10,576 research outputs found
Adapting the interior point method for the solution of LPs on serial, coarse grain parallel and massively parallel computers
In this paper we describe a unified scheme for implementing an interior point algorithm (IPM) over a range of computer architectures. In the inner iteration of the IPM a search direction is computed using Newton's method. Computationally this involves solving a sparse symmetric positive definite (SSPD) system of equations. The choice of direct and indirect methods for the solution of this system, and the design of data structures to take advantage of serial, coarse grain parallel and massively parallel computer architectures, are considered in detail. We put forward arguments as to why integration of the system within a sparse simplex solver is important and outline how the system is designed to achieve this integration
Distributed Robust Stability Analysis of Interconnected Uncertain Systems
This paper considers robust stability analysis of a large network of
interconnected uncertain systems. To avoid analyzing the entire network as a
single large, lumped system, we model the network interconnections with
integral quadratic constraints. This approach yields a sparse linear matrix
inequality which can be decomposed into a set of smaller, coupled linear matrix
inequalities. This allows us to solve the analysis problem efficiently and in a
distributed manner. We also show that the decomposed problem is equivalent to
the original robustness analysis problem, and hence our method does not
introduce additional conservativeness.Comment: This paper has been accepted for presentation at the 51st IEEE
Conference on Decision and Control, Maui, Hawaii, 201
Parameter Selection and Pre-Conditioning for a Graph Form Solver
In a recent paper, Parikh and Boyd describe a method for solving a convex
optimization problem, where each iteration involves evaluating a proximal
operator and projection onto a subspace. In this paper we address the critical
practical issues of how to select the proximal parameter in each iteration, and
how to scale the original problem variables, so as the achieve reliable
practical performance. The resulting method has been implemented as an
open-source software package called POGS (Proximal Graph Solver), that targets
multi-core and GPU-based systems, and has been tested on a wide variety of
practical problems. Numerical results show that POGS can solve very large
problems (with, say, more than a billion coefficients in the data), to modest
accuracy in a few tens of seconds. As just one example, a radiation treatment
planning problem with around 100 million coefficients in the data can be solved
in a few seconds, as compared to around one hour with an interior-point method.Comment: 28 pages, 1 figure, 1 open source implementatio
Stable Camera Motion Estimation Using Convex Programming
We study the inverse problem of estimating n locations (up to
global scale, translation and negation) in from noisy measurements of a
subset of the (unsigned) pairwise lines that connect them, that is, from noisy
measurements of for some pairs (i,j) (where the
signs are unknown). This problem is at the core of the structure from motion
(SfM) problem in computer vision, where the 's represent camera locations
in . The noiseless version of the problem, with exact line measurements,
has been considered previously under the general title of parallel rigidity
theory, mainly in order to characterize the conditions for unique realization
of locations. For noisy pairwise line measurements, current methods tend to
produce spurious solutions that are clustered around a few locations. This
sensitivity of the location estimates is a well-known problem in SfM,
especially for large, irregular collections of images.
In this paper we introduce a semidefinite programming (SDP) formulation,
specially tailored to overcome the clustering phenomenon. We further identify
the implications of parallel rigidity theory for the location estimation
problem to be well-posed, and prove exact (in the noiseless case) and stable
location recovery results. We also formulate an alternating direction method to
solve the resulting semidefinite program, and provide a distributed version of
our formulation for large numbers of locations. Specifically for the camera
location estimation problem, we formulate a pairwise line estimation method
based on robust camera orientation and subspace estimation. Lastly, we
demonstrate the utility of our algorithm through experiments on real images.Comment: 40 pages, 12 figures, 6 tables; notation and some unclear parts
updated, some typos correcte
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