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96
Complete Orthogonal Decomposition for Weighted Least Squares
 SIAM J. Matrix Anal. Appl
, 1995
"... Consider a fullrank weighted leastsquares problem in which the weight matrix is highly illconditioned. Because of the illconditioning, standard methods for solving leastsquares problems, QR factorization and the nullspace method for example, break down. G. W. Stewart established a norm bound fo ..."
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Cited by 16 (4 self)
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Consider a fullrank weighted leastsquares problem in which the weight matrix is highly illconditioned. Because of the illconditioning, standard methods for solving leastsquares problems, QR factorization and the nullspace method for example, break down. G. W. Stewart established a norm bound for such a system of equations, indicating that it may be possible to find an algorithm that gives an accurate solution. S. A. Vavasis proposed a new definition of stability that is based on this result. He also defined the NSH algorithm for solving this leastsquares problem and showed that it satisfies his definition of stability. In this paper, we propose a complete orthogonal decomposition algorithm to solve this problem and show that it is also stable. This new algorithm is simpler and more efficient than the NSH method. 1 Introduction We consider solving the problem min y2R n kD \Gamma1=2 (Ay \Gamma b) k (1) for y, where D is a symmetric positive definite m \Theta m matrix, A is an ...
Global Optimization for the Biaffine Matrix Inequality Problem
, 1995
"... It has recently been shown that an extremely wide array of robust controller design problems may be reduced to the problem of finding a feasible point under a Biaffine Matrix Inequality (BMI) constraint. The BMI feasibility problem is the bilinear version of the Linear (Affine) Matrix Inequality (L ..."
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Cited by 16 (0 self)
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It has recently been shown that an extremely wide array of robust controller design problems may be reduced to the problem of finding a feasible point under a Biaffine Matrix Inequality (BMI) constraint. The BMI feasibility problem is the bilinear version of the Linear (Affine) Matrix Inequality (LMI) feasibility problem, and may also be viewed as a bilinear extension to the Semidefinite Programming (SDP) problem. The BMI problem may be approached as a biconvex global optimization problem of minimizing the maximum eigenvalue of a biaffine combination of symmetric matrices. This paper presents a branch and bound global optimization algorithm for the BMI. A simple numerical example is included. The robust control problem, i.e., the synthesis of a controller for a dynamic physical system which guarantees stability and performance in the face of significant modelling error and worstcase disturbance inputs, is frequently encountered in a variety of complex engineering applications including the design of aircraft, satellites, chemical plants, and other precision positioning and tracking systems.
Application of barrier function base model predictive control to an edible oil refining process. Provisionally accepted for the Journal of Process Control
, 2004
"... March, 2003I hereby certify that the work embodied in this thesis is the result of original research and has not been submitted for a higher degree to any other University or Institution. Adrian WillsAcknowledgements I would like to thank my supervisor Dr. Will Heath for his exceptional patience, hi ..."
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March, 2003I hereby certify that the work embodied in this thesis is the result of original research and has not been submitted for a higher degree to any other University or Institution. Adrian WillsAcknowledgements I would like to thank my supervisor Dr. Will Heath for his exceptional patience, his willingness to sacrifice, his genuine and pragmatic approach to research and for his friendship which I hope continues. I am indebted to Will for more than I can recall and I am truly grateful for all of his help and support. Thanks. A special thanks to Dr. Liuping Wang, who helped establish my scholarship and the industrial partnership. A further special thanks to Professors Graham Goodwin and Rick Middleton for their technical and financial support. Thanks to Dr. Charlie Chessari and Jay Selahewa who established my scholarship through
Adaptive Use Of Iterative Methods In Interior Point Methods For Linear Programming
, 1995
"... In this work we devise efficient algorithms for finding the search directions for interior point methods applied to linear programming problems. There are two innovations. The first is the use of updating of preconditioners computed for previous barrier parameters. The second is an adaptive automate ..."
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Cited by 15 (3 self)
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In this work we devise efficient algorithms for finding the search directions for interior point methods applied to linear programming problems. There are two innovations. The first is the use of updating of preconditioners computed for previous barrier parameters. The second is an adaptive automated procedure for determining whether to use a direct or iterative solver, whether to reinitialize or update the preconditioner, and how many updates to apply. These decisions are based on predictions of the cost of using the different solvers to determine the next search direction, given costs in determining earlier directions. These ideas are tested by applying a modified version of the OB1R code of Lustig, Marsten, and Shanno to a variety of problems from the NETLIB and other collections. If a direct method is appropriate for the problem, then our procedure chooses it, but when an iterative procedure is helpful, substantial gains in efficiency can be obtained.
REGULARIZED SEQUENTIAL QUADRATIC PROGRAMMING METHODS
, 2011
"... We present the formulation and analysis of a new sequential quadratic programming (SQP) method for general nonlinearly constrained optimization. The method pairs a primaldual generalized augmented Lagrangian merit function with a flexible line search to obtain a sequence of improving estimates of t ..."
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Cited by 15 (3 self)
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We present the formulation and analysis of a new sequential quadratic programming (SQP) method for general nonlinearly constrained optimization. The method pairs a primaldual generalized augmented Lagrangian merit function with a flexible line search to obtain a sequence of improving estimates of the solution. This function is a primaldual variant of the augmented Lagrangian proposed by Hestenes and Powell in the early 1970s. A crucial feature of the method is that the QP subproblems are convex, but formed from the exact second derivatives of the original problem. This is in contrast to methods that use a less accurate quasiNewton approximation. Additional benefits of this approach include the following: (i) each QP subproblem is regularized; (ii) the QP subproblem always has a known feasible point; and (iii) a projected gradient method may be used to identify the QP active set when far from the solution.
Zonotopes as Bounding Volumes
 Proceedings of 14 t h Annual ACMSIAM Symposium on Discrete Algorithms
, 2003
"... Zonotopes are centrally symmetric polytopes with a very special structure: they are the Minkowski sum of line segments. In this paper we propose to use zonotopes as bounding volumes for geometry in collision detection and other applications where the spatial relationship between two pieces of geomet ..."
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Zonotopes are centrally symmetric polytopes with a very special structure: they are the Minkowski sum of line segments. In this paper we propose to use zonotopes as bounding volumes for geometry in collision detection and other applications where the spatial relationship between two pieces of geometry is important. We show how to construct optimal, or approximately optimal zonotopes enclosing given set of points or other geometry. We also show how zonotopes can be used for efficient collision testing, based on their description via their defining line segments — without ever building their explicit description as polytopes. This implicit representation adds flexibility, power, and economy to the use of zonotopes as bounding volumes. 1
Global Optimization For Constrained Nonlinear Programming
, 2001
"... In this thesis, we develop constrained simulated annealing (CSA), a global optimization algorithm that asymptotically converges to constrained global minima (CGM dn ) with probability one, for solving discrete constrained nonlinear programming problems (NLPs). The algorithm is based on the necessary ..."
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Cited by 14 (2 self)
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In this thesis, we develop constrained simulated annealing (CSA), a global optimization algorithm that asymptotically converges to constrained global minima (CGM dn ) with probability one, for solving discrete constrained nonlinear programming problems (NLPs). The algorithm is based on the necessary and sufficient condition for constrained local minima (CLM dn ) in the theory of discrete constrained optimization using Lagrange multipliers developed in our group. The theory proves the equivalence between the set of discrete saddle points and the set of CLM dn, leading to the firstorder necessary and sufficient condition for CLM dn. To find
Properties of the LogBarrier Function on Degenerate Nonlinear Programs
 MATH. OPER. RES
, 1999
"... We examine the sequence of local minimizers of the logbarrier function for a nonlinear program near a solution at which secondordersufficient conditions and the MangasarianFromovitz constraint qualifications are satisfied, but the active constraint gradients are not necessarily linearly independ ..."
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We examine the sequence of local minimizers of the logbarrier function for a nonlinear program near a solution at which secondordersufficient conditions and the MangasarianFromovitz constraint qualifications are satisfied, but the active constraint gradients are not necessarily linearly independent. When a strict complementarity condition is satisfied, we show uniqueness of the local minimizer of the barrier function in the vicinity of the nonlinear program solution, and obtain a semiexplicit characterization of this point. When strict complementarity does not hold, we obtain several other interesting characterizations, in particular, an estimate of the distance between the minimizers of the barrier function and the nonlinear program in terms of the barrier parameter, and a result about the direction of approach of the sequence of minimizers of the barrier function to the nonlinear programming solution.
A globally convergent Lagrangian barrier algorithm for optimization with general inequality constraints and simple bounds
 MATH. OF COMPUTATION
, 1997
"... We consider the global and local convergence properties of a class of Lagrangian barrier methods for solving nonlinear programming problems. In such methods, simple bound constraints may be treated separately from more general constraints. The objective and general constraint functions are combine ..."
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Cited by 13 (1 self)
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We consider the global and local convergence properties of a class of Lagrangian barrier methods for solving nonlinear programming problems. In such methods, simple bound constraints may be treated separately from more general constraints. The objective and general constraint functions are combined in a Lagrangian barrier function. A sequence of such functions are approximately minimized within the domain defined by the simple bounds. Global convergence of the sequence of generated iterates to a firstorder stationary point for the original problem is established. Furthermore, possible numerical difficulties associated with barrier function methods are avoided as it is shown that a potentially troublesome penalty parameter is bounded away from zero. This paper is a companion to previous work of ours on augmented Lagrangian methods.