# Smart Grid Seminar

**Geometry and convexity of quadratic maps**

**Speaker:**Anatoly Dymarsky, Center for Energy Systems, Skolkovo Institute of Science and Technology

**Location:**Moore 139

Quadratic maps (or quadratic mappings), general vector-valued mutivariate quadratic functions y=f(x), are ubiquitous and appear in many practical applications. One such example is the Power Flow equations where powers being the quadratic functions of voltages. Often it is desirable to know if the image of a given quadratic map F possesses particular geometric properties, in particular if it is convex. In the latter case many questions of practical importance, e.g. solvability of f(x)=y for some given y, or various optimization problems with a convex utility function, can be easily addressed. At the same time it is known that establishing convexity of F is NP hard. In the talk I will review a number of recent results which allow to establish convexity of F analytically, provided the map satisfies a number of additional conditions. Applying this toward Power Flow equations we will show convexity of feasibility set for the balanced distribution networks. In the case when F is non-convex our approach allows to identify a compact subset within F which is convex. Finally, I will introduce a Matlab library which implements stochastic algorithms to verify convexity of F and find the convex subpart of it numerically.

**BIO: **Anatoly graduated from Princeton University with a PhD in Physics in 2007. He continued as a postdoc at Stanford University and then at the Institute for Advanced Study and University of Cambridge. Starting 2013 Anatoly is with the Center for Energy Systems at Skolkovo Institute of Science and Technology (Skoltech). From 2016 Anatoly is also an assistant professor at the University of Kentucky.

**Series**Smart Grid Seminar

**Contact:** Daniel Guo lguo@caltech.edu