We study the convex hull of SO(n), thought of as the set of n×n orthogonal matrices with unit determinant, from the point of view of semidefinite programming. We show that the convex hull of SO(n) is doubly spectrahedral, i.e. both it and its polar have a description as the intersection of the cone of positive semidefinite matrices with an affine subspace. Our spectrahedral representations are explicit, and are of minimum size, in the sense that there are no smaller spectrahedral representations of these convex bodies.

As an application, we consider the problem of jointly estimating the attitude and spin-rate of a spinning spacecraft (known as Psiaki’s first generalization of Wahba’s problem), and show that it can be reformulated exactly as a semidefinite programming problem, which allows us to globally solve it using convex optimization techniques.

Joint work with James Saunderson and Alan Willsky.  Preprints available at arXiv:1403.4914 and 1410.2841.

Bio: Pablo A. Parrilo is a Professor of Electrical Engineering and Computer Science at the Massachusetts Institute of Technology. He is currently Associate Director of the Laboratory for Information and Decision Systems (LIDS), and is also affiliated with the Operations Research Center (ORC). Past appointments include Assistant Professor at the Automatic Control Laboratory of the Swiss Federal Institute of Technology (ETH Zurich), Visiting Associate Professor at the California Institute of Technology, as well as short-term research visits at the University of California at Santa Barbara (Physics), Lund Institute of Technology (Automatic Control), and University of California at Berkeley (Mathematics). He received an Electronics Engineering undergraduate degree from the University of Buenos Aires, and a PhD in Control and Dynamical Systems from the California Institute of Technology.
His research interests include optimization methods for engineering applications, control and identification of uncertain complex systems, robustness analysis and synthesis, and the development and application of computational tools based on convex optimization and algorithmic algebra to practically relevant engineering problems.

Prof. Parrilo has received several distinctions, including a Finmeccanica Career Development Chair, the Donald P. Eckman Award of the American Automatic Control Council, the SIAM Activity Group on Control and Systems Theory (SIAG/CST) Prize, the IEEE Antonio Ruberti Young Researcher Prize, and the Farkas Prize of the INFORMS Optimization Society.