ResearchMy research interests include
Solving largescale quadratic systems in linear timeThe problem of solving systems of quadratic equations has a plethora of applications ranging from mixed linear regressions to the wellknown phase retrieval. Under Gaussian random sampling/feature vectors, we develop simple, scalable, and efficient iterative optimization algorithms that are able to solve a quadratic system when there are about as many equations as unknowns in linear time. It is known in statistical inference and learning that convex formulations are unbounded and thus sensitive to outliers, yet nonconvex ones that are difficult to optimize lead to computationally more scalable and statistically more accurate solution algorithms. We formulate the problem of solving quadratic equations as a nonconvex optimization, and develop twostage iterative optimization algorithms, that consist of obtaining an orthogonalitypromoting initialization first and refining the initialization via truncated/stochastic gradienttype iterations. Empirically, our algorithms recover exactly any realvalued signals when the number of equations is about 3 times the number of unknowns, narrowing the gap from the informationtheoretic measurement/unknown ratio 2.
Stochastic energy management in power distribution gridsDistribution microgrids are currently being challenged by voltage fluctuations due to renewable generation, demand response, and electric vehicles. Advances in photovoltaic (PV) inverters offer new opportunities for reactive power management, provided PV owners have the right investment incentives. Accounting for the increasing timevariability of distributed generation and demand, a stochastic reactive power compensation scheme is developed. The scheme is distributionfree, and it relies solely on realtime power injection data. Numerical tests on an industrial 47bus microgrid and the residential IEEE 123bus feeder corroborate its superiority over its deterministic alternative, as well as its capability to track variations in solar generation and household demand.
