1,211 research outputs found
On Quasi-Newton Forward--Backward Splitting: Proximal Calculus and Convergence
We introduce a framework for quasi-Newton forward--backward splitting
algorithms (proximal quasi-Newton methods) with a metric induced by diagonal
rank- symmetric positive definite matrices. This special type of
metric allows for a highly efficient evaluation of the proximal mapping. The
key to this efficiency is a general proximal calculus in the new metric. By
using duality, formulas are derived that relate the proximal mapping in a
rank- modified metric to the original metric. We also describe efficient
implementations of the proximity calculation for a large class of functions;
the implementations exploit the piece-wise linear nature of the dual problem.
Then, we apply these results to acceleration of composite convex minimization
problems, which leads to elegant quasi-Newton methods for which we prove
convergence. The algorithm is tested on several numerical examples and compared
to a comprehensive list of alternatives in the literature. Our quasi-Newton
splitting algorithm with the prescribed metric compares favorably against
state-of-the-art. The algorithm has extensive applications including signal
processing, sparse recovery, machine learning and classification to name a few.Comment: arXiv admin note: text overlap with arXiv:1206.115
A differential analysis of the power flow equations
The AC power flow equations are fundamental in all aspects of power systems
planning and operations. They are routinely solved using Newton-Raphson like
methods. However, there is little theoretical understanding of when these
algorithms are guaranteed to find a solution of the power flow equations or how
long they may take to converge. Further, it is known that in general these
equations have multiple solutions and can exhibit chaotic behavior. In this
paper, we show that the power flow equations can be solved efficiently provided
that the solution lies in a certain set. We introduce a family of convex
domains, characterized by Linear Matrix Inequalities, in the space of voltages
such that there is at most one power flow solution in each of these domains.
Further, if a solution exists in one of these domains, it can be found
efficiently, and if one does not exist, a certificate of non-existence can also
be obtained efficiently. The approach is based on the theory of monotone
operators and related algorithms for solving variational inequalities involving
monotone operators. We validate our approach on IEEE test networks and show
that practical power flow solutions lie within an appropriately chosen convex
domain.Comment: arXiv admin note: text overlap with arXiv:1506.0847
Conic Optimization Theory: Convexification Techniques and Numerical Algorithms
Optimization is at the core of control theory and appears in several areas of
this field, such as optimal control, distributed control, system
identification, robust control, state estimation, model predictive control and
dynamic programming. The recent advances in various topics of modern
optimization have also been revamping the area of machine learning. Motivated
by the crucial role of optimization theory in the design, analysis, control and
operation of real-world systems, this tutorial paper offers a detailed overview
of some major advances in this area, namely conic optimization and its emerging
applications. First, we discuss the importance of conic optimization in
different areas. Then, we explain seminal results on the design of hierarchies
of convex relaxations for a wide range of nonconvex problems. Finally, we study
different numerical algorithms for large-scale conic optimization problems.Comment: 18 page
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