Structure Exploitation in Semidefinite Programming for Control
Abstract: Many control problems can be cast as semideﬁnite programs. However, since the size of these problems grow quite quickly, the computational time to solve them can be quite substantial. In order to reduce the computational time, many proposals of how to tailormake algorithms to various types of control problems can be found in the literature. In this thesis, two papers with similar ambitions are presented.The ﬁrst paper deals with the case where the constraints of the optimization problem are of the type that stems from the Kalman-Yakubovic-Popov lemma, and where some of these constraints are so called complicating constraints. This means the optimization problem will be greatly simpliﬁed if these constraints were not present. By the use of Lagrangian relaxation, the optimization problem is decomposed into smaller ones, which can be solved independently of each other. Computational results show that for some classes of problems, this algorithm can reduce the computational time compared to using a solver which does not take into account the nature of the complicating constraints.In the second paper, the fact that many control-related semideﬁnite programs have matrix-valued variables is utilized to speed up computations. This implies that the corresponding basis matrices have a certain low-rank structure which can be exploited when formulating the equations for the search directions, something that was discovered in the 90s and is implemented in LMI Lab. However, much has happened in the area of semidefinite programming since the release of LMI Lab, and new, faster algorithms have been developed. However, the idea of using the lowrank structure in the basis matrices can still be used. We implement this, using the publicly available solver SDPT3 in combination with our code for formulating the system of equations for the search directions. In order to facilitate for potential users, we also describe how the modeling language YALMIP is changed so that this lowrank structure can be tracked, and how the code can be easily interfaced. Computational results show that the computational time is decreased.
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