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Sequential Quadratic Programming
, 1995
"... this paper we examine the underlying ideas of the SQP method and the theory that establishes it as a framework from which effective algorithms can ..."
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Cited by 114 (2 self)
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this paper we examine the underlying ideas of the SQP method and the theory that establishes it as a framework from which effective algorithms can
A New Trust Region Algorithm For Equality Constrained Optimization
, 1995
"... . We present a new trust region algorithm for solving nonlinear equality constrained optimization problems. At each iterate a change of variables is performed to improve the ability of the algorithm to follow the constraint level sets. The algorithm employs L 2 penalty functions for obtaining global ..."
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Cited by 51 (7 self)
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. We present a new trust region algorithm for solving nonlinear equality constrained optimization problems. At each iterate a change of variables is performed to improve the ability of the algorithm to follow the constraint level sets. The algorithm employs L 2 penalty functions for obtaining global convergence. Under certain assumptions we prove that this algorithm globally converges to a point satisfying the second order necessary optimality conditions; the local convergence rate is quadratic. Results of preliminary numerical experiments are presented. 1. Introduction. We consider the equality constrained optimization problem minimize f(x) subject to c(x) = 0 (1:1) where x 2 ! n and f : ! n ! !, and c : ! n ! ! m are smooth nonlinear functions. Problem (1.1) is often solved by successive quadratic programming (SQP) methods. At a current point x k 2 ! n , SQP methods determine a search direction d k by solving a quadratic programming problem minimize rf(x k ) T d + 1 2 ...
On Smooth Decompositions Of Matrices
, 1999
"... . In this paper we consider smooth orthonormal decompositions of smooth time varying matrices. Among others, we consider QR, Schur, and singular value decompositions, and their block analogues. Sufficient conditions for existence of such decompositions are given and differential equations for th ..."
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Cited by 26 (14 self)
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. In this paper we consider smooth orthonormal decompositions of smooth time varying matrices. Among others, we consider QR, Schur, and singular value decompositions, and their block analogues. Sufficient conditions for existence of such decompositions are given and differential equations for the factors are derived. Also generic smoothness of these factors is discussed. 1. Introduction For very good reasons, orthogonal matrices (unitary in the complex case) are the backbone of modern matrix computation. They can be computed stably, and provide some of the most successful algorithmic procedures for a number of familiar tasks: finding orthonormal bases, solving least squares problems, eigenvalues and singular values computations, and so forth. The purpose of this work is to consider orthogonal decompositions for matrices depending on a real parameter. Thus we consider k times continuously differentiable matrix functions, i.e., A 2 C k (R; C m\Thetan ) ; k 0 : We consider a n...
Computation Of Orthonormal Factors For Fundamental Solution Matrices
 Numer. Math
, 1999
"... . In this work, we introduce and analyze two new techniques for obtaining the Q factor in the QR factorization of some (or all) columns of a fundamental solution matrix Y of a linear differential system. These techniques are based on elementary Householder and Givens transformations. We implement a ..."
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Cited by 9 (4 self)
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. In this work, we introduce and analyze two new techniques for obtaining the Q factor in the QR factorization of some (or all) columns of a fundamental solution matrix Y of a linear differential system. These techniques are based on elementary Householder and Givens transformations. We implement and compare these new techniques with existing approaches on some examples. 1. INTRODUCTION Consider the linear system Y = A(t)Y; t 0 ; Y (0) = Y 0 ; Y 0 full rank (1:1) where A : t 2 [0; t f ] ! A(t) 2 IR n\Thetan is a C k\Gamma1 function, k 1, so that Y 2 C k , and Y (t); Y 0 2 IR n\Thetap ; n p. The task is to determine a C k QR factorization of Y (t); 8t 2 [0; t f ]. Let k \Delta k always denote the 2norm for vectors or matrices. There are many known instances in which one needs to compute a QR factorization of Y , and often only Q and the diagonal of R are required (e.g., see the continuous orthonormalization technique in [AsMaRu], and the computation of Lyapunov expone...
On the Convergence Theory of TrustRegionBased Algorithms for EqualityConstrained Optimization
, 1995
"... In this paper we analyze incxact trust region interior point (TRIP) sequential quadr tic programming (SOP) algorithms for the solution of optimization problems with nonlinear equality constraints and simple bound constraints on some of the variables. Such problems arise in many engineering applicati ..."
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Cited by 8 (0 self)
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In this paper we analyze incxact trust region interior point (TRIP) sequential quadr tic programming (SOP) algorithms for the solution of optimization problems with nonlinear equality constraints and simple bound constraints on some of the variables. Such problems arise in many engineering applications, in particular in optimal control problems with bounds on the control. The nonhnear constraints often come from the discretization of partial differential equations. In such cases the calculation of derivative information and the solution of hncarizcd equations is expensive. Often, the solution of hncar systems and derivatives arc computed incxactly yielding nonzero residuals. This paper
Vleck, Orthonormal integrators based on Householder and Givens transformations, Future Generation Computer Systems 19(3) (2003) Special issue: Geometric numerical algorithms
"... Abstract. We carry further our work [DV2] on orthonormal integrators based on Householder and Givens transformations. We propose new algorithms and pay particular attention to appropriate implementation of these techniques. We also present a suite of Fortran codes and provide numerical testing to sh ..."
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Cited by 4 (2 self)
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Abstract. We carry further our work [DV2] on orthonormal integrators based on Householder and Givens transformations. We propose new algorithms and pay particular attention to appropriate implementation of these techniques. We also present a suite of Fortran codes and provide numerical testing to show the efficiency and accuracy of our techniques. AMS(MOS) subject classifications. Primary 65L Key words. Continuous Householder and Givens transformations, orthonormal integrators.
Convergence to a SecondOrder Point of a TrustRegion Algorithm with a Nonmonotonic Penalty Parameter for Constrained Optimization
 Rice University
, 1996
"... In a recent paper, the author (Ref. 1) proposed a trustregion algorithm for solving the problem of minimizing a nonlinear function subject to a set of equality constraints. The main feature of the algorithm is that the penalty parameter in the merit function can be decreased whenever it is warrant ..."
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Cited by 2 (0 self)
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In a recent paper, the author (Ref. 1) proposed a trustregion algorithm for solving the problem of minimizing a nonlinear function subject to a set of equality constraints. The main feature of the algorithm is that the penalty parameter in the merit function can be decreased whenever it is warranted. He studied the behavior of the penalty parameter and proved several global and local convergence results. One of these results is that there exists a subsequence of the iterates generated by the algorithm, that converges to a point that satisfies the firstorder necessary conditions. In the current paper, we show that, for this algorithm, there exists a subsequence of iterates that converges to a point that satisfies both the firstorder and the secondorder necessary conditions. Key Words : Constrained optimization, equality constrained, penalty parameter, nonmonotonic penalty parameter, convergence, trustregion methods, firstorder point, secondorder point, necessary conditions. B 1...
A QuasiNewton L2Penalty Method for Minimization Subject to Nonlinear Equality Constraints
"... . We present a modified L 2 penalty function method for equality constrained optimization problems. The pivotal feature of our algorithm is that at every iterate we invoke a special change of variables to improve the ability of the algorithm to follow the constraint level sets. This change of variab ..."
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. We present a modified L 2 penalty function method for equality constrained optimization problems. The pivotal feature of our algorithm is that at every iterate we invoke a special change of variables to improve the ability of the algorithm to follow the constraint level sets. This change of variables gives rise to a suitable block diagonal approximation to the Hessian which is then used to construct a quasiNewton method. We show that the complete algorithm is globally convergent with a local Qsuperlinearly convergence rate. Preliminary computational results are given for a few problems. 1. Introduction. We construct a quasiNewton L 2 penalty method for solving the equality constrained optimization problem minimize f(x) subject to c(x) = 0 (1:1) where x 2 ! n , and f : ! n ! ! and c : ! n ! ! m are smooth nonlinear functions. This method possesses both strong global convergence properties and a local superlinear convergence rate by combining an L 2 penalty function method ...
Numerical Decomposition of a Convex Function 1
"... Abstract. Given the n xp orthogonal matrix A and the convex function f: R" ~ R, we find two orthogonal matrices P and Q such that f is almost constant on the convex hull of ± the columns of P, f is sufficiently nonconstant on the column space of Q, and the column spaces of P and Q provide an orthog ..."
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Abstract. Given the n xp orthogonal matrix A and the convex function f: R" ~ R, we find two orthogonal matrices P and Q such that f is almost constant on the convex hull of ± the columns of P, f is sufficiently nonconstant on the column space of Q, and the column spaces of P and Q provide an orthogonal direct sum decomposition of the column space of A. This provides a numerically stable algorithm for calculating the cone of directions of constancy, at a point x, of a convex function. Applications to convex programming are discussed. Key Words. Convex functions, convex programming, cone of directions of constancy, numerical stability. 1.