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Semidefinite and robust optimization and their economic applications


Robustness to modeling and estimation errors is an issue of critical importance for financial optimization problems because of the serious consequences of making wrong bets. Surprisingly, however, robust optimization has not been widely explored in financi al engineering. The research proposed here formulates robust dynamical models for financial problems and develops semidefmite programming (SDP) based methods for solving them. These models systematically account for parameter uncertainty and robustly updat e error-bounds as more information becomes available over time. In addition, this research extends the semidefmite relaxation methodology to probabilistically robust optimization problems that naturally emerge in the financial context. The other research f ocus of this proposal is on developing semidefmite models for graph theoretic problems such as the traveling salesman problem and network design. These models employ linear matrix inequalities (LMI) to represent 'geometric' constraints, such as graph conne ctivity, specified number of edge/vertex disjoint paths, etc. The optimization problems resulting from these LMI models are, typically, mixed integer semidefmite programs,Currently, mixed semidefmite programs are approximately solved by relaxing the integr ality constraints. However, as computational power increases and the interior point methods for solving semidefmite programs become more efficient, there grows a trend for developing systematic methods of tightening the relaxations - as in the case of line ar programming relaxations of mixed integer programs. As a first step in this direction, I propose to develop several cutting plane strategies for mixed semidefmite programs. Although the problems of interest belong to various application areas, they are l inked in that linear matrix inequalities and semidefmite programming provide the necessary tools to efficiently model and solve them.

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