507 research outputs found
On the Geometry and Refined Rate of Primal-Dual Hybrid Gradient for Linear Programming
We study the convergence behaviors of primal-dual hybrid gradient (PDHG) for
solving linear programming (LP). PDHG is the base algorithm of a new
general-purpose first-order method LP solver, PDLP, which aims to scale up LP
by taking advantage of modern computing architectures. Despite its numerical
success, the theoretical understanding of PDHG for LP is still very limited;
the previous complexity result relies on the global Hoffman constant of the KKT
system, which is known to be very loose and uninformative. In this work, we aim
to develop a fundamental understanding of the convergence behaviors of PDHG for
LP and to develop a refined complexity rate that does not rely on the global
Hoffman constant. We show that there are two major stages of PDHG for LP: in
Stage I, PDHG identifies active variables and the length of the first stage is
driven by a certain quantity which measures how close the non-degeneracy part
of the LP instance is to degeneracy; in Stage II, PDHG effectively solves a
homogeneous linear inequality system, and the complexity of the second stage is
driven by a well-behaved local sharpness constant of the system. This finding
is closely related to the concept of partial smoothness in non-smooth
optimization, and it is the first complexity result of finite time
identification without the non-degeneracy assumption. An interesting
implication of our results is that degeneracy itself does not slow down the
convergence of PDHG for LP, but near-degeneracy does
Optimization and Applications
Proceedings of a workshop devoted to optimization problems, their theory and resolution, and above all applications of them. The topics covered existence and stability of solutions; design, analysis, development and implementation of algorithms; applications in mechanics, telecommunications, medicine, operations research
A preconditioned nullspace method for liquid crystal director modelling
We present a preconditioned nullspace method for the numerical solution of large sparse linear systems that arise from discretizations of continuum models for the orientational properties of liquid crystals. The approach effectively deals with pointwise unit-vector constraints, which are prevalent in such models. The indefinite, saddle-point nature of such problems, which can arise from either or both of two sources (pointwise unit-vector constraints, coupled electric fields), is illustrated. Both analytical and numerical results are given for a model problem
Implementing Quantum Gates by Optimal Control with Doubly Exponential Convergence
We introduce a novel algorithm for the task of coherently controlling a
quantum mechanical system to implement any chosen unitary dynamics. It performs
faster than existing state of the art methods by one to three orders of
magnitude (depending on which one we compare to), particularly for quantum
information processing purposes. This substantially enhances the ability to
both study the control capabilities of physical systems within their coherence
times, and constrain solutions for control tasks to lie within experimentally
feasible regions. Natural extensions of the algorithm are also discussed.Comment: 4+2 figures; to appear in PR
Euclidean distance geometry and applications
Euclidean distance geometry is the study of Euclidean geometry based on the
concept of distance. This is useful in several applications where the input
data consists of an incomplete set of distances, and the output is a set of
points in Euclidean space that realizes the given distances. We survey some of
the theory of Euclidean distance geometry and some of the most important
applications: molecular conformation, localization of sensor networks and
statics.Comment: 64 pages, 21 figure
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