Computer-Aided Nonsmooth Analysis and Applications

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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.