。
英文摘要
In this paper, we study dual semismooth Newton (SSN) methods for degenerate polyhedral projection problems, where generalized Jacobians of the dual residual may remain singular even arbitrarily close to the solution set. Rather than regularizing these singular systems, we exploit the nonuniqueness of the dual representation. We introduce a primal--dual lifted projection-equivalent set that always possesses extreme points without additional structural assumptions on the polyhedron, and show that its extreme-point geometry identifies dual representatives at which nonsingular generalized Jacobians of the dual residual can be constructed. This geometry is further linked to a full-column-rank condition and a generalized weak strict Robinson constraint qualification, showing that the regularity required by the Newton step can be recovered rather than imposed \emph{a priori}. We also establish displacement bounds that connect representative selection throughout the algorithm with the local Newton mechanism. Building on this variational framework, we develop an inexact dual SSN method with local superlinear convergence and a globalized version combining monotone representative selection with a Wolfe line search. The resulting method is globally convergent and eventually recovers the fast local rate. Numerical experiments on regularized optimal transport, battery-scheduling feasibility restoration, and occupation-measure projection demonstrate its robustness in highly degenerate settings.
