55 research outputs found
Faster Approximate Multicommodity Flow Using Quadratically Coupled Flows
The maximum multicommodity flow problem is a natural generalization of the
maximum flow problem to route multiple distinct flows. Obtaining a
approximation to the multicommodity flow problem on graphs is a well-studied
problem. In this paper we present an adaptation of recent advances in
single-commodity flow algorithms to this problem. As the underlying linear
systems in the electrical problems of multicommodity flow problems are no
longer Laplacians, our approach is tailored to generate specialized systems
which can be preconditioned and solved efficiently using Laplacians. Given an
undirected graph with m edges and k commodities, we give algorithms that find
approximate solutions to the maximum concurrent flow problem and
the maximum weighted multicommodity flow problem in time
\tilde{O}(m^{4/3}\poly(k,\epsilon^{-1}))
Hardness Results for Structured Linear Systems
We show that if the nearly-linear time solvers for Laplacian matrices and
their generalizations can be extended to solve just slightly larger families of
linear systems, then they can be used to quickly solve all systems of linear
equations over the reals. This result can be viewed either positively or
negatively: either we will develop nearly-linear time algorithms for solving
all systems of linear equations over the reals, or progress on the families we
can solve in nearly-linear time will soon halt
Conic optimization of electric power systems
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mechanical Engineering, 2011.Cataloged from PDF version of thesis.Includes bibliographical references (p. 103-115).The electric power grid is recognized as an essential modern infrastructure that poses numerous canonical design and operational problems. Perhaps most critically, the inherently large scale of the power grid and similar systems necessitates fast algorithms. A particular complication distinguishing problems in power systems from those arising in other large infrastructures is the mathematical description of alternating current power flow: it is nonconvex, and thus excludes power systems from many frameworks benefiting from theoretically and practically efficient algorithms. However, advances over the past twenty years in optimization have led to broader classes possessing such algorithms, as well as procedures for transferring nonconvex problem to these classes. In this thesis, we approximate difficult problems in power systems with tractable, conic programs. First, we formulate a new type of NP-hard graph cut arising from undirected multicommodity flow networks. An eigenvalue bound in the form of the Cheeger inequality is proven, which serves as a starting point for deriving semidefinite relaxations. We next apply a lift-and-project type relaxation to transmission system planning. The approach unifies and improves upon existing models based on the DC power flow approximation, and yields new mixed-integer linear, second-order cone, and semidefinite models for the AC case. The AC models are particularly applicable to scenarios in which the DC approximation is not justified, such as the all-electric ship. Lastly, we consider distribution system reconfiguration. By making physically motivated simplifications to the DistFlow equations, we obtain mixed-integer quadratic, quadratically constrained, and second-order cone formulations, which are accurate and efficient enough for near-optimal, real-time application. We test each model on standard benchmark problems, as well as a new benchmark abstracted from a notional shipboard power system. The models accurately approximate the original formulations, while demonstrating the scalability required for application to realistic systems. Collectively, the models provide tangible new tradeoffs between computational efficiency and accuracy for fundamental problems in power systems.by Joshua Adam Taylor.Ph.D
Linear Coupling: An Ultimate Unification of Gradient and Mirror Descent
First-order methods play a central role in large-scale machine learning. Even
though many variations exist, each suited to a particular problem, almost all
such methods fundamentally rely on two types of algorithmic steps: gradient
descent, which yields primal progress, and mirror descent, which yields dual
progress.
We observe that the performances of gradient and mirror descent are
complementary, so that faster algorithms can be designed by LINEARLY COUPLING
the two. We show how to reconstruct Nesterov's accelerated gradient methods
using linear coupling, which gives a cleaner interpretation than Nesterov's
original proofs. We also discuss the power of linear coupling by extending it
to many other settings that Nesterov's methods cannot apply to.Comment: A new section added; polished writin
Approximate Gaussian Elimination for Laplacians: Fast, Sparse, and Simple
We show how to perform sparse approximate Gaussian elimination for Laplacian
matrices. We present a simple, nearly linear time algorithm that approximates a
Laplacian by a matrix with a sparse Cholesky factorization, the version of
Gaussian elimination for symmetric matrices. This is the first nearly linear
time solver for Laplacian systems that is based purely on random sampling, and
does not use any graph theoretic constructions such as low-stretch trees,
sparsifiers, or expanders. The crux of our analysis is a novel concentration
bound for matrix martingales where the differences are sums of conditionally
independent variables
- …