Semidefinite programming (SDP), i.e., convex optimization with linear matrix inequality (LMI) constraints, is now widely recognized as a powerful numerical tool in several areas of engineering, including control, communication, signal processing, and circuit design. However, some limitations of SDP have become apparent. Most important, the reduction of an engineering problem to an SDP often requires a large number of auxiliary variables, so that the resulting SDP problem can be very large, even though the underlying engineering problem is not particularly large-scale. This limits the problem sizes that can be handled by general-purpose SDP solvers, and hence the applicability of SDP in practical engineering problems. In order to have an impact on engineering practice, future SDP solvers will have to be able to take advantage of problem structure, without overly restricting their scope of applicability or sacrificing reliability. Motivated by these observations, we have identified a class of engineering problems with a rich enough structure that much faster special SDP algorithms can be devised for it, but with wide enough applicability that the benefit is widespread. Specifically, we address the fast solution of SDP problems where the underlying LMIs have a special form that is typically encountered with the application of the Kalman-Yakubovich-Popov Lemma. The research component of the proposal will contribute to the propagation of SDP into the mainstream of engineering practice. The educational component will provide graduate student researchers with an interdisciplinary training in control, optimization, and numerical computing.
