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  • 1
    Monograph available for loan
    Monograph available for loan
    Handbook for matrix computations (1988)
    Philadelphia : Society for Industrial and Applied Mathematics
    Associated volumes
    Details Availability
    Call number: 19/M 95.0079
    In: Frontiers in applied mathematics
    Type of Medium: Monograph available for loan
    Pages: vii, 264 S.
    ISBN: 0898712270
    Series Statement: Frontiers in applied mathematics 4
    Classification:
    C.1.8.
    Language: English
    Location: Reading room
    Branch Library: GFZ Library
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  • 2
    Electronic Resource
    Electronic Resource
    On the convergence of interior-reflective Newton methods for nonlinear minimization subject to bounds (1994)
    Springer
    Mathematical programming 67 (1994), S. 189-224 
    Details
    ISSN: 1436-4646
    Keywords: Box constraints ; Interior-point method ; Nonlinear minimization
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract We consider a new algorithm, an interior-reflective Newton approach, for the problem of minimizing a smooth nonlinear function of many variables, subject to upper and/or lower bounds on some of the variables. This approach generatesstrictly feasible iterates by using a new affine scaling transformation and following piecewise linear paths (reflection paths). The interior-reflective approach does not require identification of an “activity set”. In this paper we establish that the interior-reflective Newton approach is globally and quadratically convergent. Moreover, we develop a specific example of interior-reflective Newton methods which can be used for large-scale and sparse problems.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01582221
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  • 3
    Electronic Resource
    Electronic Resource
    A chordal preconditioner for large-scale optimization (1988)
    Springer
    Mathematical programming 40 (1988), S. 265-287 
    Details
    ISSN: 1436-4646
    Keywords: Large-scale optimization ; chordal graph ; preconditioned conjugate-gradients ; unconstrained minimization ; perfect elimination graph ; sparse Cholesky factorization
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract We propose an automatic preconditioning scheme for large sparse numerical optimization. The strategy is based on an examination of the sparsity pattern of the Hessian matrix: using a graph-theoretic heuristic, a block-diagonal approximation to the Hessian matrix is induced. The blocks are submatrices of the Hessian matrix; furthermore, each block is chordal. That is, under a positive definiteness assumption, the Cholesky factorization can be applied to each block without creating any new nonzeros (fill). Therefore the preconditioner is space efficient. We conduct a number of numerical experiments to determine the effectiveness of the preconditioner in the context of a linear conjugate-gradient algorithm for optimization.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01580736
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  • 4
    Electronic Resource
    Electronic Resource
    Partitioned quasi-Newton methods for nonlinear equality constrained optimization (1992)
    Springer
    Mathematical programming 53 (1992), S. 17-44 
    Details
    ISSN: 1436-4646
    Keywords: 65H10 ; 65K05 ; 65K10 ; Constrained optimization ; equality constraints ; numerical optimization ; quasi-Newton method ; secant method ; sequential quadratic programming ; SQP-method ; augmented Lagrangian method ; penalty function methods
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract We derive new quasi-Newton updates for the (nonlinear) equality constrained minimization problem. The new updates satisfy a quasi-Newton equation, maintain positive definiteness on the null space of the active constraint matrix, and satisfy a minimum change condition. The application of the updates is not restricted to a small neighbourhood of the solution. In addition to derivation and motivational remarks, we discuss various numerical subtleties and provide results of numerical experiments.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01585692
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  • 5
    Electronic Resource
    Electronic Resource
    A globally and quadratically convergent affine scaling method for linearℓ 1 problems (1992)
    Springer
    Mathematical programming 56 (1992), S. 189-222 
    Details
    ISSN: 1436-4646
    Keywords: 65H10 ; 65K05 ; 65K10 ; Linearℓ 1 estimation ; linear programming ; interior-point algorithm ; simplex method ; least absolute value regression ; affine scaling method ; Karmarkar
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract Recently, various interior point algorithms related to the Karmarkar algorithm have been developed for linear programming. In this paper, we first show how this “interior point” philosophy can be adapted to the linear ℓ1 problem (in which there are no feasibility constraints) to yield a globally and linearly convergent algorithm. We then show that the linear algorithm can be modified to provide aglobally and ultimatelyquadratically convergent algorithm. This modified algorithm appears to be significantly more efficient in practise than a more straightforward interior point approach via a linear programming formulation: we present numerical results to support this claim.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01580899
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  • 6
    Electronic Resource
    Electronic Resource
    A direct active set algorithm for large sparse quadratic programs with simple bounds (1989)
    Springer
    Mathematical programming 45 (1989), S. 373-406 
    Details
    ISSN: 1436-4646
    Keywords: 65K05 ; 90C20 ; 65K10 ; 65F30 ; Quadratic programming ; large sparse minimization ; active set methods ; trust region methods ; sparse Cholesky factorization updates ; simple bounds ; box constraints
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract We show how a direct active set method for solving definite and indefinite quadratic programs with simple bounds can be efficiently implemented for large sparse problems. All of the necessary factorizations can be carried out in a static data structure that is set up before the numeric computation begins. The space required for these factorizations is no larger than that required for a single sparse Cholesky factorization of the Hessian of the quadratic. We propose several improvements to this basic algorithm: a new way to find a search direction in the indefinite case that allows us to free more than one variable at a time and a new heuristic method for finding a starting point. These ideas are motivated by the two-norm trust region problem. Additionally, we also show how projection techniques can be used to add several constraints to the active set at each iteration. Our experimental results show that an algorithm with these improvements runs much faster than the basic algorithm for positive definite problems and finds local minima with lower function values for indefinite problems.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01589112
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  • 7
    Electronic Resource
    Electronic Resource
    A trust region and affine scaling interior point method for nonconvex minimization with linear inequality constraints (2000)
    Springer
    Mathematical programming 88 (2000), S. 1-31 
    Details
    ISSN: 1436-4646
    Keywords: Key words: trust region – interior point method – Dikin-affine scaling – Newton step
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract. A trust region and affine scaling interior point method (TRAM) is proposed for a general nonlinear minimization with linear inequality constraints [8]. In the proposed approach, a Newton step is derived from the complementarity conditions. Based on this Newton step, a trust region subproblem is formed, and the original objective function is monotonically decreased. Explicit sufficient decrease conditions are proposed for satisfying the first order and second order necessary conditions.¶The objective of this paper is to establish global and local convergence properties of the proposed trust region and affine scaling interior point method. It is shown that the proposed explicit decrease conditions are sufficient for satisfy complementarity, dual feasibility and second order necessary conditions respectively. It is also established that a trust region solution is asymptotically in the interior of the proposed trust region subproblem and a properly damped trust region step can achieve quadratic convergence.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/PL00011369
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  • 8
    Electronic Resource
    Electronic Resource
    An interior Newton method for quadratic programming (1999)
    Springer
    Mathematical programming 85 (1999), S. 491-523 
    Details
    ISSN: 1436-4646
    Keywords: Key words: nonconvex quadratic programming – interior method – Newton method – trust-region method – dogleg method – quadratic convergence Mathematics Subject Classification (1991): 65K05, 90C20, 90C06
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract. We propose a new (interior) approach for the general quadratic programming problem. We establish that the new method has strong convergence properties: the generated sequence converges globally to a point satisfying the second-order necessary optimality conditions, and the rate of convergence is 2-step quadratic if the limit point is a strong local minimizer. Published alternative interior approaches do not share such strong convergence properties for the nonconvex case. We also report on the results of preliminary numerical experiments: the results indicate that the proposed method has considerable practical potential.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s101070050069
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  • 9
    Electronic Resource
    Electronic Resource
    An Exterior Newton Method for Strictly Convex Quadratic Programming (2000)
    Springer
    Computational optimization and applications 15 (2000), S. 5-32 
    Details
    ISSN: 1573-2894
    Keywords: convex quadratic programming ; exterior methods ; Newton methods ; dual problems
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Notes: Abstract We propose an exterior Newton method for strictly convex quadratic programming (QP) problems. This method is based on a dual formulation: a sequence of points is generated which monotonically decreases the dual objective function. We show that the generated sequence converges globally and quadratically to the solution (if the QP is feasible and certain nondegeneracy assumptions are satisfied). Measures for detecting infeasibility are provided. The major computation in each iteration is to solve a KKT-like system. Therefore, given an effective symmetric sparse linear solver, the proposed method is suitable for large sparse problems. Preliminary numerical results are reported.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1008773230148
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  • 10
    Electronic Resource
    Electronic Resource
    A Quasi-Newton Quadratic Penalty Method for Minimization Subject to Nonlinear Equality Constraints (2000)
    Springer
    Computational optimization and applications 15 (2000), S. 103-123 
    Details
    ISSN: 1573-2894
    Keywords: nonlinearly constrained optimization ; equality constraints ; quasi-Newton methods ; BFGS ; quadratic penalty function ; reduced Hessian approximation
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Notes: Abstract We present a modified quadratic 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 quasi-Newton method. We show that the complete algorithm is globally convergent. Preliminary computational results are reported.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1008730909894
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