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  • 1
    Electronic Resource
    Electronic Resource
    A new line-search step based on the Weierstrass $\wp$ -function for minimizing a class of logarithmic barrier functions (1994)
    Springer
    Numerische Mathematik 68 (1994), S. 81-94 
    Details
    ISSN: 0945-3245
    Keywords: Mathematics Subject Classification (1991): 65K05
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary. We define the notion of self-concordance of order two for the restriction $f$ of a logarithmic barrier function to a given line. Based on this notion we prove an inner approximation of the domain of $f$ , as well as a lower bound of the distance from a point $t$ to the minimum of $f$ . These results provide the theoretical tools to develop a simple and efficient search step for finding the minimum of the barrier function along a given line. The new bound on the size of the line-search step is better than the optimal bound known for the case of a self-concordant function (of order one). We conclude with some numerical examples that illustrate the promise of the new line-search step.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s002110050049
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    Paper (German National Licenses)
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  • 2
    Electronic Resource
    Electronic Resource
    The role of linear objective functions in barrier methods (1999)
    Springer
    Mathematical programming 84 (1999), S. 357-373 
    Details
    ISSN: 1436-4646
    Keywords: Key words: logarithmic barrier function – Newton’s method – superlinear convergence ; Mathematics Subject Classification (1991): 90C30, 65K05
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s101070050026
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    Paper (German National Licenses)
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  • 3
    Electronic Resource
    Electronic Resource
    Global and polynomial-time convergence of an infeasible-interior-point algorithm using inexact computation (1999)
    Springer
    Mathematical programming 84 (1999), S. 105-122 
    Details
    ISSN: 1436-4646
    Keywords: Key words: linear programming problem – interior-point method – primal-dual – convergence – inexact computation
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s10107980020a
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    Paper (German National Licenses)
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  • 4
    Electronic Resource
    Electronic Resource
    An interior-point method for fractional programs with convex constraints (1994)
    Springer
    Mathematical programming 67 (1994), S. 407-440 
    Details
    ISSN: 1436-4646
    Keywords: Interior-point method ; Fractional programs ; Convex constraints ; Polynomial-rate convergence
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract We present an interior-point method for a class of fractional programs with convex constraints. The proposed algorithm converges at a polynomial rate, similarly as in the case of a convex problem, even though fractional programs are only pseudo-convex. Here, the rate of convergence is measured in terms of the area of two-dimensional convex setsC k containing the origin and certain projections of the optimal points, and the area ofC k is reduced by a constant factorc 〈 1 at each iteration. The factorc depends only on the self-concordance parameter of a barrier function associated with the feasible set. We present an outline of a practical implementation of the proposed method, and we report results of some preliminary numerical experiments.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01582229
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  • 5
    Electronic Resource
    Electronic Resource
    Interior-point methods for convex programming (1992)
    Springer
    Applied mathematics & optimization 26 (1992), S. 287-311 
    Details
    ISSN: 1432-0606
    Keywords: Convex programming ; Ellipsoidal approximation ; Relative Lipschitz condition ; Self-concordance ; Primary 90C ; Secondary 65K
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This work is concerned with generalized convex programming problems, where the objective function and also the constraints belong to a certain class of convex functions. It examines the relationship of two basic conditions used in interior-point methods for generalized convex programming—self-concordance and a relative Lipschitz condition—and gives a short and simple complexity analysis of an interior-point method for generalized convex programming. In generalizing ellipsoidal approximations for the feasible set, it also allows a geometrical interpretation of the analysis.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01371086
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  • 6
    Electronic Resource
    Electronic Resource
    Optimal Design of Trusses Under a Nonconvex Global Buckling Constraint (2000)
    Springer
    Optimization and engineering 1 (2000), S. 189-213 
    Details
    ISSN: 1573-2924
    Keywords: truss design ; buckling ; nonconvex semidefinite programming
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Technology
    Notes: Abstract We propose a novel formulation of a truss design problem involving a constraint on the global stability of the structure due to the linear buckling phenomenon. The optimization problem is modelled as a nonconvex semidefinite programming problem. We propose two techniques for the numerical solution of the problem and apply them to a series of numerical examples.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1010091831812
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  • 7
    Electronic Resource
    Electronic Resource
    An Interior Method for Nonconvex Semidefinite Programs (2000)
    Springer
    Optimization and engineering 1 (2000), S. 347-372 
    Details
    ISSN: 1573-2924
    Keywords: nonlinear optimization ; semidefinite program ; predictor corrector method ; trust region method
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Technology
    Notes: Abstract In several applications, semidefinite programs arise in which the matrix depends nonlinearly on the unknown variables. We propose a new solution method for such semidefinite programs that also applies to other smooth nonconvex programs. The method is an extension of a primal predictor corrector interior method to nonconvex programs. The predictor steps are based on Dikin ellipsoids of a “convexified” domain. The corrector steps are based on quadratic subprograms that combine aspects of line search and trust region methods. Convergence results are given, and some preliminary numerical experiments suggest a high robustness of the proposed method.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1011562523132
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    Paper (German National Licenses)
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  • 8
    Electronic Resource
    Electronic Resource
    An infeasible-interior-point algorithm using projections onto a convex set (1996)
    Springer
    Annals of operations research 62 (1996), S. 59-80 
    Details
    ISSN: 1572-9338
    Keywords: Linear programming ; infeasible-interior-point method ; polynomiality ; projection ; 90C05
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Economics
    Notes: Abstract We present a new class of primal-dual infeasible-interior-point methods for solving linear programs. Unlike other infeasible-interior-point algorithms, the iterates generated by our methods lie in general position in the positive orthant of ℝ2 and are not restricted to some linear manifold. Our methods comprise the following features: At each step, a projection is used to “recenter” the variables to the domainx i s i ≥μ. The projections are separable into two-dimensional orthogonal projections on a convex set, and thus they are seasy to implement. The use of orthogonal projections allows that a full Newton step can be taken at each iteration, even if the result violates the nonnegativity condition. We prove that a short step version of our method converges in polynomial time.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02206811
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  • 9
    Book
    Book
    Optimierung / (2004)
    Berlin [u.a.] :Springer,
    Details
    Title: Optimierung /
    Author: Jarre, Florian
    Contributer: Stoer, Josef
    Publisher: Berlin [u.a.] :Springer,
    Year of publication: 2004
    Pages: XII, 475 S. : graph. Darst.
    Series Statement: Springer-Lehrbuch
    ISBN: 3-540-43575-1
    Type of Medium: Book
    Language: German
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    ZIB Catalog
    Zuse Institute Berlin get availability status
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