Computer-Aided Nonsmooth Analysis and Applications

计算机辅助非光滑分析及应用

• 批准号: RGPIN-2018-03928
• 负责人: Lucet, Yves
• 金额: $ 2.99万
• 依托单位: University of British Columbia
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=688217
• 关键词: Computer; Aided; Nonsmooth; Analysis; Applications

Our society would greatly benefit from a wider use of optimization thereby achieving better results with fewer resources. Spreading the use of optimization techniques while improving their efficiency and, at the same time, deepening our understanding of those techniques, is an ongoing task that is expected to generate huge economic and social benefits. An exponential increase in data collection and computational power allow us to routinely optimize our time (Google map computes the fastest road using real-time traffic information), our survival (the best cancer treatment takes into account individual patient history), and our costs (road costs are minimized using remote sensing ground survey collected by air). Optimization algorithms used to solve these problems are rarely able to guarantee that the best solution was found. My research program aims to improve our understanding of the core optimization techniques by developing efficient algorithms to manipulate mathematical objects in real time. Those algorithms will support a real-time interactive visualization environment that exploits the latest advances in virtual reality technology. The understanding and knowledge obtained through such interactive visualization tools will help training HQP in optimization. In the long term, the new algorithms will help developing better global optimization algorithms to provide guarantees of optimality for a larger number of optimization problems.*** While the core of the proposed research focuses on new algorithms for low dimensional functions, incorporating those techniques into deterministic global optimization algorithms will also be researched. One objective is to propose new strategies that will make an improvement of several order of magnitude in computation time and quality of the solution. Another objective is to provide some guarantee of global optimality for nonconvex difficult optimization problems by using approximation and dimension reduction techniques coupled with explicit computation of convex analysis transforms. Transforms like the Moreau envelope and the Legendre-Fenchel conjugate play fundamental roles in optimization; being able to compute them efficiently or at least approximate them has led to many improvements. Two important techniques, storing the complete graph of the function to be optimized and exploiting problem structures, have already been used to great success. The plan is to further develop their use by leveraging new algorithms that compute entire graph of functions instead of performing pointwise evaluations. HQP interested in software development will be trained on optimization techniques, software engineering, and scientific computing; while HQP with more mathematical interests will learn about nonsmooth analysis, algorithm performance, and floating point computation.

我们的社会将大大受益于更广泛地使用优化，从而实现更好的结果，用更少的资源。推广优化技术的使用，同时提高其效率，同时加深我们对这些技术的理解，是一项持续的任务，预计将产生巨大的经济和社会效益。数据收集和计算能力的指数级增长使我们能够定期优化我们的时间（谷歌地图使用实时交通信息计算最快的道路），我们的生存（最好的癌症治疗考虑到个体患者的病史）和我们的成本（使用空中收集的遥感地面调查将道路成本降至最低）。用于解决这些问题的优化算法很少能够保证找到最佳解决方案。我的研究计划旨在通过开发有效的算法来真实的时间操作数学对象，以提高我们对核心优化技术的理解。这些算法将支持利用虚拟现实技术最新进展的实时交互式可视化环境。通过这种交互式可视化工具获得的理解和知识将有助于培训HQP优化。从长远来看，新算法将有助于开发更好的全局优化算法，为大量优化问题提供最优性保证。虽然所提出的研究的核心集中在低维函数的新算法，将这些技术纳入确定性全局优化算法也将进行研究。一个目标是提出新的策略，这将提高几个数量级的计算时间和质量的解决方案。另一个目标是提供一些保证的全局最优性的非凸困难的优化问题，通过使用近似和降维技术，再加上显式计算的凸分析变换。像Moreau包络和Legendre-Fenchel共轭这样的变换在优化中起着重要的作用;能够有效地计算它们或至少近似它们已经导致了许多改进。两个重要的技术，存储完整的图形的功能进行优化和利用问题的结构，已经被用来取得巨大成功。该计划是通过利用计算整个函数图而不是执行逐点评估的新算法来进一步开发它们的使用。对软件开发感兴趣的HQP将接受优化技术、软件工程和科学计算方面的培训;而对数学感兴趣的HQP将学习非光滑分析、算法性能和浮点计算。