AI 中文总结
本文针对混合整数PDE约束优化问题，分析SLIP、Patch-SLIP算法，提出并证明Randomized-Patch-SLIP的收敛性，对比三者性能以确定最优组合，建立图像去噪等基准问题并提供实验代码。
AI 中文摘要
混合整数偏微分方程（PDE）约束优化问题因整数规划的组合特性及模型评估需求，计算极具挑战性。已有诸多算法及性能改进方案用于求解此类问题，但常受限于问题规模。本文对两种此类算法SLIP与Patch-SLIP开展数值分析，二者分别在全域或部分域上求解信赖域子问题。此外，本文提出并证明了第三种随机化算法Randomized-Patch-SLIP的收敛性，该算法在随机选取的域块上求解信赖域子问题。本文结合文献中各类改进技术对比了三种算法的性能，旨在记录这些改进与各类求解器结合的最优组合方案，并建立了图像去噪与隐身领域的基准问题。计算结果报告了算法与改进技术的组合情况，实验所用代码以软件包形式提供。
英文摘要
Mixed-integer PDE-constrained optimization problems are computationally challenging due to both the combinatorial nature of integer programming as well as evaluation of the model. Many algorithms and subsequent performance improvements have been developed to solve these problems, but they are often limited by problem size. We numerically analyze two such algorithms: SLIP and Patch-SLIP, which solve trust-region subproblems over either the full or partial domain, respectively. We additionally propose and prove convergence of a randomized third algorithm, Randomized-Patch-SLIP, which solves trust-region subproblems over randomly selected patches of the domain. We compare performance of all three algorithms with various improvement techniques found throughout the literature; the purpose of this work is to document the best combinations of these improvements in conjunction with various solvers. We additionally establish benchmark problems in image denoising and cloaking. Computational results are reported on combinations of algorithms and improvements, and code used in the experiments is provided as a package.
