Theory and computational methods of robust optimization

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This research project aims at developing efficient algorithms for a class of robust optimization problem based on its theoretical analysis. Such a class of problem is represented in general as a min-max problem involving uncertainty as parameters. We first studied (1) inexact implementations of some numerical algorithms based on the cutting plane strategy, and finally proposed (2) a bilateral cutting plane framework as an ultimate solution form, whose global convergence is shown for a class of optimization problem formulated in a measure space. Furthermore we tried some refinements of the numerical framework towards its more efficient implementation with better local convergence properties. The practical side of this research project includes (3) some applications to semi-infinite programming especially for digital filter design, (4) a theoretical analysis of the asymptotic behavior of the developed framework when applied to a special case whose decision variable is limited to a class of absolutely continuous measure, and (5) some applications to optimal control problems with state variable inequality constraints. We also investigated, as related topics, (6) a type of path-following algorithm based on continuous descent methods and (7) a numerical analysis of the integration of stiff systems of ordinary differential equations.