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  • 1
    Electronic Resource
    Electronic Resource
    On combined phase 1-phase 2 projective methods for linear programming (1993)
    Springer
    Algorithmica 9 (1993), S. 64-83 
    Details
    ISSN: 1432-0541
    Keywords: Linear programming ; Interior-point methods ; Projective methods ; Combined phase 1-phase 2
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract We compare the projective methods for linear programming due to de Ghellinck and Vial, Anstreicher, Todd, and Fraley. These algorithms have the feature that they approach feasibility and optimality simultaneously, rather than requiring an initial feasible point. We compare the directions used in these methods and the lower-bound updates employed. In many cases the directions coincide and two of the lower-bound updates give the same result. It appears that Todd's direction and Fraley's lower-bound update have slight advantages, and this is borne out in limited computational testing.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01185339
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  • 2
    Electronic Resource
    Electronic Resource
    Age-Related Changes in Human Smooth Pursuit Responses to Horizontal Step-Ramp Target Trajectories (1992)
    Oxford, UK : Blackwell Publishing Ltd
    Annals of the New York Academy of Sciences 656 (1992), S. 0 
    Details
    ISSN: 1749-6632
    Source: Blackwell Publishing Journal Backfiles 1879-2005
    Topics: Natural Sciences in General
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1111/j.1749-6632.1992.tb25264.x
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  • 3
    Electronic Resource
    Electronic Resource
    Combining phase I and phase II in a potential reduction algorithm for linear programming (1993)
    Springer
    Mathematical programming 59 (1993), S. 133-150 
    Details
    ISSN: 1436-4646
    Keywords: Linear programming ; interior-point methods ; combined phase I—phase II
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract This paper describes an affine potential reduction algorithm for linear programming that simultaneously seeks feasibility and optimality. The algorithm is closely related to a similar method of Anstreicher. The new features are that we use a two-dimensional programming problem to derive better lower bounds than Anstreicher, that our direction-finding subproblem treats phase I and phase II more symmetrically, and that we do not need an initial lower bound. Our method also allows for the generation of a feasible solution (so that phase I is terminated) during the course of the iterations, and we describe two ways to encourage this behavior.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01581241
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  • 4
    Electronic Resource
    Electronic Resource
    Interior-point algorithms for semi-infinite programming (1994)
    Springer
    Mathematical programming 65 (1994), S. 217-245 
    Details
    ISSN: 1436-4646
    Keywords: Linear programming ; Semi-infinite programming ; Interior-point methods
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract In order to study the behavior of interior-point methods on very large-scale linear programming problems, we consider the application of such methods to continuous semi-infinite linear programming problems in both primal and dual form. By considering different discretizations of such problems we are led to a certain invariance property for (finite-dimensional) interior-point methods. We find that while many methods are invariant, several, including all those with the currently best complexity bound, are not. We then devise natural extensions of invariant methods to the semi-infinite case. Our motivation comes from our belief that for a method to work well on large-scale linear programming problems, it should be effective on fine discretizations of a semi-infinite problem and it should have a natural extension to the limiting semi-infinite case.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01581697
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  • 5
    Electronic Resource
    Electronic Resource
    On Anstreicher's combined phase I—phase II projective algorithm for linear programming (1992)
    Springer
    Mathematical programming 55 (1992), S. 1-15 
    Details
    ISSN: 1436-4646
    Keywords: Linear programming ; interior-point method ; projective algorithm ; combining phase I–phase II
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract Anstreicher has proposed a variant of Karmarkar's projective algorithm that handles standard-form linear programming problems nicely. We suggest modifications to his method that we suspect will lead to better search directions and a more useful algorithm. Much of the analysis depends on a two-constraint linear programming problem that is a relaxation of the scaled original problem.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01581187
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  • 6
    Electronic Resource
    Electronic Resource
    Solving combinatorial optimization problems using Karmarkar's algorithm (1992)
    Springer
    Mathematical programming 56 (1992), S. 245-284 
    Details
    ISSN: 1436-4646
    Keywords: Combinatorial optimization ; integer programming ; Karmarkar's method
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract We describe a cutting plane algorithm for solving combinatorial optimization problems. The primal projective standard-form variant of Karmarkar's algorithm for linear programming is applied to the duals of a sequence of linear programming relaxations of the combinatorial optimization problem.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01580902
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  • 7
    Electronic Resource
    Electronic Resource
    On the complexity of approximating the maximal inscribed ellipsoid for a polytope (1993)
    Springer
    Mathematical programming 61 (1993), S. 137-159 
    Details
    ISSN: 1436-4646
    Keywords: Maximal inscribed ellipsoid ; maximal inscribed paraboloid ; path-following Newton's method ; computational complexity
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract We give a new polynomial bound on the complexity of approximating the maximal inscribed ellipsoid for a polytope.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01582144
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  • 8
    Electronic Resource
    Electronic Resource
    Containing and shrinking ellipsoids in the path-following algorithm (1990)
    Springer
    Mathematical programming 47 (1990), S. 1-9 
    Details
    ISSN: 1436-4646
    Keywords: Linear programming ; convex quadratic programming ; path-following algorithm ; Karmarkar's algorithm ; ellipsoid method
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract We describe a new potential function and a sequence of ellipsoids in the path-following algorithm for convex quadratic programming. Each ellipsoid in the sequence contains all of the optimal primal and dual slack vectors. Furthermore, the volumes of the ellipsoids shrink at the ratio $$2^{ - \Omega (\sqrt n )} $$ , in comparison to 2−Ω(1) in Karmarkar's algorithm and 2−Ω(1/n) in the ellipsoid method. We also show how to use these ellipsoids to identify the optimal basis in the course of the algorithm for linear programming.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01580848
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  • 9
    Electronic Resource
    Electronic Resource
    An O(n 3 L) adaptive path following algorithm for a linear complementarity problem (1991)
    Springer
    Mathematical programming 52 (1991), S. 587-595 
    Details
    ISSN: 1436-4646
    Keywords: Linear programming ; linear complementarity problem ; interior point algorithms ; path following algorithms
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract In this paper we propose an O(n 3 L) algorithm which is a modification of the path following algorithm [8] for a linear complementarity problem. The path following algorithm has to take a short step size in each iteration in order to bound the number of overall arithmetic operations by O(n 3 L). In practical computation, we can determine the step size adaptively. Mizuno, Yoshise, and Kikuchi [11] reported that such an adaptive algorithm required about O(L) iterations for some test problems. Here we show that we can use a rank one update technique in the adaptive algorithm so that the number of overall arithmetic operations is theoretically bounded by O(n 3 L).
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01582906
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  • 10
    Electronic Resource
    Electronic Resource
    Scaling, shifting and weighting in interior-point methods (1994)
    Springer
    Computational optimization and applications 3 (1994), S. 305-315 
    Details
    ISSN: 1573-2894
    Keywords: Linear programming ; interior-point methods
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Notes: Abstract We examine certain questions related to the choice of scaling, shifting and weighting strategies for interior-point methods for linear programming. One theme is the desire to make trajectories to be followed by algorithms into straight lines if possible to encourage fast convergence. While interior-point methods in general follow curves, this occurrence of straight lines seems appropriate to honor George Dantzig's contributions to linear programming, since his simplex method can be seen as following either a piecewise-linear path inn-space or a straight line inm-space (the simplex interpretation).
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01299206
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