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Multimodal and nonsmooth optimization on matrix manifolds

Author: Borckmans Pierre
Publication venue
Publication date: 01/01/2013
Field of study

Optimization is undoubtedly one of the most important and developed fields of applied mathematics. Getting familiar with this field can be overwhelming for the engineer who is looking for simple and effective solutions to a given problem. On the other hand, the pure theoretician is often dealing with highly specialized classes of optimization problems and he is mostly interested in providing convergence analysis. As a consequence of this admittedly caricatured dichotomy, it is not uncommon to observe practitioners making relatively poor choices of solvers. Furthermore, it is also quite rare to find optimization-related literature presenting a balanced trade-off between theoretical developments and more pragmatical approaches. In this thesis, we tend to occupy this in-between research area. Through a series of well-known applications, originating from diverse engineering fields such as computer vision, machine learning and electrical power generation, we adopt a practical point of view and aim at designing specialized algorithms for these problems, based on advanced optimization techniques. The applications we study present several challenges including: - piecewise smooth and non-differentiable objective functions ; - non-linearly constrained admissible sets ; - multimodal landscapes. We resort to derivative-free techniques, exploration heuristics, optimization on manifolds and nonsmooth methods. Finally, we combine these techniques with ideas coming from the meta-heuristics community to design hybrid solutions, providing global exploration of the search space, while ensuring robust local convergence properties.(FSA - Sciences de l'ingénieur) -- UCL, 201

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