193 research outputs found
An algorithm for the solution of dynamic linear programs
The algorithm's objective is to efficiently solve Dynamic Linear Programs (DLP) by taking advantage of their special staircase structure. This algorithm constitutes a stepping stone to an improved algorithm for solving Dynamic Quadratic Programs, which, in turn, would make the nonlinear programming method of Successive Quadratic Programs more practical for solving trajectory optimization problems. The ultimate goal is to being trajectory optimization solution speeds into the realm of real-time control. The algorithm exploits the staircase nature of the large constraint matrix of the equality-constrained DLPs encountered when solving inequality-constrained DLPs by an active set approach. A numerically-stable, staircase QL factorization of the staircase constraint matrix is carried out starting from its last rows and columns. The resulting recursion is like the time-varying Riccati equation from multi-stage LQR theory. The resulting factorization increases the efficiency of all of the typical LP solution operations over that of a dense matrix LP code. At the same time numerical stability is ensured. The algorithm also takes advantage of dynamic programming ideas about the cost-to-go by relaxing active pseudo constraints in a backwards sweeping process. This further decreases the cost per update of the LP rank-1 updating procedure, although it may result in more changes of the active set that if pseudo constraints were relaxed in a non-stagewise fashion. The usual stability of closed-loop Linear/Quadratic optimally-controlled systems, if it carries over to strictly linear cost functions, implies that the saving due to reduced factor update effort may outweigh the cost of an increased number of updates. An aerospace example is presented in which a ground-to-ground rocket's distance is maximized. This example demonstrates the applicability of this class of algorithms to aerospace guidance. It also sheds light on the efficacy of the proposed pseudo constraint relaxation scheme
Regularized Decomposition of Stochastic Programs: Algorithmic Techniques and Numerical Results
A finitely convergent non-simplex method for large scale structured linear programming problems arising in stochastic programming is presented. The method combines the ideas of the Dantzig-Wolfe decomposition principle and modern nonsmooth optimization methods. Algorithmic techniques taking advantage of properties of stochastic programs are described and numerical results for large real world problems reported
Active set Methods for Problems in Column Block Angular Form
We study active set methods for optimization problems in Block Angular Form (BAF). We begin by reviewing some standard basis factorizations, including Saunders' orthogonal factorization and updates for the simplex method that do not impose any restriction on the pivot sequence and maintain the basis factorization structured in BAF throughout the algorithm. We then suggest orthogonal factorization and updating procedures that allow coarse grain parallelization, pivot updates local to the affected blocks, and independent block reinversion. A simple parallel environment appropriate to the description and complexity analysis of test procedures is defined in Section 5. The factorization and updating procedures are presented in Sections 6 and 7. Our update procedure outperforms conventional Updating procedures even in a purely sequential environment
A comparison of three numerical methods for updating regressions.
Three numerical procedures are presented for updating regressions. All three
methods are based on QR factorization, but after that they use different philosophies
to update the regression coefficients. Elden's algorithm updates using only the
triangular matrix R. This procedure does not use orthogonal transformations, but it
uses hyperbolic rotations. The modified Gram-Schmidt QR process is used by Gragg-
Leveque-Trangenstein's method where the matrix with orthonormal columns is stored
and updated. Chan's algorithm computes a column permutation n and a QR
factorization of a matrix A such that a rank deficiency of A will be revealed. Although
the three methods are based on different ideas and can be used for different purposes
their comparison shows that Chan's algorithm is the only accurate one in the rank
deficient case, and that Gragg-Leveque-Trangenstein's method is the cheapest and the
most stable.http://archive.org/details/comparisonofthre00raptLieutenant Commander, Hellenic NavyApproved for public release; distribution is unlimited
Design and analysis of numerical algorithms for the solution of linear systems on parallel and distributed architectures
The increasing availability of parallel computers is having a very significant impact on
all aspects of scientific computation, including algorithm research and software
development in numerical linear algebra. In particular, the solution of linear systems,
which lies at the heart of most calculations in scientific computing is an important
computation found in many engineering and scientific applications.
In this thesis, well-known parallel algorithms for the solution of linear systems are
compared with implicit parallel algorithms or the Quadrant Interlocking (QI) class of
algorithms to solve linear systems. These implicit algorithms are (2x2) block
algorithms expressed in explicit point form notation. [Continues.
Quadtree Structured Approximation Algorithms
The success of many image restoration algorithms is often due to their ability to sparsely describe the original signal. Many sparse promoting transforms exist, including wavelets, the so called âletsâ family of transforms and more recent non-local learned transforms. The first part of this thesis reviews sparse approximation theory, particularly in relation to 2-D piecewise polynomial signals. We also show the connection between this theory and current state of the art algorithms that cover the following image restoration and enhancement applications: denoising, deconvolution, interpolation and multi-view super resolution.
In [63], Shukla et al. proposed a compression algorithm, based on a sparse quadtree decomposition model, which could optimally represent piecewise polynomial images. In the second part of this thesis we adapt this model to image restoration by changing the rate-distortion penalty to a description-length penalty. Moreover, one of the major drawbacks of this type of approximation is the computational complexity required to find a suitable subspace for each node of the quadtree. We address this issue by searching for a suitable subspace much more efficiently using the mathematics of updating matrix factorisations. Novel algorithms are developed to tackle the four problems previously mentioned. Simulation results indicate that we beat state of the art results when the original signal is in the model (e.g. depth images) and are competitive for natural images when the degradation is high.Open Acces
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