120 research outputs found
Sparse Inverse Covariance Estimation for Chordal Structures
In this paper, we consider the Graphical Lasso (GL), a popular optimization
problem for learning the sparse representations of high-dimensional datasets,
which is well-known to be computationally expensive for large-scale problems.
Recently, we have shown that the sparsity pattern of the optimal solution of GL
is equivalent to the one obtained from simply thresholding the sample
covariance matrix, for sparse graphs under different conditions. We have also
derived a closed-form solution that is optimal when the thresholded sample
covariance matrix has an acyclic structure. As a major generalization of the
previous result, in this paper we derive a closed-form solution for the GL for
graphs with chordal structures. We show that the GL and thresholding
equivalence conditions can significantly be simplified and are expected to hold
for high-dimensional problems if the thresholded sample covariance matrix has a
chordal structure. We then show that the GL and thresholding equivalence is
enough to reduce the GL to a maximum determinant matrix completion problem and
drive a recursive closed-form solution for the GL when the thresholded sample
covariance matrix has a chordal structure. For large-scale problems with up to
450 million variables, the proposed method can solve the GL problem in less
than 2 minutes, while the state-of-the-art methods converge in more than 2
hours
PowerModels.jl: An Open-Source Framework for Exploring Power Flow Formulations
In recent years, the power system research community has seen an explosion of
novel methods for formulating and solving power network optimization problems.
These emerging methods range from new power flow approximations, which go
beyond the traditional DC power flow by capturing reactive power, to convex
relaxations, which provide solution quality and runtime performance guarantees.
Unfortunately, the sophistication of these emerging methods often presents a
significant barrier to evaluating them on a wide variety of power system
optimization applications. To address this issue, this work proposes
PowerModels, an open-source platform for comparing power flow formulations.
From its inception, PowerModels was designed to streamline the process of
evaluating different power flow formulations on shared optimization problem
specifications. This work provides a brief introduction to the design of
PowerModels, validates its implementation, and demonstrates its effectiveness
with a proof-of-concept study analyzing five different formulations of the
Optimal Power Flow problem
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