发表机构
University of California, Los Angeles; University of Central Florida（加州大学洛杉矶分校; 中佛罗里达大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对内点法计算瓶颈，提出pdLIP，将学习预条件与原始-对偶投影搜索结合，用循环神经网络预测对角预条件器，避免海森计算和牛顿求解，在多种约束问题上实现63-67%的迭代减少。
AI 中文摘要
内点法（IPMs）是约束优化中应用最广泛的算法之一，然而其基于牛顿法的搜索方向需要昂贵的二阶信息和大型线性方程组求解。学习优化（Learning to optimize）提供了从数据中学习的更廉价更新，但对数障碍函数在约束边界附近的奇异行为使得内点法对扰动高度敏感，这给热启动和学习可靠更新带来了复杂性。我们提出了pdLIP，一种用于光滑非线性规划的内点法，它将学习预条件与pdProj（一种全移位的原始-对偶投影搜索内点法）相结合。一个共享的坐标级循环神经网络预测一个正的对角预条件矩阵，该矩阵缩放原始步的简化牛顿系统右侧，而剩余的松弛变量和乘子方向通过解析方法恢复。学习迭代避免了海森矩阵计算和牛顿系统求解，仅使用一阶和坐标级操作，便于GPU并行化。训练是自监督的，损失函数基于惩罚-障碍罚函数和扰动最优性条件的残差，既不需要目标方向也不需要预计算的解。原始和双移位缓解了障碍函数在约束边界附近对扰动的敏感性，从而实现有效的热启动。在四类200维凸和非凸约束问题上，与相同KKT残差容差$10^{-8}$下的冷启动相比，pdLIP热启动将pdProj精化迭代次数减少了63-67%，且相对于后续pdProj求解，热启动生成成本可忽略不计。改进在具有1000个变量的盒约束二次规划（QPs）上持续存在，并扩展到包括投资组合优化、支持向量机和非线性控制示例在内的应用中。
英文摘要
Interior-point methods (IPMs) are among the most widely used algorithms for constrained optimization, yet their Newton-based search directions require costly second-order information and large linear-system solves. Learning to optimize offers cheaper updates learned from data, but the singular behavior of logarithmic barriers near constraint boundaries makes IPMs highly sensitive to perturbations, complicating both warm starting and learning reliable updates. We introduce pdLIP, an IPM for smooth nonlinear programs that integrates learned preconditioning with pdProj, an all-shifted primal-dual projected-search IPM. A shared coordinate-wise recurrent network predicts a positive diagonal preconditioner that scales the right-hand side of the reduced Newton system for the primal step, and the remaining slack and multiplier directions are recovered analytically. The learned iterations avoid Hessian evaluations and Newton-system solves, using only first-order and coordinate-wise operations amenable to GPU parallelization. Training is self-supervised, with a loss based on a penalty-barrier merit function and the residual of perturbed optimality conditions, requiring neither target directions nor precomputed solutions. Primal and dual shifts mitigate the barrier's sensitivity to perturbations near constraint boundaries, enabling effective warm starting. Across four classes of 200-dimensional convex and nonconvex constrained problems, pdLIP warm starts reduce pdProj refinement iterations by 63-67% compared with cold starts at the same KKT residual tolerance of $10^{-8}$, with negligible warm-start generation cost relative to the subsequent pdProj solve. Improvements persist on box-constrained QPs with 1000 variables and extend to applications including portfolio optimization, support vector machines, and a nonlinear control example.
