IFAC MICNON 2015, Saint Petersburg - Lecture by B. Polyak

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Boris T. Polyak - Quadratic Transformations in Control and Optimization.
Quadratic transformations arise in numerous fields of optimization and control: general quadratic programming, relaxations for discrete optimization, multiobjective minimization, S‐theorem for absolute stability, mu‐analysis, ellipsoidal techniques and so on. Recent applications in physics (quantum mechanics and power systems) are highly promising. The key problem is convexity of quadratic maps. It has been established for some particular classes of problems, mainly for maps into low‐dimensional space. We address a different approach to the problem: given a quadratic transformation, how to recognize if the corresponding image is convex or not. We provide some «convexity/nonconvexity sertificates» which can be effectively verified. Another issue is an approximate description of the image or its boundary even for nonconvex case. These techniques are the basis for solving nonconvex quadratic optimization problems.

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