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: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=658060
• 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.

我们的社会将从更广泛地使用优化中受益匪浅，从而用更少的资源取得更好的结果。推广优化技术的使用，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）、我们的生存率（最好的癌症治疗考虑到个体患者的病史）和我们的成本（使用通过空中收集的遥感地面调查可以最大限度地降低道路成本）。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实时操纵数学对象的有效算法来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。通过此类交互式可视化工具获得的理解和知识将有助于培训 HQP 进行优化。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。***虽然拟议研究的核心侧重于低维函数的新算法，但结合了这些技术还将研究确定性全局优化算法。一个目标是提出新的策略，将计算时间和解决方案的质量提高几个数量级。另一个目标是通过使用近似和降维技术以及凸分析变换的显式计算，为非凸困难优化问题提供全局最优性的一些保证。 Moreau 包络和 Legendre-Fenchel 共轭等变换在优化中发挥着重要作用；能够有效地计算它们或至少近似它们已经带来了许多改进。存储待优化函数的完整图形和利用问题结构这两项重要技术已经取得了巨大成功。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用途。对软件开发感兴趣的总部将接受优化技术、软件工程和科学计算方面的培训；而对数学更感兴趣的 HQP 将学习非光滑分析、算法性能和浮点计算。