Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization

@inproceedings{Parrilo2000StructuredSP,
  title={Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization},
  author={Pablo A. Parrilo},
  year={2000}
}
In the first part of this thesis, we introduce a specific class of Linear Matrix Inequalities (LMI) whose optimal solution can be characterized exactly. This family corresponds to the case where the associated linear operator maps the cone of positive semidefinite matrices onto itself. In this case, the optimal value equals the spectral radius of the operator. It is shown that some rank minimization problems, as well as generalizations of the structured singular value ($mu$) LMIs, have exactly… 

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Convex algebraic geometry and semidefinite optimization

  • P. Parrilo
  • Mathematics, Computer Science
    ISSAC '13
  • 2013
This tutorial will focus on basic and recent developments in convex algebraic geometry, and the associated computational methods based on semidefinite programming for optimization problems involving polynomial equations and inequalities.

Computing optimal fixed order H∞-synthesis values by matrix sum of squares relaxations

  • C. HolC. Scherer
  • Mathematics, Computer Science
    2004 43rd IEEE Conference on Decision and Control (CDC) (IEEE Cat. No.04CH37601)
  • 2004
Two approaches to solve robust LMI problems based on sos matrix decompositions, a direct approach and one based on the S-procedure are presented, which lead to an asymptotically exact family of LMI relaxations for computing lower bounds on the optimal fixed-order H∞-norm whose size only grow quadratically in the dimension of the system state.
...

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