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  • 1
    Electronic Resource
    Electronic Resource
    On a monotone Newton-like method (1987)
    Springer
    Computing 39 (1987), S. 233-246 
    Details
    ISSN: 1436-5057
    Keywords: 65H10 ; Systems of nonlinear equations ; Newton-like methods ; monotone convergence
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Description / Table of Contents: Zusammenfassung Wir stellen ein Iterationsverfahren zur Auflösung nichtlinearer Gleichungssysteme vor welches für eine große Klasse von Problemen monoton und überlinear konvergiert. Zur Konstruktion des Verfahrens werden feste Schranken für die zweite Ableitung verwendet. Für konvexe Abbildungen geht das Verfahren in das Newton-Fourier-Verfahren über.
    Notes: Abstract We present an iterative algorithm for solving nonlinear systems of equations which is monotonically and superlinearly convergent for a large class of problems. In the construction of the algorithm we use some fixed bounds of the second derivatives. For convex mappings the algorithm reduces to the Newton-Fourier method.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02309557
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    Paper (German National Licenses)
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  • 2
    Electronic Resource
    Electronic Resource
    Interior-point algorithm for quadratically constrained entropy minimization problems (1993)
    Springer
    Journal of optimization theory and applications 77 (1993), S. 79-95 
    Details
    ISSN: 1573-2878
    Keywords: Entropy minimization ; quadratic constraints ; analytic center ; global linear convergence
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In this paper, we develop an interior point algorithm for quadratically constrained entropy problems. The algorithm uses a variation of Newton's method to follow a central path trajectory in the interior of the feasible set. The primal-dual gap is made less than a given ɛ in at most $$O(|\log \in |\sqrt {m + m} )$$ steps, wheren is the dimension of the problem andm is the number of quadratic inequality constraints.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00940780
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  • 3
    Electronic Resource
    Electronic Resource
    Interior-point methods for nonlinear complementarity problems (1996)
    Springer
    Journal of optimization theory and applications 88 (1996), S. 617-642 
    Details
    ISSN: 1573-2878
    Keywords: Complementarity problems ; interior-point methods ; convergence rates
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We present a potential reduction interior-point algorithm for monotone nonlinear complementarity problems. At each iteration, one has to compute an approximate solution of a nonlinear system such that a certain accuracy requirement is satisfied. For problems satisfying a scaled Lipschitz condition, this requirement is satisfied by the approximate solution obtained by applying one Newton step to that nonlinear system. We discuss the global and local convergence rates of the algorithm, convergence toward a maximal complementarity solution, a criterion for switching from the interior-point algorithm to a pure Newton method, and the complexity of the resulting hybrid algorithm.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02192201
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  • 4
    Electronic Resource
    Electronic Resource
    Superlinear Convergence of Interior-Point Algorithms for Semidefinite Programming (1998)
    Springer
    Journal of optimization theory and applications 99 (1998), S. 103-119 
    Details
    ISSN: 1573-2878
    Keywords: Semidefinite programming ; path-following algorithms ; infeasible interior-point algorithms ; polynomiality ; superlinear convergence
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We prove the superlinear convergence of the primal-dual infeasible interior-point path-following algorithm proposed recently by Kojima, Shida, and Shindoh and by the present authors, under two conditions: (i) the semidefinite programming problem has a strictly complementary solution; (ii) the size of the central path neighborhood approaches zero. The nondegeneracy condition suggested by Kojima, Shida, and Shindoh is not used in our analysis. Our result implies that the modified algorithm of Kojima, Shida, and Shindoh, which enforces condition (ii) by using additional corrector steps, has superlinear convergence under the standard assumption of strict complementarity. Finally, we point out that condition (ii) can be made weaker and show the superlinear convergence under the strict complementarity assumption and a weaker condition than (ii).
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1021700210959
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  • 5
    Electronic Resource
    Electronic Resource
    Predictor-corrector method for linear complementarity problems with polynomial complexity and superlinear convergence (1995)
    Springer
    Journal of optimization theory and applications 85 (1995), S. 187-199 
    Details
    ISSN: 1573-2878
    Keywords: Linear complementarity problems ; predictor-corrector method ; interior-point algorithms ; polynomiality ; superlinear convergence
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We extend the Mizuno-Todd-Ye predictor-corrector algorithm for solving monotone linear complementarity problems. We prove that the extended algorithm is globally Q-linearly convergent and solves problems with integer data of bitlengthL in at most $$O(\sqrt n L)$$ iterations. We also prove that the duality gap converges to zero Q-superlinearly for problems having strictly complementary solutions. Our results generalize the results obtained by Ye, Tapia, and Zhang for linear programming.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02192304
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  • 6
    Electronic Resource
    Electronic Resource
    Efficient line search algorithm for unconstrained optimization (1995)
    Springer
    Journal of optimization theory and applications 85 (1995), S. 677-704 
    Details
    ISSN: 1573-2878
    Keywords: Unconstrained optimization ; line search algorithms
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A new line search algorithm for smooth unconstrained optimization is presented that requires only one gradient evaluation with an inaccurate line search and at most two gradient evaluations with an accurate line search. It terminates in finitely many operations and shares the same theoretical properties as the standard line search rules like the Armijo-Goldstein-Wolfe-Powell rules. This algorithm is especially appropriate for the situation when gradient evaluations are very expensive relative to function evaluations.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02193062
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    Paper (German National Licenses)
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  • 7
    Electronic Resource
    Electronic Resource
    Superlinearly Convergent Infeasible-Interior-Point Algorithm for Degenerate LCP (1998)
    Springer
    Journal of optimization theory and applications 97 (1998), S. 249-269 
    Details
    ISSN: 1573-2878
    Keywords: Linear complementarity problems ; sufficient matrices ; P *-matrices ; path-following algorithm ; infeasible-interior-point algorithm ; polynomiality ; superlinear convergence
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A large-step infeasible path-following method is proposed for solving general linear complementarity problems with sufficient matrices. If the problem has a solution, the algorithm is superlinearly convergent from any positive starting points, even for degenerate problems. The algorithm generates points in a large neighborhood of the central path. Each iteration requires only one matrix factorization and at most three (asymptotically only two) backsolves. It has been recently proved that any sufficient matrix is a P *(κ)-matrix for some κ≥0. The computational complexity of the algorithm depends on κ as well as on a feasibility measure of the starting point. If the starting point is feasible or close to being feasible, then the iteration complexity is $$O((1 + {\kappa)}\sqrt {nL})$$ . Otherwise, for arbitrary positive and large enough starting points, the iteration complexity is O((1 + κ)2 nL). We note that, while computational complexity depends on κ, the algorithm itself does not.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1022670415661
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  • 8
    Electronic Resource
    Electronic Resource
    Formulating Dynamic Multi-Rigid-Body Contact Problems with Friction as Solvable Linear Complementarity Problems (1997)
    Springer
    Nonlinear dynamics 14 (1997), S. 231-247 
    Details
    ISSN: 1573-269X
    Keywords: Impacts with friction ; multi-body dynamics ; complementarity problems ; contact constraints
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A linear complementarity formulation for dynamic multi-rigid-body contact problems with Coulomb friction is presented. The formulation, based on explicit Euler integration and polygonal approximation of the friction cone, is guaranteed to have a solution for any number of contacts and contact configuration. A model with the same property, based on the Poisson hypothesis, is formulated for impact problems with friction and nonzero restitution coefficients. An explicit Euler scheme based on these formulations is presented and is proved to have uniformly bounded velocities as the stepsize tends to zero for the Newton–Euler formulation in body co-ordinates.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1008292328909
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  • 9
    Electronic Resource
    Electronic Resource
    OnQ-order andR-order of convergence (1989)
    Springer
    Journal of optimization theory and applications 63 (1989), S. 415-431 
    Details
    ISSN: 1573-2878
    Keywords: Nonlinear iterations ; order of convergence
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We give sufficient conditions for a sequence to have theQ-order and/or theR-order of convergence greater than one. If an additional condition is satisfied, then the sequence has an exactQ-order of convergence. We show that our results are sharp and we compare them with older results.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00939805
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  • 10
    Book
    Book
    Nondiscrete induction and iteractive processes; 103 (1984)
    London u.a. :Pitman,
    Details
    Title: Nondiscrete induction and iteractive processes; 103
    Author: Potra, F-A
    Contributer: Ptak, V
    Publisher: London u.a. :Pitman,
    Year of publication: 1984
    Pages: 207 S.
    Series Statement: Research notes in applied mathematics 103
    Type of Medium: Book
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    ZIB Catalog
    Zuse Institute Berlin get availability status
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