Computer-Aided Nonsmooth Analysis and Applications

我们的社会将大大受益于更广泛地使用优化，从而用更少的资源取得更好的结果。在提高效率的同时推广优化技术的使用，同时加深我们对这些技术的理解，是一项预计将产生巨大经济和社会效益的持续任务。数据收集和计算能力的指数级增长使我们能够例行公事地优化我们的时间(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单个患者的病史)和我们的成本(使用航空收集的遥感地面调查将道路成本降至最低)。用来解决这些问题的优化算法很少能保证找到最优解。我的研究计划旨在通过开发实时操作数学对象的高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的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，为更多的优化问题提供最优性保证。*虽然所提出的研究的核心是针对低维函数的新算法，但将这些技术结合到确定性全局优化算法中也将被研究。一个目标是提出新的策略，使计算时间和解的质量提高几个数量级。另一个目标是利用逼近和降维技术，结合显式的凸分析变换计算，为非凸困难优化问题提供全局最优性的保证。像莫罗包络和勒让德-芬切尔共轭这样的变换在优化中扮演着基本的角色；能够有效地计算它们，或者至少是近似它们，已经带来了许多改进。存储待优化函数的完整图和开发问题结构这两项重要技术已经获得了巨大的成功。我们的计划是通过利用计算整个函数图的新算法来进一步开发它们的用途，而不是执行逐点评估。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；而对数学更感兴趣的HQP将学习非光滑分析、算法性能和浮点计算。

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.