2 research outputs found
Fast Nonconvex SDP Solvers for Large-scale Power System State Estimation
Fast power system state estimation (SE) solution is of paramount importance
for achieving real-time decision making in power grid operations. Semidefinite
programming (SDP) reformulation has been shown effective to obtain the global
optimum for the nonlinear SE problem, while suffering from high computational
complexity. Thus, we leverage the recent advances in nonconvex SDP approach
that allows for the simple first-order gradient-descent (GD) updates. Using the
power system model, we can verify that the SE objective function enjoys nice
properties (strongly convex, smoothness) which in turn guarantee a linear
convergence rate of the proposed GD-based SE method. To further accelerate the
convergence speed, we consider the accelerated gradient descent (AGD)
extension, as well as their robust versions under outlier data and a hybrid
GD-based SE approach with additional synchrophasor measurements. Numerical
tests on the IEEE 118-bus, 300-bus and the synthetic ACTIVSg2000-bus systems
have demonstrated that FGD-SE and AGD-SE, can approach the near-optimal
performance of the SDP-SE solution at significantly improved computational
efficiency, especially so for AGD-SE
Provably convergent acceleration in factored gradient descent with applications in matrix sensing
We present theoretical results on the convergence of \emph{non-convex}
accelerated gradient descent in matrix factorization models with -norm
loss. The purpose of this work is to study the effects of acceleration in
non-convex settings, where provable convergence with acceleration should not be
considered a \emph{de facto} property. The technique is applied to matrix
sensing problems, for the estimation of a rank optimal solution . Our contributions can be summarized as follows.
We show that acceleration in factored gradient descent converges at a
linear rate; this fact is novel for non-convex matrix factorization settings,
under common assumptions. Our proof technique requires the acceleration
parameter to be carefully selected, based on the properties of the problem,
such as the condition number of and the condition number of objective
function. Currently, our proof leads to the same dependence on the
condition number(s) in the contraction parameter, similar to recent results on
non-accelerated algorithms. Acceleration is observed in practice, both in
synthetic examples and in two real applications: neuronal multi-unit activities
recovery from single electrode recordings, and quantum state tomography on
quantum computing simulators.Comment: 23 page