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  • 1
    Electronic Resource
    Electronic Resource
    Algorithms for bound constrained quadratic programming problems (1989)
    Springer
    Numerische Mathematik 55 (1989), S. 377-400 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65k05 ; CR: G1.6
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We present an algorithm which combines standard active set strategies with the gradient projection method for the solution of quadratic programming problems subject to bounds. We show, in particular, that if the quadratic is bounded below on the feasible set then termination occurs at a stationary point in a finite number of iterations. Moreover, if all stationary points are nondegenerate, termination occurs at a local minimizer. A numerical comparison of the algorithm based on the gradient projection algorithm with a standard active set strategy shows that on mildly degenerate problems the gradient projection algorithm requires considerable less iterations and time than the active set strategy. On nondegenerate problems the number of iterations typically decreases by at least a factor of 10. For strongly degenerate problems, the performance of the gradient projection algorithm deteriorates, but it still performs better than the active set method.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01396045
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  • 2
    Electronic Resource
    Electronic Resource
    On the use of directions of negative curvature in a modified newton method (1979)
    Springer
    Mathematical programming 16 (1979), S. 1-20 
    Details
    ISSN: 1436-4646
    Keywords: Unconstrained Optimization ; Modified Newton's Method ; Descent Pairs ; Directions of Negative Curvature ; Symmetric Indefinite Factorization ; Steplength Algorithm
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract We present a modified Newton method for the unconstrained minimization problem. The modification occurs in non-convex regions where the information contained in the negative eigenvalues of the Hessian is taken into account by performing a line search along a path which is initially tangent to a direction of negative curvature. We give termination criteria for the line search and prove that the resulting iterates are guaranteed to converge, under reasonable conditions, to a critical point at which the Hessian is positive semidefinite. We also show how the Bunch and Parlett decomposition of a symmetric indefinite matrix can be used to give entirely adequate directions of negative curvature.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01582091
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  • 3
    Electronic Resource
    Electronic Resource
    On the solution of concave knapsack problems (1990)
    Springer
    Mathematical programming 49 (1990), S. 397-411 
    Details
    ISSN: 1436-4646
    Keywords: Concave functions ; knapsack problems ; strict minimizers ; NP-hard ; nonconvex ; local minimizers
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract We consider a version of the knapsack problem which gives rise to a separable concave minimization problem subject to bounds on the variables and one equality constraint. We characterize strict local miniimizers of concave minimization problems subject to linear constraints, and use this characterization to show that although the problem of determining a global minimizer of the concave knapsack problem is NP-hard, it is possible to determine a local minimizer of this problem with at most O(n logn) operations and 1+[logn] evaluations of the function. If the function is quadratic this algorithm requires at most O(n logn) operations.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01588800
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  • 4
    Electronic Resource
    Electronic Resource
    Convergence properties of trust region methods for linear and convex constraints (1990)
    Springer
    Mathematical programming 47 (1990), S. 305-336 
    Details
    ISSN: 1436-4646
    Keywords: Trust region ; linear constraints ; convex constraints ; global convergence ; local convergence ; degeneracy ; rate of convergence ; identification of active constraints ; Newton's method ; sequential quadratic programming ; gradient projection
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract We develop a convergence theory for convex and linearly constrained trust region methods which only requires that the step between iterates produce a sufficient reduction in the trust region subproblem. Global convergence is established for general convex constraints while the local analysis is for linearly constrained problems. The main local result establishes that if the sequence converges to a nondegenerate stationary point then the active constraints at the solution are identified in a finite number of iterations. As a consequence of the identification properties, we develop rate of convergence results by assuming that the step is a truncated Newton method. Our development is mainly geometrical; this approach allows the development of a convergence theory without any linear independence assumptions.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01580867
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  • 5
    Electronic Resource
    Electronic Resource
    Classes of functions and feasibility conditions in nonlinear complementarity problems (1974)
    Springer
    Mathematical programming 6 (1974), S. 327-338 
    Details
    ISSN: 1436-4646
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract Given a mappingF from real Euclideann-space into itself, we investigate the connection between various known classes of functions and the nonlinear complementarity problem: Find anx * such thatFx * ⩾ 0 andx * ⩾ 0 are orthogonal. In particular, we study the extent to which the existence of au ⩾ 0 withFu ⩾ 0 (feasible point) implies the existence of a solution to the nonlinear complementarity problem, and extend, to nonlinear mappings, known results in the linear complementarity problem on P-matrices, diagonally dominant matrices with non-negative diagonal elements, matrices with off-diagonal non-positive entries, and positive semidefinite matrices.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01580248
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  • 6
    Electronic Resource
    Electronic Resource
    Impact of Partial Separability on Large-Scale Optimization (1997)
    Springer
    Computational optimization and applications 7 (1997), S. 27-40 
    Details
    ISSN: 1573-2894
    Keywords: large-scale optimization ; partial separability ; automatic differentiation
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Notes: Abstract ELSO is an environment for the solution oflarge-scale optimization problems. With ELSO the user is required to provide only code for the evaluation of a partially separable function. ELSO exploits the partialseparability structure of the function to computethe gradient efficiently using automatic differentiation.We demonstrate ELSO's efficiency by comparing thevarious options available in ELSO.Our conclusion is that the hybrid option in ELSOprovides performance comparable to the hand-coded option, while having the significantadvantage of not requiring a hand-coded gradient orthe sparsity pattern of the partially separable function.In our test problems, which have carefully coded gradients,the computing time for the hybrid AD option is within a factor of two of thehand-coded option.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1008628114432
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  • 7
    Electronic Resource
    Electronic Resource
    Distance Geometry Optimization for Protein Structures (1999)
    Springer
    Journal of global optimization 15 (1999), S. 219-234 
    Details
    ISSN: 1573-2916
    Keywords: Distance geometry ; Distance constraints ; Protein structures
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We study the performance of the dgsol code for the solution of distance geometry problems with lower and upper bounds on distance constraints. The dgsol code uses only a sparse set of distance constraints, while other algorithms tend to work with a dense set of constraints either by imposing additional bounds or by deducing bounds from the given bounds. Our computational results show that protein structures can be determined by solving a distance geometry problem with dgsol and that the approach based on dgsol is significantly more reliable and efficient than multi-starts with an optimization code.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1008380219900
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  • 8
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    Optimization in SciDAC applications (2007)
    Institute of Physics
    Details
    Publication Date: 2007-07-01
    Print ISSN: 1742-6588
    Electronic ISSN: 1742-6596
    Topics: Physics
    Published by Institute of Physics
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  • 9
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    On the Local and Superlinear Convergence of Quasi-Newton Methods (1973)
    Oxford University Press
    Details
    Publication Date: 1973-01-01
    Print ISSN: 0272-4960
    Electronic ISSN: 1464-3634
    Topics: Mathematics
    Published by Oxford University Press on behalf of Institute of Mathematics and its Applications.
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  • 10
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    Do you trust derivatives or differences? (2014)
    Elsevier
    Details
    Publication Date: 2014-09-01
    Print ISSN: 0021-9991
    Electronic ISSN: 1090-2716
    Topics: Computer Science , Physics
    Published by Elsevier
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