发表机构
University of Konstanz; University of the Bundeswehr Munich（康斯坦茨大学; 联邦国防军慕尼黑大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出一种非精确内点近端方法，用于求解Hilbert空间中带锥序约束的非光滑非凸优化问题，证明了收敛性并给出复杂度界，并应用于最优控制与字典学习。
AI 中文摘要
我们研究了一类具有锥不等式约束的非光滑、非凸优化问题的非精确内点方法。目标函数由光滑（可能非凸）项与具有可计算近端映射的凸（可能非光滑）项之和给出。约束通过Banach格中的序锥来表述。该设定涵盖了具有分量约束的有限维非光滑非线性问题，以及具有逐点状态约束的无限维PDE约束优化问题。该方法基于障碍正则化子问题，这些子问题通过近端梯度法非精确求解。我们考虑了对数型和幂型障碍函数，并推导了收敛性和复杂性分析所需的可微性和曲率估计。在标准约束规范条件下，我们建立了原始问题的KKT型最优性条件，并证明了原始迭代点和障碍诱导乘子满足近似KKT条件。对于内点方案，我们证明了收敛到稳定点，并推导了达到近似稳定性所需的总近端梯度迭代次数的界。特别地，我们表明总复杂度由最终外部迭代主导，因为障碍曲率增长。在凸设定下，我们获得了更强的收敛结果：整个序列弱收敛到全局解，并在二次增长条件下强收敛。我们将抽象结果应用于具有状态约束的半线性椭圆最优控制问题（所有假设均得到验证），以及具有非线性侧约束的有限维稀疏字典学习问题。数值实验支持理论发现。
英文摘要
We study an inexact interior-point method for nonsmooth, possibly nonconvex optimization in a Hilbert space with inequality constraints ordered by a cone in a Banach lattice, with particular emphasis on infinite-dimensional state-constrained optimal control. The objective function is given by the sum of a smooth, possibly nonconvex term and a convex, possibly nonsmooth term with a computable proximal mapping. The constraints are formulated by means of an order cone in a Banach lattice. This setting covers finite-dimensional nonsmooth nonlinear problems with componentwise constraints as well as infinite-dimensional PDE-constrained optimization problems with pointwise state constraints. The method is based on barrier-regularized subproblems, which are solved inexactly by a proximal-gradient method. We consider logarithmic and power-type barriers and derive the differentiability and curvature estimates needed for the convergence and complexity analysis. Under suitable constraint qualifications and compactness assumptions, we establish approximate KKT conditions for the original problem and convergence of the inexact interior-point sequence. For logarithmic and power-type barriers, we derive complementarity estimates and outer iteration bounds; the corresponding power-barrier rates and total inner-outer complexity bounds are stated under explicit barrier-path and uniform smoothness assumptions. For convex problems, we obtain stronger convergence results. We apply the framework to state-constrained semilinear elliptic optimal control and sparse dictionary learning with nonlinear side constraints. Numerical experiments illustrate the proposed method.
Comments37 pages, 7 figures, 3 tables, 2 algorithms
