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  • Articles  (1,591)
  • Springer  (1,395)
  • Wiley  (196)
  • 1975-1979  (1,591)
  • 1950-1954
  • 1978  (817)
  • 1977  (774)
  • Computer Science  (1,591)
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  • Articles  (1,591)
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  • 1975-1979  (1,591)
  • 1950-1954
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  • 1
    Electronic Resource
    Electronic Resource
    Bilinear programming: An exact algorithm (1977)
    Springer
    Mathematical programming 12 (1977), S. 173-194 
    Details
    ISSN: 1436-4646
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract The Bilinear Programming Problem is a structured quadratic programming problem whose objective function is, in general, neither convex nor concave. Making use of the formal linearity of a dual formulation of the problem, we give a necessary and sufficient condition for optimality, and an algorithm to find an optimal solution.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01593787
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  • 2
    Electronic Resource
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    On the existence and uniqueness of solutions in nonlinear complementarity theory (1977)
    Springer
    Mathematical programming 12 (1977), S. 110-130 
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    ISSN: 1436-4646
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract A complementarity problem is said to be globally uniquely solvable (GUS) if it has a unique solution, and this property will not change, even if any constant term is added to the mapping generating the problem. A characterization of the GUS property which generalizes a basic theorem in linear complementarity theory is given. Known sufficient conditions given by Cottle, Karamardian, and Moré for the nonlinear case are also shown to be generalized. In particular, several open questions concerning Cottle's condition are settled and a new proof is given for the sufficiency of this condition. A simple characterization for the two-dimensional case and a necessary condition for then-dimensional case are also given.
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    URL: http://dx.doi.org/10.1007/BF01593774
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  • 3
    Electronic Resource
    Electronic Resource
    Quadratic termination properties of Davidon's new variable metric algorithm (1977)
    Springer
    Mathematical programming 12 (1977), S. 141-147 
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    ISSN: 1436-4646
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract Davidon [1] provides a variable metric algorithm for minimization calculations that achieves quadratic termination without any line searches by a method that uses positive definite second derivative approximations. We give a new and elementary proof of these termination properties. It shows the great freedom allowed by the algorithm in the choice of search directions and it explains the effect of a restart procedure.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01593779
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  • 4
    Electronic Resource
    Electronic Resource
    The Mathematical Programming Society (1977)
    Springer
    Mathematical programming 12 (1977), S. 155-155 
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    ISSN: 1436-4646
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01593784
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  • 5
    Electronic Resource
    Electronic Resource
    The zero pivot phenomenon in transportation and assignment problems and its computational implications (1977)
    Springer
    Mathematical programming 12 (1977), S. 226-240 
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    ISSN: 1436-4646
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract Transportation and assignment models have been widely used in many applications. Their use was motivated, among other reasons, by the existence of efficient solution methods and their occurrence as sub-problems in the solution of combinatorial problems. A previous study [10] observed that, in large-scale Transportation and Assignment Problems, 95 percent of the pivots were zero or degenerate pivots. This study investigates the ratio of zero pivots to the total number of pivots and verifies the above observation under conditions of small rim variability. Rules are introduced that pay special attention to the zero pivot phenomenon, and significantly reduce CPU time in both phase-1 (generating the initial basic feasible solution) and in phase-2 (selecting the variable leaving the base and the variable entering the base). When these rules were applied, they reduced the CPU time substantially: a 500×500 assignment problem was solved in 1.3 seconds.
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    URL: http://dx.doi.org/10.1007/BF01593789
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  • 6
    Electronic Resource
    Electronic Resource
    Matrix factorizations in optimization of nonlinear functions subject to linear constraints — an addendum (1977)
    Springer
    Mathematical programming 12 (1977), S. 279-280 
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    ISSN: 1436-4646
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01593793
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  • 7
    Electronic Resource
    Electronic Resource
    The use of augmented lagrangian functions for sensitivity analysis in nonlinear programming (1977)
    Springer
    Mathematical programming 12 (1977), S. 281-284 
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    ISSN: 1436-4646
    Keywords: Nonlinear programming ; Augmented Lagrangian functions ; Sensitivity analysis
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01593794
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  • 8
    Electronic Resource
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    Constructive proofs of theorems relating to:F(x) = y, with applications (1977)
    Springer
    Mathematical programming 12 (1977), S. 328-343 
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    ISSN: 1436-4646
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract Some theorems are given which relate to approximating and establishing the existence of solutions to systemsF(x) = y ofn equations inn unknowns, for variousy, in a region of euclideann-space E n . They generalize known theorems. Viewing complementarity problems and fixed-point problems as examples, known results or generalizations of known results are obtained. A familiar use is made of homotopies H: E n × [0, 1]→E n of the formH(x, t) = (1 −t)F 0 (x) + t[F(x) − y] where theF 0 in this paper is taken to be linear. Simplicial subdivisionsT k of E n × [0, 1] furnish piecewise linear approximatesG k toH. The basic computation is via the generation of piecewise linear curvesP k which satisfyG k (x, t) = 0. Visualizing a sequence {T k } of such subdivisions, with mesh size going to zero, arguments are made on connected, compact limiting curvesP on whichH(x, t) = 0. This paper builds upon and continues recent work of C.B. Garcia.
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    URL: http://dx.doi.org/10.1007/BF01593801
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  • 9
    Electronic Resource
    Electronic Resource
    A practicable steepest-edge simplex algorithm (1977)
    Springer
    Mathematical programming 12 (1977), S. 361-371 
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    ISSN: 1436-4646
    Keywords: Simplex Method ; Linear Programming ; Steepest-edge ; LU Factorization
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract It is shown that suitable recurrences may be used in order to implement in practice the steepest-edge simplex linear programming algorithm. In this algorithm each iteration is along an edge of the polytope of feasible solutions on which the objective function decreases most rapidly with respect to distance in the space of all the variables. Results of computer comparisons on medium-scale problems indicate that the resulting algorithm requires less iterations but about the same overall time as the algorithm of Harris [8], which may be regarded as approximating the steepest-edge algorithm. Both show a worthwhile advantage over the standard algorithm.
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    URL: http://dx.doi.org/10.1007/BF01593804
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  • 10
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    The Mathematical Programming Society (1977)
    Springer
    Mathematical programming 12 (1977), S. 423-423 
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    ISSN: 1436-4646
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01593809
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  • 11
    Electronic Resource
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    On monotonicity in parametric linear complementarity problems (1977)
    Springer
    Mathematical programming 12 (1977), S. 60-66 
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    ISSN: 1436-4646
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract This paper generalizes the answers that were given by R.W. Cottle to questions that were originally raised by G. Maier. Essentially, we give necessary and sufficient conditions for some notions of monotonicity of solutions for the parametric linear complementarity problem. Our proofs are direct ones and not algorithmic, as Cottle's proofs are, and also cover a broader class of matrices.
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    URL: http://dx.doi.org/10.1007/BF01593769
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  • 12
    Electronic Resource
    Electronic Resource
    Progressive global random search of continuous functions (1978)
    Springer
    Mathematical programming 15 (1978), S. 330-342 
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    ISSN: 1436-4646
    Keywords: Global Optimization ; Random Search ; Convergence ; Sequential Minimization ; Lipschitz Functions ; Stochastic Programming
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract A sequential random search method for the global minimization of a continuous function is proposed. The algorithm gradually concentrates the random search effort on areas neighboring the global minima. A modification is included for the case that the function cannot be exactly evaluated. The global convergence and the asymptotical optimality of the sequential sampling procedure are proved for both the stochastic and deterministic optimization problem.
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    URL: http://dx.doi.org/10.1007/BF01609037
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  • 13
    Electronic Resource
    Electronic Resource
    Tenth International Symposium on Mathematical Programming (1978)
    Springer
    Mathematical programming 15 (1978), S. 364-364 
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    ISSN: 1436-4646
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
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    URL: http://dx.doi.org/10.1007/BF01609044
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  • 14
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    Least distance methods for the scheme of polytopes (1978)
    Springer
    Mathematical programming 15 (1978), S. 1-11 
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    ISSN: 1436-4646
    Keywords: APL-Codes ; Polytope ; Point Set ; Vertices ; Faces ; Facets ; Quadratic Programming ; Distance ; Decomposition
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract Methods are described and APL-codes are supplied to find vertices, edges, other faces and facets of polytopes given by point sets. The basic subroutine is a simplicial decomposition version of least distance, i.e. quadratic, programming. Computational experience indicates high efficiency.
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    URL: http://dx.doi.org/10.1007/BF01608995
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  • 15
    Electronic Resource
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    An algorithm for determining all extreme points of a convex polytope (1977)
    Springer
    Mathematical programming 12 (1977), S. 81-96 
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    ISSN: 1436-4646
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract An algorithm for determining all the extreme points of a convex polytope associated with a set of linear constraints, via the computation of basic feasible solutions to the constraints, is presented. The algorithm is based on the product-form revised simplex method and as such can be readily linked onto standard linear programming codes. Applications of such an algorithm are reviewed and limited computational experience given.
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    URL: http://dx.doi.org/10.1007/BF01593771
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  • 16
    Electronic Resource
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    A monotone complementarity problem with feasible solutions but no complementary solutions (1977)
    Springer
    Mathematical programming 12 (1977), S. 131-132 
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    ISSN: 1436-4646
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
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    URL: http://dx.doi.org/10.1007/BF01593775
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  • 17
    Electronic Resource
    Electronic Resource
    On some properties of L1 estimators (1977)
    Springer
    Mathematical programming 12 (1977), S. 139-140 
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    ISSN: 1436-4646
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
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    URL: http://dx.doi.org/10.1007/BF01593778
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  • 18
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    The Committee on Algorithms of the mathematical programming society: (1977)
    Springer
    Mathematical programming 12 (1977), S. 151-153 
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    ISSN: 1436-4646
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
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    URL: http://dx.doi.org/10.1007/BF01593782
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  • 19
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    The complementary unboundedness of dual feasible solution sets in convex programming (1977)
    Springer
    Mathematical programming 12 (1977), S. 392-405 
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    ISSN: 1436-4646
    Keywords: Complementary Unboundedness ; Dual Feasible Solution Sets ; Convex Programming ; Geometric Programming ; Fenchel Duality ; Rockafellar Duality ; Ordinary Duality ; Quadratic Programming ; Optimal Location ; Traffic Equilibria
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract F.E. Clark has shown that if at least one of the feasible solution sets for a pair of dual linear programming problems is nonempty then at least one of them is both nonempty and unbounded. Subsequently, M. Avriel and A.C. Williams have obtained the same result in the more general context of (prototype posynomial) geometric programming. In this paper we show that the same result is actually false in the even more general context of convex programming — unless a certain regularity condition is satisfied. We also show that the regularity condition is so weak that it is automatically satisfied in linear programming (prototype posynomial) geometric programming, quadratic programming (with either linear or quadratic constraints),l p -regression analysis, optimal location, roadway network analysis, and chemical equilibrium analysis. Moreover, we develop an equivalent regularity condition for each of the usual formulations of duality.
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    URL: http://dx.doi.org/10.1007/BF01593806
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  • 20
    Electronic Resource
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    Finite horizon approximations of infinite horizon linear programs (1977)
    Springer
    Mathematical programming 12 (1977), S. 1-17 
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    ISSN: 1436-4646
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract This paper describes the class of infinite horizon linear programs that have finite optimal values. A sequence of finite horizon (T period) problems is shown to approximate the infinite horizon problems in the following sense: the optimal values of theT period problems converge monotonically to the optimal value of the infinite problem and the limit of any convergent subsequence of initialT period optimal decisions is an optimal decision for the infinite horizon problem.
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    URL: http://dx.doi.org/10.1007/BF01593765
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  • 21
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    Isotone solutions of parametric linear complementarity problems (1977)
    Springer
    Mathematical programming 12 (1977), S. 48-59 
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    ISSN: 1436-4646
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract The parametric linear complementarity problem is given by the conditions:q + αp + Mz ⩾ 0,α ⩾ 0,z ⩾ 0,z T (q + αp + Mz) = 0. Under the assumption thatM is a P-matrix, Cottle proved that the solution mapz(α) of the above problem is montonically nondecreasing in the parameterα for every nonnegativeq and everyp if and only ifM is a Minkowski matrix. This paper examines whether a similar result holds in various other settings including a nonlinear case.
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    URL: http://dx.doi.org/10.1007/BF01593768
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  • 22
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    Optimality condition: Constraint regularization (1977)
    Springer
    Mathematical programming 13 (1977), S. 38-48 
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    ISSN: 1436-4646
    Keywords: Constraint regularization ; Determinant cone ; Kuhn—Tucker constraint-qualification ; Lagrange regularity ; Optimality condition ; x 0-redundant constraints
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract Necessity and sufficiency conditions are given for the existence of a finite set of redundant constraints which together with a given set of constraints in an optimization problem satisfy the Kuhn—Tucker, and the Guignard Constraint qualifications when the original set of constraints fails to satisfy either or both of them.
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    URL: http://dx.doi.org/10.1007/BF01584322
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  • 23
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    Ergodicity and symmetric mathematical programs (1977)
    Springer
    Mathematical programming 13 (1977), S. 81-87 
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    ISSN: 1436-4646
    Keywords: Symmetric mathematical programs ; Ergodicity ; S-concavity and majorization ; Stochastic matrices ; Cyclic symmetry
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract The paper provides new conditions ensuring the optimality of a symmetric feasible point of certain mathematical programs. It is shown that these conditions generalize and unify most of the known results dealing with optimality of symmetric policies (e.g. [2, 4, 6, 11]). The generalization is based on certain ergodic properties of nonnegative matrices. An application to a socio-economic model dealing with optimization of a welfare function is presented.
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    URL: http://dx.doi.org/10.1007/BF01584325
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  • 24
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    Editorial policy (1978)
    Springer
    Mathematical programming 14 (1978), S. 127-127 
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    ISSN: 1436-4646
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
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    URL: http://dx.doi.org/10.1007/BF01588960
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  • 25
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    Algorithms for nonlinear constraints that use lagrangian functions (1978)
    Springer
    Mathematical programming 14 (1978), S. 224-248 
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    ISSN: 1436-4646
    Keywords: Augmented Lagrangian ; Lagrangian Function ; Nonlinear Constraints ; Nonlinear Programming ; Optimization Algorithm
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract Lagrangian functions are the basis of many of the more successful methods for nonlinear constraints in optimization calculations. Sometimes they are used in conjunction with linear approximations to the constraints and sometimes penalty terms are included to allow the use of algorithms for unconstrained optimization. Much has been discovered about these techniques during the last eight years and this paper gives a view of the progress and understanding that has been achieved and its relevance to practical algorithms. A particular method is recommended that seems to be more powerful than the author believed to be possible at the beginning of 1976.
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    URL: http://dx.doi.org/10.1007/BF01588967
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  • 26
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    The Mathematical Programming Society (1978)
    Springer
    Mathematical programming 14 (1978), S. 262-262 
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    ISSN: 1436-4646
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
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    URL: http://dx.doi.org/10.1007/BF01588969
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  • 27
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    The reformulation of two mixed integer programming problems (1978)
    Springer
    Mathematical programming 14 (1978), S. 325-331 
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    ISSN: 1436-4646
    Keywords: Integer Programming ; Formulation ; Models
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract Two practical problems are described, each of which can be formulated in more than one way as a mixed integer programming problem. The computational experience with two formulations of each problem is given. It is pointed out how in each case a reformulation results in the associated linear programming problem being more constrained. As a result the reformulated mixed integer problem is easier to solve. The problems are a multi-period blending problem and a mining investment problem.
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  • 28
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    Numerically stable methods for quadratic programming (1978)
    Springer
    Mathematical programming 14 (1978), S. 349-372 
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    ISSN: 1436-4646
    Keywords: Indefinite Quadratic Programming ; Matrix Factorizations ; Numerical Software
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract Numerically stable algorithms for quadratic programming are discussed. A new algorithm is described for indefinite quadratic programming which utilizes methods for updating positivedefinite factorizations only. Consequently all the updating procedures required are common to algorithms for linearly-constrained optimization. The new algorithm can be used for the positive-definite case without loss of efficiency.
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    URL: http://dx.doi.org/10.1007/BF01588976
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  • 29
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    A mathematical programming system for preference and compatibility maximized menu planning and scheduling (1978)
    Springer
    Mathematical programming 15 (1978), S. 63-76 
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    ISSN: 1436-4646
    Keywords: Menu Planning ; Separation Scheduling ; Menu Scheduling ; Decomposition ; Non-linear Programming ; Binary Knapsack Problem ; Lagrangian Relaxation ; Transportation Problem ; Branch and Bound
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract In this paper, the analytical representation of food preference is used in a separable non-linear program to yield the serving frequencies of menu items for a finite time horizon. The frequencies obtained in this way insure cost and nutritional control. Subsequently, the scheduling problem dealing with item assignments to meals and days is formulated as an integer program consisting of several transportation problems linked by weekly nutritional constraints. This problem is solved using a branch and bound algorithm which employs Lagrangian relaxation to obtain bounds and to decide on branching strategy.
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    URL: http://dx.doi.org/10.1007/BF01609000
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  • 30
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    On equivalent representations of certain multicommodity networks as single commodity flow problems (1978)
    Springer
    Mathematical programming 15 (1978), S. 92-99 
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    ISSN: 1436-4646
    Keywords: Network Flows ; Multicommodity Networks
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract Several classes of multicommodity networks have been shown to have the property that they can be transformed to equivalent uncapacitated single commodity flow problems. We show that many of these networks can be further reduced to smaller, semi-capacitated flow problems using the inverse of a result of Ford and Fulkerson. This appears to be a useful computationally-oriented tool for developing practically efficient algorithms. These concepts are also used to establish a generalization of a previous result concerning multicommodity transportation problems.
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    URL: http://dx.doi.org/10.1007/BF01609003
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  • 31
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    Formation of the international association for statistical computing (1978)
    Springer
    Mathematical programming 15 (1978), S. 119-120 
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    Source: Springer Online Journal Archives 1860-2000
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    URL: http://dx.doi.org/10.1007/BF01609008
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  • 32
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    A parametric linear complementarity problem involving derivatives (1978)
    Springer
    Mathematical programming 15 (1978), S. 146-154 
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    ISSN: 1436-4646
    Keywords: Linear Complementarity Problem ; Principal Pivoting Algorithm
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract The problem considered in this paper is given by the conditions:w = q + tp + Mz, w ≥ 0,ż ≥ 0,w T ż = 0, where a dot denotes the derivative with respect to the scalar parametert ≥ 0. In this problem,q, p aren-vectors withq ≥ 0 andM is an byn P-matrix. This problem arises in a certain basic problem in the field of structural mechanics. The main result in this paper is the existence and uniqueness theorem of a solution to this problem. The existence proof is constructive providing a computational method of obtaining the solution asymptotically.
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    URL: http://dx.doi.org/10.1007/BF01609013
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  • 33
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    A subgradient algorithm for certain minimax and minisum problems (1978)
    Springer
    Mathematical programming 15 (1978), S. 130-145 
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    ISSN: 1436-4646
    Keywords: Minimax Problems ; Minisum Problems ; Nondifferentiable Optimization ; Subgradients ; Location Problems ; Linear Approximation Problems
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract We present a subgradient algorithm for minimizing the maximum of a finite collection of functions. It is assumed that each function is the sum of a finite collection of basic convex functions and that the number of different subgradient sets associated with nondifferentiable points of each basic function is finite on any bounded set. Problems belonging to this class include the linear approximation problem and both the minimax and minisum problems of location theory. Convergence of the algorithm to an epsilon-optimal solution is proven and its effectiveness is demonstrated by solving a number of location problems and linear approximation problems.
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    Restart procedures for the conjugate gradient method (1977)
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    Mathematical programming 12 (1977), S. 241-254 
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    Notes: Abstract The conjugate gradient method is particularly useful for minimizing functions of very many variables because it does not require the storage of any matrices. However the rate of convergence of the algorithm is only linear unless the iterative procedure is “restarted” occasionally. At present it is usual to restart everyn or (n + 1) iterations, wheren is the number of variables, but it is known that the frequency of restarts should depend on the objective function. Therefore the main purpose of this paper is to provide an algorithm with a restart procedure that takes account of the objective function automatically. Another purpose is to study a multiplying factor that occurs in the definition of the search direction of each iteration. Various expressions for this factor have been proposed and often it does not matter which one is used. However now some reasons are given in favour of one of these expressions. Several numerical examples are reported in support of the conclusions of this paper.
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    Line perfect graphs (1977)
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    Mathematical programming 12 (1977), S. 255-259 
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    Notes: Abstract The concept of line perfection of a graph is defined so that a simple graph is line perfect if and only if its line graph is perfect in the usual sense. Line perfect graphs are characterized as those which contain no odd cycles of size larger than 3. Two well-know theorems of König for bipartite graphs are shown to hold also for line perfect graphs; this extension provides a reinterpretation of the content of these theorems.
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    Testing for complete efficiency in a vector maximization problem (1977)
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    Mathematical programming 12 (1977), S. 285-288 
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    Keywords: Vector maximization ; Efficient points
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    An algebraic approach to assignment problems (1977)
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    Mathematical programming 12 (1977), S. 318-327 
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    Keywords: Combinatorial Optimization ; Assignment Problem ; Ordered Semigroups ; Bottleneck Objective ; Lexicographical Objective ; Labeling Method ; Admissible Transformation
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    Notes: Abstract For assignment problems a class of objective functions is studied by algebraic methods and characterized in terms of an axiomatic system. It says essentially that the coefficients of the objective function can be chosen from a totally ordered commutative semigroup, which obeys a divisibility axiom. Special cases of the general model are the linear assignment problem, the linear bottleneck problem, lexicographic multicriteria problems,p-norm assignment problems and others. Further a polynomial bounded algorithm for solving this generalized assignment problem is stated. The algebraic approach can be extended to a broader class of combinatorial optimization problems.
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    Examples of non-convergence of conjugate descent algorithms with exact line-searches (1977)
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    Mathematical programming 12 (1977), S. 356-360 
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    Keywords: Conjugate ; Convergence ; Exact Line-search ; Optimization
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    Notes: Abstract Examples are given in which quasi-Newton and other conjugate descent algorithms fail to converge to a minimum of the object function. In the first there is convergence to a point where the gradient is infinite; in the second, a region characterized by a fine terraced structure causes the iterates to spiral indefinitely. A modification of the second construction gives convergence to a point where the gradient is non-zero, but the gradient is not continuous at this point. The implication of these examples is discussed.
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    Mathematical programming study 5 and 6 (1977)
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    Mathematical programming 12 (1977), S. 415-422 
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    Using shortest paths in some transshipment problems with concave costs (1977)
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    Mathematical programming 12 (1977), S. 18-25 
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    Notes: Abstract This paper deals with the problem of solving an uncapacitated transshipment problem with either one source and several sinks or one sink and several sources. The cost function of the problem is concave in the amount shipped on each arc and thus local optima are possible. A characterization of adjacent extreme flows in terms of corresponding arborescences is given for this type of networks. This characterization together with shortest path methods is then used to attack the problems of finding local optima and of ranking extreme points. A real-world problem and computational evidence for the usefulness of the method are produced.
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    On optimality conditions in nondifferentiable programming (1978)
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    Mathematical programming 14 (1978), S. 73-86 
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    Keywords: Gradient ; Subdifferential ; Generalized Gradient ; Optimality Conditions ; Nondifferentiability
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    Notes: Abstract This paper is devoted to necessary optimality conditions in a mathematical programming problem without differentiability or convexity assumptions on the data. The main tool of this study is the concept of generalized gradient of a locally Lipschitz function (and more generally of a lower semi-continuous function). In the first part, we consider local extremization problems in the unconstrained case for objective functions taking values in (−∞, +∞]. In the second part, the constrained case is considered by the way of the cone of adherent displacements. In the presence of inequality constraints, we derive in the third part optimality conditions in the Kuhn—Tucker form under a constraint qualification.
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    A quadratic network optimization model for equilibrium single commodity trade flows (1978)
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    Mathematical programming 14 (1978), S. 98-107 
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    Keywords: Network Flows ; Equilibrium Trade ; Quadratic Programming
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    Notes: Abstract When supply and demand curves for a single commodity are approximately linear in each ofN regions and interregional transportation costs are linear, then equilibrium trade flows can be computed by solving a quadratic program of special structure. An equilibrium trade flow exists in which the routes carrying positive flow form a forest, and this solution can be efficiently computed by a tree growing algorithm.
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    Referees 1977 (1978)
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    Mathematical programming 14 (1978), S. 125-125 
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    The Mathematical Programming Society (1978)
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    Mathematical programming 14 (1978), S. 126-126 
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    Second order characterizations of pseudoconvex functions (1978)
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    Mathematical programming 14 (1978), S. 170-185 
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    Keywords: Nonlinear Programming ; Pseudoconvexity ; Second order Characterizations ; Extended Hessians ; Bordered Determinants ; Quadratic Functions
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    Notes: Abstract Second order characterizations for (strictly) pseudoconvex functions are derived in terms of extended Hessians and bordered determinants. Additional results are presented for quadratic functions.
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    Editorial policy (1978)
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    Mathematical programming 14 (1978), S. 263-263 
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    Symposium on constructive approaches to Mathematical Models (1978)
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    Mathematical programming 14 (1978), S. 378-378 
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    International Symposium on systems optimization and analysis (1978)
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    Mathematical programming 14 (1978), S. 380-380 
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    The Mathematical Programming Society (1978)
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    Mathematical programming 14 (1978), S. 381-381 
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    Convex optimization and lagrange multipliers (1978)
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    Mathematical programming 15 (1978), S. 87-91 
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    Keywords: Lagrange Multipliers ; Constrained Convex Optimization ; Kuhn—Tucker Theorem ; Duality
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    Notes: Abstract The duality theorem of linear programming is used to prove several results on convex optimization. This is done without using separating hyerplane theorems.
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    A note on a theorem of Yamamuro (1978)
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    Mathematical programming 15 (1978), S. 110-113 
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    Keywords: Fixed Point Computation
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    Notes: Abstract Conditions are presented which are necessary for the existence of a regular fixed point of aC 1 map.
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    Editorial policy (1978)
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    Mathematical programming 15 (1978), S. 122-122 
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    A generalized knapsack problem with variable coefficients (1978)
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    Mathematical programming 15 (1978), S. 162-176 
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    Keywords: Mixed Integer Programming ; Knapsack Problem ; Branch and Bound Method
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    Notes: Abstract The ordinary knapsack problem is to find the optimal combination of items to be packed in a knapsack under a single constraint on the total allowable resources, where all coefficients in the objective function and in the constraint are constant. In this paper, a generalized knapsack problem with coefficients depending on variable parameters is proposed and discussed. Developing an effective branch and bound algorithm for this problem, the concept of relaxation and the efficiency function introduced here will play important roles. Furthermore, a relation between the algorithm and the dynamic programming approach is discussed, and subsequently it will be shown that the ordinary 0–1 knapsack problem, the linear programming knapsack problem and the single constrained linear programming problem with upper-bounded variables are special cases of the interested problem. Finally, practical applications of the problem and its computational experiences will be shown.
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    A continuous variable representation of the traveling salesman problem (1978)
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    Mathematical programming 15 (1978), S. 211-213 
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    Keywords: Traveling Salesman Problem ; Cardinality Constraints ; Hamiltonian Circuits ; Network Flow Problems
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    Notes: Abstract The classic traveling salesman problem is characterized in terms of continuous flows on a specially constructed non-conservative network, in 2n − 1 linear constraints and a cardinality constraint. It is shown that every solution to the network problem is a hamiltonian circuit.
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    A note on ‘monotone optimal policies for markov decision processes’ (1978)
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    Mathematical programming 15 (1978), S. 220-222 
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    Keywords: Dynamic Decision Model ; Monotone Optimal Decision Functions
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    Notes: Abstract The purpose of this short note is to correct some oversights in [1]. More precisely, we point out that stronger assumptions have to be imposed on the decision model (in order to use results in [2]) and present a counterexample to a comment to [1, Theorem 3.1].
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    A note on ‘new algorithms for constrained minimax optimization’ (1978)
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    Mathematical programming 15 (1978), S. 239-242 
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    Keywords: Minimax Optimization ; Nonlinear Programming
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    Notes: Abstract An unconstrained minimax algorithm of Charalambous and Conn is easily modified to solve the constrained case. Here we present some numerical results and find that this algorithm compares favourably to those of Dutta and Vidyasagar.
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    Optimal conditioning in the convex class of rank two updates (1978)
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    Mathematical programming 15 (1978), S. 247-260 
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    Keywords: Quasi-Newton Method ; Optimal Conditioning ; Rank-two Update
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    Notes: Abstract Davidon's new quasi-Newton optimization algorithm selects the new inverse Hessian approximation $$\bar H$$ at each step to be the “optimally conditioned” member of a certain one-parameter class of rank two updates to the last inverse Hessian approximationH. In this paper we show that virtually the same goals of conditioning can be achieved while restricting $$\bar H$$ to the convex class of updates, which are bounded by the popular DFP and BFGS updates. This suggests the computational testing of alternatives to the “optimal conditioning” strategy.
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    An implementation of some implicit function theorems with applications to sensitivity analysis (1978)
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    Mathematical programming 15 (1978), S. 261-267 
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    Keywords: Sensitivity Analysis in Nonlinear Programming ; Computational Aspects ; Chemical Equilibrium Problems ; Entropy Maximization Problems
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    Notes: Abstract This paper presents an implementation of some recent results of Bigelow and Shapiro [1]. These implicit function theorems are shown to provide a convenient means of performing certain types of sensitivity analysis, in particular updating the lagrange multipliers, associated with particular classes of problems. As a result we extend the usual sensitivity analysis results to include improving estimates of the effect of changing the right-hand sides of constraints. Examples of chemical equilibrium and entropy maximization models are used to illustrate the results.
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    A lower bound for the controlling parameters of the exact penalty functions (1978)
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    Mathematical programming 15 (1978), S. 278-290 
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    Keywords: Constrained Optimization ; Exact Penalty Functions ; Nondifferentiable Functions
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    Notes: Abstract The purpose of this paper is to present new exact penalty functions and discuss their properties. A lower bound on the controlling parameters is given, for which above this value, the optimum of the exact penalty function coincides with the optimum of the nonlinear programming problem.
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    Testing of a large-scale network optimization program (1978)
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    Mathematical programming 15 (1978), S. 291-314 
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    Keywords: Network Flow Problems ; Computational Results—Efficiency—Comparison ; Specific problems in mathematical programming ; Computational Experiments ; Pivotal Selection Methods ; Starting Strategies ; Scaling Procedure
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    Notes: Abstract This paper describes the experimental results of testing a “large-scale” program for solving minimum-cost network flow problems. With this program, general structure transshipment problems with over ten thousand nodes and thirty thousand arcs have been easily solved without resorting to auxiliary storage. The algorithm is a variant of the primal revised simplex method; the computer code is called LPNET illustrating the close connection between linear programming and network graphs. This approach substantially improves computer processing timeand core storage, especially for relatively large network problems. The results of these experiments are provided. It is emphasized that an organized experimental design and a detailed series of empirical tests are crucial for an efficient implementation.
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    The decomposition of nonlinear problems (1978)
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    Mathematical programming 15 (1978), S. 360-362 
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    Keywords: Nonlinear Decomposition ; Nonlinear Programming
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    Notes: Abstract We show that under certain conditions nonlinear programming problems can be decomposed into a series of smaller problems. A Decomposition Theorem and example are presented.
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    Nominations for 1979 elections (1978)
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    Mathematical programming 15 (1978), S. 363-363 
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    Mathematical programming study 9 Mathematical Programming in use (1978)
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    Mathematical programming 15 (1978), S. 365-368 
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    Optimal design of a manpower system (1978)
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    Mathematical programming 15 (1978), S. 26-35 
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    Keywords: Manpower Planning ; Column Generation ; Network Flows ; Shortest Path ; System Design
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    Notes: Abstract An equilibrium model of a manpower system is developed based on the notion of a career flow. Institutional constraints and measures of system performance are linear functions of the career flow. A typical optimal design problem is formulated and a solution procedure is developed. The optimization problem is a generalized linear program in which columns are generated by solving a shortest path problem. Upper and lower bounds on the optimal value function can be developed at each stage of the calculations.
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    Piecewise convex programs (1978)
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    Mathematical programming 15 (1978), S. 53-62 
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    Keywords: Convex Program ; Decomposition ; Cutting Plane Algorithm ; Stochastic Quadratic Program with Recourse
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    Notes: Abstract A piecewise convex program is a convex program such that the constraint set can be decomposed in a finite number of closed convex sets, called the cells of the decomposition, and such that on each of these cells the objective function can be described by a continuously differentiable convex function. In a first part, a cutting hyperplane method is proposed, which successively considers the various cells of the decomposition, checks whether the cell contains an optimal solution to the problem, and, if not, imposes a convexity cut which rejects the whole cell from the feasibility region. This elimination, which is basically a dual decomposition method but with an efficient use of the specific structure of the problem is shown to be finitely convergent. The second part of this paper is devoted to the study of some special cases of piecewise convex program and in particular the piecewise quadratic program having a polyhedral constraint set. Such a program arises naturally in stochastic quadratic programming with recourse, which is the subject of the last section.
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    Estimating errors of numerical approximation for periodic analytic functions (1977)
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    Computing 18 (1977), S. 241-248 
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    Topics: Computer Science
    Description / Table of Contents: Zusammenfassung Ausgehend von der Cauchyschen Integralformel beschreiben wir ein einfaches Verfahren zur Fehlerabschätzung für numerische Approximationen bei periodischen analytischen Funktionen. Anschließend werden Fehlerabschätzungen angegeben für die Quadraturformel nach Chawla und Ramakrishnan [1] zur numerischen Berechnung des Cauchyschen Hauptwerts $$I\left( {f;a} \right) = \int\limits_0^{2\pi } {f(x)\cot \left( {\left( {x - a} \right)/2} \right)dx,} $$ unf für die Quadraturformel nach Garrick [2] für die Berechnung vonI f; o). Dabei ergeben sich für die Garricksche Formel günstigere Fehlerschranken als für die Formel von Chawla und Ramakrishnan. Schließlich betrachten wir eine Erweiterung der Garrickschen Formel zur Berechnung vonI(f;a) für beliebigesa∈[0,2 π), welche für allea die gleiche Fehlerschranke hat. Während füra=x j die erweiterte Formel mit der Quadraturformel nach Wittich [3] übereinstimmt, ist sie füra≠x j vorteilhafter, weil sie für die gleiche Genauigkeit nur die Hälfte an Funktionsauswertungen benötigt gegenüber der Wittichschen Formel.
    Notes: Abstract We give a simple method based on Cauchy's integral formula for estimating the errors of numerical approximation for periodic analytic functions. We then obtain error estimates for the quadrature formulas of Chawla and Ramakrishnan [1] for the numerical evaluation of the Cauchy principal value integral $$I\left( {f;a} \right) = \int\limits_0^{2\pi } {f(x)\cot \left( {\left( {x - a} \right)/2} \right)dx,} $$ and for the quadrature formula of Garrick [2] for the evaluation ofI (f;o), Based on these error estimates, we are led to conclude that for the evaluation ofI (f;o), Garrick's formula has a better error estimate than the formula of Chawla and Ramakrishnan with the same number of function evaluations. Finally, we extend Garrick's formula for the evaluation ofI (f;a) for arbitrarya∈[0,2π); the extended formula has, for alla, the same error estimate as Garrick's formula. While, for a=xj, the extended formula is identical with the quadrature formula of Wittich [3], fora≠x j, the extended formula is much better in that it uses only half the number of function evaluations of Wittich's formula for the same accuracy.
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    Auswertungsalgorithmen fast-optimaler numerischer Stabilität für Polynome (1977)
    Springer
    Computing 17 (1977), S. 289-296 
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    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Description / Table of Contents: Abstract Within quite a general family of evaluation-schemes for polynomials, Clenshaw's algorithm based on the Chebyshevpolynomials of the second kind is of almost maximum numerical stability.
    Notes: Zusammenfassung Innerhalb einer recht allgemeinen Familie von Auswertungsalgorithmen für Polynome zeichnet sich der auf Clenshaw zurückgehende Algorithmus, welcher auf Entwicklung nach Tschebyscheff-Polynomen 2. Art beruht, durch fast-optimale numerische Stabilität aus.
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    Kriterien für die Liapunovstabilität und ihre numerische Anwendung (1977)
    Springer
    Computing 17 (1977), S. 297-308 
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    Description / Table of Contents: Abstract There are considered three different approaches for solving numerically the stability problem, i.e. the question whether the eigenvalues of a matrix A have negative real parts: Calculating the characteristic polynomial and applying the Routh scheme, direct solution of the Liapunov equation and transforming A to the so-called Schwarz form. From the latter a Routh scheme for complex polynomials is derived. Further, by an appropriate Möbius transformation, an inclusion method for the eigenvalues of a complex matrix is proposed and applied to a matrix resulting from an econometric model.
    Notes: Zusammenfassung Drei verschiedene Methoden zur numerischen Lösung des Stabilitätsproblems, d. h. der Frage, ob die Eigenwerte einer reellen oder komplexen Matrix A in der linken Halbebene liegen, werden verglichen: Die Berechnung des charakteristischen Polynoms von A mit der Anwendung des Routh-Kriteriums, die direkte Lösung der Liapunovgleichung und die Transformation von A auf die sogenannte Schwarzsche Form; aus der letzteren wird ein Routh-Schema für komplexe Polynome hergeleitet. Ferner läßt sich durch eine geeignete Möbius-Transformation eine Methode zur Einschließung von Eigenwerten komplexer Matrizen angeben, die bei einer Matrix aus einem ökonometrischen Modell erprobt wird.
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    Alternative forms with orders of the lax equivalence theorem in Banach spaces (1977)
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    Computing 17 (1977), S. 335-342 
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    Description / Table of Contents: Zusammenfassung Der Äquivalenzsatz von Lax über die Konvergenz der Lösung des diskretisierten Problems gegen die des gegebenen sachgemäß gestellten Anfangswertproblems zeigt, daß Stabilität notwendig und hinreichend für die Konvergenz ist, falls das Differenzenschema konsistent ist. In einer früheren Arbeit von Butzer-Weis wurde dieser Satz im Rahmen allgemeiner Banachräume mit Ordnungen versehen, im Sinne, daß Konsistenz, Stabilität und Konvergenz mit Ordnungen betrachtet werden. Nach geeigneter Abänderung der benutzten Definitionen besagt nun eine alternative Form des Äquivalenzsatzes von Lax, daß Konsistenz und Stabilität äquivalent zur Konvergenz sind. Dieses Ergebnis wird ebenfalls auf den Fall mit Ordnungen verallgemeinert. In der Tat sind beide Formen des Satzes von Lax gültig unter den gleichen Definitionen von Konsistenz, Stabilität und Konvergenz mit Ordnungen. Ein Beispiel zeigt, daß der letztere Satz in einem gewissen Sinne bestmöglich ist.
    Notes: Abstract The Lax equivalence theorem on the convergence of the solution of the discrete problem to that of the given properly posed initial-value problem states that if the difference scheme is consistent, then stability is necessary and sufficient for convergence. In a recent paper by Butzer-Weis this theorem was equipped with orders in the setting of arbitrary Banach spaces in the sense that consistency, stability and convergence were considered with orders. By modifying the concepts involved suitably, an alternative form of the Lax theorem reads that consistency and stability is equivalent to convergence. This result is also generalized to one containing orders, in fact, both forms of the Lax theorem are valid under the same definitions of consistency, stability, and convergence with orders. An example is given showing that the latter theorem is in a certain sense best possible.
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    Penalty-Methode bei der numerischen Behandlung von quasilinearen elliptischen Differentialgleichungen vierter und höherer Ordnung (1977)
    Springer
    Computing 17 (1977), S. 343-350 
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    Description / Table of Contents: Abstract In the following paper we treat the numerical solution of quasilinear elliptic differential equations of fourth and higher order which are Euler-equations of certain variational problems We reduce the differential equation to a system of equations of the second order and solve this system by the method of finite differences. Existence and uniqueness of a minimal solution of the discrete problem and convergence to the solution of the variational problem under the assumptions of consistency and stability are established as the mesh size and the Penalty-parameter tend to zero.
    Notes: Zusammenfassung Es werden quasilineare elliptische Differentialgleichungen vierter und höherer Ordnung behandelt, die Euler-Gleichungen gewisser Variationsprobleme sind. Die Differentialgleichung wird in ein System von Gleichungen zweiter Ordnung umgeformt und mittels Differenzenverfahren gelöst. Existenz und Eindeutigkeit einer Minimallösung des diskreten Problems und Konvergenz gegen die Lösung des kontinuierlichen Problems unter der Voraussetzung von Konsistenz und Stabilität werden bewiesen, falls die Gitterkonstante und der Penalty-Parameter gegen Null streben.
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    Über die einseitige Randsteuerung einer schwingenden Saite in einen Zustand minimaler Energie (1977)
    Springer
    Computing 17 (1977), S. 351-359 
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    Description / Table of Contents: Abstract We consider the problem of steering a vibrating string which is fixed at the left end within a given time into a state of minimal energy by controlling at the right-hand side. First it is shown that the solution of this problem equals the control that transfers the string into rest in minimal time. Second a weak bang-bang-principle for optimal controls having a bounded acceleration is derived, and is demonstrated by an example.
    Notes: Zusammenfassung Betrachtet wird das Problem, eine linksseitig eingespannte schwingende Saite durch Steuerung am rechten Rand innerhalb vorgegebener Zeit in einen Zustand minimaler Energie überzuführen. Zunächst wird gezeigt, daß die Lösung dieses Problems gleich der Steuerung ist, die die Saite in minimaler Zeit in die Ruhelage bringt. Sodann wird für optimale Steuerungen mit beschränkter Beschleunigung ein schwaches Bang-Bang-Prinzip hergeleitet und an einem Beispiel demonstriert.
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    Ein Verfahren zur iterativen Einschließung des positiven Eigenvektors einer irreduziblen, nichtnegativen Matrix (1978)
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    Computing 20 (1978), S. 27-34 
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    Description / Table of Contents: Abstract We develop a new iteration scheme giving bounds for the positive eigenvector of a nonnegative and irreducible matrix and the corresponding eigenvalue and test it on some examples.
    Notes: Zusammenfassung Wir geben ein neues Iterationsverfahren an, das Schranken für den positiven Eigenvektor einer nichtnegativen, irreduziblen Matrix und des zugehörigen Eigenwerts liefert.
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    Über eine Wahl des Parameters beim Parallelenverfahren (Parallel-Chord-Method) (1978)
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    Computing 20 (1978), S. 17-26 
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    Description / Table of Contents: Abstract Due to the free parameter in the fixed-point equation of the parallel-chord-method, it is possible to reduce the Lipschitz constant of the contraction mapping. By use of this reduction, it is possible in suitable cases to generate the contraction property of the mapping. If the mapping possesses this property, then this is also true for the mapping with the parameter determined here. In addition, both an improved error estimate for the sequences of iterates and a smaller convergence factor are obtained. An example is presented.
    Notes: Zusammenfassung Mit Hilfe des freien Parameters in der Fixpunktgleichung des Parallelenverfahrens ist es möglich, die Lipschitz-Konstante der kontrahierenden Abbildung zu verkleinern. Durch diese Verkleinerung kann in geeigneten Fällen eine kontrahierende Abbildung erzeugt werden. Ist die Abbildung bereits kontrahierend, ist sie auch kontrahierend für den berechneten Wert des Parameters. Ferner ergibt sich neben einer verbesserten Fehlerabschätzung für die Iterationsfolgen auch ein kleinerer Konvergenzfaktor. Das Verfahren wird an Hand eines Beispiels erläutert.
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    Zur Approximation von Schalen mit hybriden Elementen (1978)
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    Computing 20 (1978), S. 75-94 
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    Description / Table of Contents: Abstract A theory of hybrid finite element approximation is developed for a class of shell problems. Though it is applied only to the special case of clamped shallow shells with regular triangularization the results may be of larger interest comparing the predicted convergence rate with the numerical outcome of some applications of the hybrid finite element method. The convergence speed can only be increased by higher degrees of the approximation and the stresses at the edges correspondingly. The use of a so-called rank condition plays a fundamental role in the study. Weak coerciveness of the under lyigg bilinear form for the derivation of hybrid elements is proved by showing the existence of a stationary point.
    Notes: Zusammenfassung Dieser Beitrag liefert eine Theorie der Finite-Element-Approximation für eine Klasse von Schalenproblemen. Obgleich diese nur für den Fall der eingespannten flachen Schale mit regulärer Triangulierung ausführlich dargestellt wird, sind die Ergebnisse darüberhinaus von Interesse, da die abgeleiteten Konvergenzordnungen das Verhalten numerischer Näherungslösungen aus anderen Anwendungen der hybriden Finite-Element-Methode erklären helfen. Die Konvergenzgeschwindigkeit kann nur verbessert werden durch gleichzeitige Erhöhung der Ansatzgrade für die Verschiebungen im Elementinnern und für die Spannungen auf den Rändern. Die Einhaltung einer sogenannten Rangbedingung hat zentrale Bedeutung. Die schwache Koerzitivität der für die Ableitung hybrider Elemente verwendeten Bilinearform ist durch den Nachweis der Existenz eines stationären Werts gesichert.
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    Interval forms of Newtons method (1978)
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    Computing 20 (1978), S. 153-163 
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    Description / Table of Contents: Zusammenfassung Eine verbesserte Darstellung Newtonscher Intervall-Verfahren wird diskutiert. Es wird dabei gezeigt, daß man bestimmte Intervalle in den Verfahren durch reelle Zahlen ersetzen kann. Dadurch werden die Konvergenzeigenschaften der Verfahren verbessert.
    Notes: Abstract Improved forms of some interval Newton Methods are given. It is shown that certain intervals in the methods can be replaced by real numbers. This improves the convergence properties of the methods.
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    Eine rundungsgenaue Formel zur maschinellen Berechnung der Prager-Oettli-Schranke (1978)
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    Computing 20 (1978), S. 177-182 
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    Description / Table of Contents: Abstract The bound of Prager-Oettli for the input error of a linear system of equations is computed via a new formula which guarantees strictness of the bound in spite of roundoff errors.
    Notes: Zusammenfassung Die Schranke von Prager-Oettli für den Eingangsfehler eines linearen Gleichungssystems wird über eine neue Formel berechnet, die trotz der Rundungsfehler die Schärfe der Schranke sichert.
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    Iterated defect correction for differential equations part I: theoretical results (1978)
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    Computing 20 (1978), S. 207-228 
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    ISSN: 1436-5057
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    Topics: Computer Science
    Description / Table of Contents: Zusammenfassung Die Iterierte Defektkorrektur (IDeC) ist ein Verfahren zur schrittweisen Verbesserung einer Näherungslösung eines gegebenen ProblemsFy=0. Eines der wichtigsten Anwendungsgebiete dieses Prinzips sind Differentialgleichungen. Die IDeC kann dort als Methode zur Verbesserung der Ordnung eines Diskretisierungsverfahrens, und damit zur Verbesserung der Genauigkeit eingesetzt werden. In der vorliegenden Arbeit wird ein Metaalgorithmus für die Klasse, der IDeC-Verfahren für Differential-gleichungen vorgestellt und analysiert. Für jeden “Baustein” dieses Metaalgorithmus werden Bedingungen angegeben, die es gewährleisten, daß eine bestimmte Ordnung erreicht wird. Diese Bedingungen sind von großer praktischer Bedeutung, wenn IDeC-Verfahren als Computer-Programme implementiert werden sollen.
    Notes: Abstract Iterated Defect Correction (IDeC) is a technique for improving successively an approximate solution of a given problemFy=0. One of the most important fields of application of this principle are differential equations. Here, IDeC can be used as a technique for increasing the order of a discretization method and thus for improving the accuracy. In this paper a metalgorithm for the class of IDeC-methods for differential equations is presented and analyzed. For every component of this metalgorithm conditions are given which guarantee a certain order of accuracy. These conditions are of particular importance for practical applications, as far as the implementation of IDeC-methods is concerned.
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    Ein Abbrechkriterium für einen numerischen Algorithmus (1978)
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    Computing 20 (1978), S. 257-265 
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    Topics: Computer Science
    Description / Table of Contents: Abstract A termination criterion by Nickel (Theorem 6, [1]) for guaranteeing the numerical convergence for locally stable and consistent algorithms is generalized. The assumption |x ν+2,x ν+1|≤L|x ν+1,x ν| (0≤L〈1,L real constant) of the approximation sequence {x ν} to the solution is replaced by the convergence of the progression $$\sum\limits_{v = 1}^\infty {|x,x_{v + 1} |/Q^v (0〈 Q〈 1)} $$ . Therefore the theorem of this paper is applicable to a large number of numerical procedures, for which untill now no termination criterion has been known (for example: Rombergprocedure). In particular this weakening is important for the computation of approximation solutions for integral equations.
    Notes: Zusammenfassung Ein Abbrechkriterium von Nickel (Satz 6, [1]) zur Sicherung der numerischen Konvergenz bei lokal stabilen und konsistenten Algorithmen wird verallgemeinert. Statt der Eigenschaft |x ν+2,x ν+1|≤L|x ν+1,x ν| (0≤L〈1,L von ν ∈ ℕ unabhängig) der Näherungsfolge {x ν} zur Lösungx reicht die Konvergenz der Reihe $$\sum\limits_{v = 1}^\infty {|x,x_{v + 1} |/Q^v (0〈 Q〈 1)} $$ aus. Damit ist der Satz dieser Arbeit bei einer großen Klasse von numerischen Verfahren anwendbar, für die bisher noch kein Abbrechkriterium bekannt ist (z. B. dem Romberg-Verfahren). Insbesondere ist diese Abschwächung zur Berechnung von Näherungslösungen bei Integralgleichungen wichtig.
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    The nonlinear Volterra equation of Abel's kind and its numerical treatment (1978)
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    Computing 20 (1978), S. 307-324 
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    Topics: Computer Science
    Description / Table of Contents: Zusammenfassung Ausgehend von einem Existenz- und Eindeutigkeitssatz für die Lösung einer verallgemeinerten Volterra-Integralgleichung zweiter Art wird die Existenz und Eindeutigkeit der Lösung der hier behandelten nichtlinearen, schwach singulären Volterra-Integralgleichung erster Art untersucht. Es wird ein numerisches Verfahren der Ordnung 2 bzw. 3 angegeben. Der Konvergenzbeweis basiert in beiden Fällen auf einem Lemma über die Beschränktheit einer speziellen Differenzenungleichung. An zwei numerischen Beispielen werden die theoretischen Ergebnisse demonstriert.
    Notes: Abstract Starting from an existence and uniqueness theorem for a generalized nonsingular second kind Volterra equation existence and uniqueness for the solution of the nonlinear, weakly singular first kind Volterra equation is examined. A new type of numerical method is developed. A basic lemma concerning the boundedness of a special difference inequality is given and order two or three convergence of the method is shown. Two numerical examples illustrate the theoretical results.
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    Über A (α)-stabile Verfahren hoher Konsistenzordnung (1978)
    Springer
    Computing 20 (1978), S. 343-350 
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    Topics: Computer Science
    Description / Table of Contents: Abstract For each α ε (0, π/2), the existence ofA (α)-stable linear multistep methods with arbitrary order of consistency is shown by an explicit construction. Some characteristic data of the methods and numerical results are given.
    Notes: Zusammenfassung Es wird in einer expliziten Konstruktion gezeigt, daß für jedes α ε (0, π/2)A (α)-stabile lineare Mehrschrittverfahren beliebiger Konsistenzordnung existieren. Einige charakteristische Daten der Verfahren und numerische Rechnungen werden angegeben.
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    A performance model for preplanned disk sorting (1978)
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    Computing 21 (1978), S. 17-35 
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    ISSN: 1436-5057
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    Topics: Computer Science
    Description / Table of Contents: Zusammenfassung Der Ansatz, Strings, die zusammengemischt werden, auf Magnetplatten vorzuplanen, wird unter Leistungsgesichtspunkten untersucht. Konzepte für die interne Pufferzuordnung, für die Erzeugung der anfänglichen Strings durch ein internes Sortierverfahren, und für die Stringverteilung auf Magnetplatten werden ausgewertet. Ein Algorithmus beschreibt die Konstruktion von suboptimalen Mischbäumen, die planbare Mischbäume genannt werden. Ein Kostenmodell, das auf detaillierte Annahmen der Zuordnung vonk Eingabeplatten und der Planung einesr-Wege-Mischens beruht, wird für das exakte Vorplanen aufgestellt. Zeitbetrachtungen für Sortieren und Mischen, die Hardware-Eigenschaften von Magnetplatten einschließen, zeigen signifikante Zeitgewinne verglichen mit weitverbreiteten Sortier- und Mischverfahren.
    Notes: Abstract The idea of preplanning strings on disks which are merged together is investigated from a performance point of view. Schemes of internal buffer allocation, initial string creation by an internal sort, and string distribution on disks are evaluated. An algorithm is given for the construction of suboptimal merge trees called plannable merge trees. A cost model is presented for accurate preplanning which consists of detailed assumptions on disk allocation fork input disks andr-way merge planning. Timing considerations for sort and merge including hardware characteristics of moveable head disks show a significant gain of time compared to widely used sort/merge applications.
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    Convergence acceleration on a general class of power series (1978)
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    Computing 21 (1978), S. 53-69 
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    Description / Table of Contents: Zusammenfassung In dieser Arbeit werden mehrere effiziente Methoden zur Berechnung von Funktionen, die durch Potenzenreihen definiert sind, präsentiert. Einfache Computerprogramme für zwei schnelle Algorithmen werden in einem eigenen Beitrag ([9]) angegeben. Die Konvergenzgeschwindigkeiten der vorgeschlagenen Verfahren werden theoretisch untersucht und die erhaltenen Ergebnisse werden an numerischen Beispielen erläutert.
    Notes: Abstract In this paper we present several efficient methods for evaluating functions defined by power series expansions. Simple computer codes for two rapid algorithms are given in a companion paper. The convergence rates of the proposed computational schemes are investigated theoretically and the results are illustrated by numerical examples.
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    On the determination of the weights in multi-dimensional numerical quadrature (1978)
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    Computing 21 (1978), S. 71-79 
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    Description / Table of Contents: Zusammenfassung Mit Hilfe der Pseudoinversen eines beschränkten linearen Operators von ℝ n inl 2 wird eine Methode zur Konstruktion von Quadraturformeln für die Integration über einem beliebigen beschränktenm-dimensionalen GebietB⊂ℝ m hergeleitet, mit der Eigenschaft, daß der mittlere Fehler in einer vorgeschriebenen FamilieF sowie die Varianz der Rundungsfehler gemäß Sard [8] minimal werden. Sodann wählen wirF als gewichtete Monome und behandeln als Beispiel Integration auf der Oberfläche derm-Kugel.
    Notes: Abstract Using the concept of the generalized inverse of a bounded linear transformation between ℝ n andl 2, a method is given for constructing quadrature rules for integration over an arbitrary boundedm-dimensional regionB⊂ℝ m with the property that the average error over the prescribed familyF of the functions continuous inB as well as the variance of the rounding errors according to Sard [8] are minimal. Then we specializeF to the weighted monomials and treat as an example integration on the surface of them-sphere.
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    Algorithm 38 two computer codes for convergence acceleration (1978)
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    Computing 21 (1978), S. 87-91 
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    ISSN: 1436-5057
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Description / Table of Contents: Zusammenfassung In dieser Arbeit werden zwei Computerprogramme für die Berechnung der Summe einer Potenzenreihe, die einer allgemeinen Klasse zugehört, beschrieben. Die Verwendung der Programme ist mit einem Beispiel erläutert.
    Notes: Abstract This paper describes two computer codes for calculating the sum of power series, belonging to a certain general class. The use of the codes for an illustrative test example is explained.
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    URL: http://dx.doi.org/10.1007/BF02252197
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    Runge-Kutta integration algorithms with built-in estimates of the accumulated truncation error (1978)
    Springer
    Computing 20 (1978), S. 1-16 
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    ISSN: 1436-5057
    Keywords: Integration ; Runge-Kutta integration ; truncation error ; error estimates ; accumulated error ; accumulated error estimates ; accumulated truncation error ; true error
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Description / Table of Contents: Zusammenfassung Es wird eine Methode für die Entwicklung von Runge-Kutta-Integrationsalgorithmen angegeben, die die Schätzung des globalen Verfahrensfehlers ermöglichen. Mehrere neue Algorithmen 2., 3. und 4. Ordnung werden angeführt. Die Rechenarbeit pro Schritt ist identisch für die neuen Algorithmen und für Algorithmen, die nur eine Schätzung des lokalen Verfahrensfehlers ermöglichen. Numerische Versuche mit den neuen Algorithmen ergeben, daß der geschätzte Fehler den wahren akkumulierten Fehler gut wiedergibt. Außerdem ist der Fehler von derselben Ordnung wie bei gewöhnlichen Runge-Kutta-Algorithmen.
    Notes: Abstract A method is presented for developing Runge-Kutta integration algorithms with built-in estimates of the accumulated truncation error. Several new 2-nd, 3-rd, and 4-th order algorithms are given. The computation per step of the new algorithms is identical to that of algorithms which provide only an estimate of the local truncation error. Numerical experimentation with the new algorithms shows that the estimated error compares very well with the true accumulated error. Further, the error is of the same order as that incurred using traditional Runge-Kutta algorithms.
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    URL: http://dx.doi.org/10.1007/BF02241897
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    On a method of characteristics for solving a hyperbolic equation of second order (1978)
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    Computing 20 (1978), S. 47-60 
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    ISSN: 1436-5057
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Description / Table of Contents: Zusammenfassung Das Anfangsrandwertproblem für die hyperbolische Differentialgleichunga(t,x,u)u tt +2b(t,x,u)u tx +c(t,x,u)u xx =d(t,x,u,u t ,u x ) zweiter Ordnung wird mit Hilfe eines Charakteristikenverfahrens gelöst, das keine Differenzengleichungen füru t undu x benutzt. Die Lösung des diskretisierten Problems besitzt eine asymptotische Entwicklung nach geraden Potenzen der Schrittweite. Daher können die numerischen Ergebnisse durch Extrapolation verbessert werden.
    Notes: Abstract The hyperbolic initial-boundary value problem for the second order equationa(t,x,u)u tt +2b(t,x,u)u tx +c(t,x,u)u xx =d(t,x,u,u t ,u x ) is solved by a special method of characteristics involving no difference equations foru t andu x . The discrete solution has an asymptotic expansion in even powers of the step size. Therefore, the numerical results can be improved by extrapolation to the limit.
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    URL: http://dx.doi.org/10.1007/BF02241901
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    Polynomial least square interval approximation (1978)
    Springer
    Computing 20 (1978), S. 165-176 
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    ISSN: 1436-5057
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Description / Table of Contents: Zusammenfassung Das folgende Problem wird behandelt: gegeben sei eine Menge vonm rechteckigen Gebieten in der Ebene, man finde ein Intervall-Polynom vom Gradn〉m durch diese Gebiete. Dies ist eine Verallgemeinerung des diskreten Problems der kleinsten Quadrate. Drei Verallgemeinerungen der gewöhnlichen Methode der kleinsten Quadrate für Polynome werden betrachtet und an numerischen Beispielen verglichen. Eine dieser Methoden wird empfohlen, da sie in allen Testbeispielen bessere Ergebnisse gibt.
    Notes: Abstract The following problem is treated: given a set ofm rectangular regions in the plane, fit an interval polynomial of degreen〉m through the regions. This is a generalization of the discrete polynomial least squares problem. Three generalizations of the standard methods for the discrete polynomial least squares are considered and compared on numerical examples. One of the methods is recommended since it gives superior results in all cases tested.
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    URL: http://dx.doi.org/10.1007/BF02252345
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    On scheduling multi-processor systems with algebraic objectives (1978)
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    Computing 20 (1978), S. 189-205 
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    ISSN: 1436-5057
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Description / Table of Contents: Zusammenfassung Eines der am besten beschriebenen Modelle kombinatorischer Optimierung ist das Scheduling Problem, bei dem eine endliche Anzahl von Tätigkeiten auf einer festen Anzahl von Maschinen so ausgeführt werden muß, daß eine gegebene Zielfunktion minimiert wird. Jede Tätigkeit benötigt charakteristische Daten wie Bearbeitungszeit, Fertigsteillungstermin, Strafkosten und technologische Nachfolgebeziehungen. Ein algebraischer Ansatz für die Zielfunktion führt zu einem allgemeinen Problem, das alle in der Literatur bekannten klassischen Fälle von Summen und Maximum Zielfunktionen einschließt. Durch die Lösung eines algebraischen Transportproblems wird eine untere Schranke für den Zielfunktionswert bestimmt. Um eine Optimallösung zu erhalten, verwenden wir ein Branch and Bound Verfahren. Weiterhin betrachten wir das allgemeine Job Shop Scheduling Problem mit algebraischer Zielfunktion.
    Notes: Abstract One of the well-studied models of combinatorial optimization is the scheduling problem dealing with a finite set of tasks, which have to be executed on a fixed number of machines so that a given objective is minimized. Each task requires a set of characteristic data like operating time, due date, penalty cost and technological requirements. An algebraic approach to the objective leads to a general problem which includes all classical cases of sum and bottleneck objectives known in literature. By solving an algebraic transportation problem a lower bound for the objective value can be determined. To obtain an optimal solution we employ a branch and bound procedure. Furthermore we consider the general job shop scheduling problem with algebraic objective function.
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    URL: http://dx.doi.org/10.1007/BF02251945
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    Minimierung der Rechenzeit bei der Berechnung von Lösungen nichtlinearer Integralgleichungen (1978)
    Springer
    Computing 20 (1978), S. 325-331 
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    ISSN: 1436-5057
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Description / Table of Contents: Abstract The result of this paper is the economizing of cpu-time by identical guaranteed precision of the approximate solution. This is possible by an optimal selection ofn j in every iteration stepj, wheren j is the number of subintervals for the numerical integration. The determination ofn 1,n 2, ... is reduced to the solution of a solvable convex optimization job.
    Notes: Zusammenfassung Es wird die Einsparung von Rechenzeit bei gleicher garantierter Genauigkeit der Näherungslösung ermöglicht. Dies wird erreicht durch eine optimale Wahl vonn j bei jedem Iterationsschrittj, wobein j die Zahl der Teilintervalle für die Quadratur ist. Die Bestimmung vonn 1,n 2,... wird auf die Lösung einer lösbaren konvexen Optimierungsaufgabe zurückgeführt.
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    URL: http://dx.doi.org/10.1007/BF02252380
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    Komplexitätserfordernisse von Modellen zur optimalen Indexselektion für Datenbanken (1978)
    Springer
    Computing 20 (1978), S. 279-290 
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    ISSN: 1436-5057
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Description / Table of Contents: Abstract It is one of the basic questions in designing a database system to determine which attributes to invert. The literature presents a variety of models to solve this question. This paper attempts to give a guideline, under which conditions a primitive model considering only retrieve operations, suffices to determine an optimal solution. The relevance of that question lies in the smaller costs of simpler models.
    Notes: Zusammenfassung Eine wesentliche Fragestellung beim Aufbau eines Datenbanksystems ist die Auswahl der Schlüsselattribute. Bei Inverted-File Systemen ist dies äquivalent zur Auswahl der zu invertierenden Attribute. In diesem Aufsatz wird versucht darzustellen, unter welchen Bedingungen Modelle, die lediglich das Retrieve-Verhalten des Benutzers berücksichtigen, zuverlässige Resultate liefern können. Die Relevanz dieser Fragestellung liegt vor allem in den durch das einfachere Modell verminderten Kosten der Datenbeschaffung und Datenanalyse im DB-Designprozeß.
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    URL: http://dx.doi.org/10.1007/BF02252377
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    On the convergence of multistep methods for the Cauchy problem for ordinary differential equations (1978)
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    Computing 20 (1978), S. 351-361 
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    ISSN: 1436-5057
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Description / Table of Contents: Zusammenfassung Es wird ein allgemeines, quasilineares, nichtstationäresk-Schrittverfahren für die Lösung des Cauchy-Problems für gewöhnliche Differentialgleichungen untersucht. Ein Konvergenzsatz mit ziemlich schwachen Bedingungen wird angegeben. Die Inkrementfunktion muß nicht Lipschitz-stetig sein; es genügt, wenn diese Funktion die Perron-Bedingung aus der Eindeutigkeitstheorie für das Cauchy-Problem mit nichtabnehmender Vergleichfunktion erfüllt. Das Ergebnis ist eine Erweiterung der Theorie von G. Dahlquist und des letzten Resultats von K. Taubert.
    Notes: Abstract The general form of a quasilinear nonstationaryk-step method for solving of the Cauchy problem for ordinary differential equations is discussed. The convergence theorem is stated under rather weak conditions. It is not assumed that the increment function is Lipschitz-continuous but only that it satisfies the Perron type condition appearing in the uniqueness theory for the Cauchy problem with a nondecreasing comparison function. The result established in the paper is an extension of the theory given by G. Dahlquist and the recent result of K. Taubert.
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    URL: http://dx.doi.org/10.1007/BF02252383
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    Algorithm 36 SNIFF: Efficient self-tuning algorithm for numerical integration (1978)
    Springer
    Computing 20 (1978), S. 363-375 
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    ISSN: 1436-5057
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Description / Table of Contents: Zusammenfassung Hier wird ein neuer selbsteichender Algorithmus für die Integration der analytischen Funktionen dargestellt. Der Algorithmus wirkt auf das Integrationsintervall indem er örtliche Unterintervalle hervorruft, deren Länge durch ein neues System überwacht wird. Die Überwachung ergibt sich aus einer Formel, die von einer analytischen Basis abgeleitet wird und für jede beliebige Integrationsregel gilt: mit zwei verschiedenen Annäherungen eines Integrals wird die Intervallänge berechnet, die notwendig ist, um eine Integralannäherung mit der innerhalb zugelassener Fehlergrenzen notwendigen Genauigkeit zu erreichen. Die daraus entstehende Methode für die Erzeugung der Unterintervalle und eine wirksame Auswahl der Fehlerverteilung unter die Unterintervalle bringen einen selbstanpassenden Algorithmus hervor, der über einen genauen und höchst günstigen Integrationsprozeß verfügt. Darauf wird der besondere Algorithmus betrachtet, der sich mit Hilfe der 6-Punkte-Gauß-Legendre-Integrationsregel ergibt. Außerdem wird er mit den besten unter den anderen Integrationsalgorithmen verglichen.
    Notes: Abstract A new adaptive algorithm for the integration of analytic functions is presented. The algorithm processes the integration interval by generating local subintervals whose length is controlled through a feedback loop. Control is performed by means of a relation derived on an analytical basis and valid for an arbitrary integration rule: two different estimates of an integral are used to compute the interval length necessary to obtain an integral estimate with accuracy within the assigned error bounds. The implied method for local generation of subintervals and an effective assumption of error partition among subintervals give rise to an adaptive algorithm provided with an accurate and very efficient integration process. The particular algorithm obtained by choosing the 6-point Gauß-Legendre integration rule is considered and extensive comparisons are made with other outstanding integration algorithms.
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    URL: http://dx.doi.org/10.1007/BF02252384
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    Zur praktischen Behandlung linearer diskreter Approximationsprobleme in der Maximumsnorm (1978)
    Springer
    Computing 21 (1978), S. 37-52 
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    ISSN: 1436-5057
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Description / Table of Contents: Abstract Direct methods for the solution of linear approximation problems tend to fail in practice because of numerical instabilities. Difficulties arise from an undesired accumulation of round-off errors. In [3] and in chapter 2 of this paper effective and inexpensive tests are developed to recognize numerical difficulties and stabilize numerical methods by iterative refinement and restart procedures, if necessary. On the ground of this, a stabilized version of a modified revised dual simplex algorithm can be developed capable of solving the approximation problems under consideration. The technical details of this method and a FORTRAN implementation are given in [4]. Numerical examples are discussed to illustrate that the developed method is superior to commonly used algorithms not only by a broader range of applications and more stability, but also by considerably less storage requirement (for large problems), while the execution times are comparable or less.
    Notes: Zusammenfassung Anhäufungen von Rundungsfehlern führen in der Praxis häufig dazu, daß direkte simplexartige Methoden zur Lösung linearer Approximationsprobleme in der Maximumsnorm versagen. Vor dem approximationstheoretischen Hintergrund des Problems werden in der vorliegenden Arbeit einfach durchzuführende Tests entwickelt, die numerische Schwierigkeitenim voraus anzeigen und so Gegenmaßnahmen ermöglichen. Anhand einer Reihe von Beispielen werden Stabilität, Anwendungsbereich und Grenzen darauf aufbauender Algorithmen diskutiert.
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    URL: http://dx.doi.org/10.1007/BF02252193
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    Algorithm 37 Algorithm for the solution of the 0–1 single knapsack problem (1978)
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    Computing 21 (1978), S. 81-86 
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    ISSN: 1436-5057
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Description / Table of Contents: Zusammenfassung Für das „0–1 single Knapsack Problem” wird eine FORTRAN-Implementierung eines wirkungsvollen Lösungsverfahrens angegeben. Die angeführten Rechenergebnisse zeigen, daß die vorgeschlagene Methode den derzeit besten bekannten Algorithmen überlegen ist.
    Notes: Abstract The FORTRAN implementation of an efficient algorithm which solves the 0–1 single knapsack problem is given. Computational results are presented, showing the proposed method to be generally superior to the best known algorithms.
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    URL: http://dx.doi.org/10.1007/BF02252196
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    A property of continuous unbounded algorithms (1978)
    Springer
    Mathematical programming 14 (1978), S. 373-377 
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    ISSN: 1436-4646
    Keywords: Nonlinear Programming ; Algorithms ; Continuous Unbounded Algorithms ; Global Convergence of Algorithms
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
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    URL: http://dx.doi.org/10.1007/BF01588977
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    Summer School in Combinatorial Optimization (1978)
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    Mathematical programming 14 (1978), S. 379-379 
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    ISSN: 1436-4646
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
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    URL: http://dx.doi.org/10.1007/BF01588979
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    On solving optimization problems with proportion-constraints (1978)
    Springer
    Mathematical programming 15 (1978), S. 77-86 
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    ISSN: 1436-4646
    Keywords: Proportion-Constraints ; Symmetry ; Converging Matrices ; Mathematical Programming ; Bounded Matrices
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract The purpose of this paper is to establish sufficient conditions for the existence of solutions to mathematical programs where the variables of the solution satisfy given proportions. These conditions rely on convergence properties of powers of nonnegative matrices when these powers form a bounded sequence. We assume that if an arbitrary vectorx is premultiplied by elements of this sequence, the limit of the sequence (which might be a Cesaro (C, 1) limit) gives an improvement of the objective.
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    URL: http://dx.doi.org/10.1007/BF01609001
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    An alternative method for a global analysis of quadratic programs in a finite number of steps (1978)
    Springer
    Mathematical programming 15 (1978), S. 105-109 
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    ISSN: 1436-4646
    Keywords: Quadratic Programming
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract This paper considers the global analysis of general quadratic programs in a finite number of steps. A procedure is presented for recursively finding either the global minimum or a halfline of the constraint set along which the minimand is unbounded below.
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    URL: http://dx.doi.org/10.1007/BF01609005
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    Mathematical programming study 7 complementarity and fixed point problems (1978)
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    Mathematical programming 15 (1978), S. 114-118 
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    ISSN: 1436-4646
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01609007
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    On a conjecture of Dixon and other topics in variable metric methods (1978)
    Springer
    Mathematical programming 15 (1978), S. 123-129 
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    ISSN: 1436-4646
    Keywords: Unconstrained Optimization ; Variable Metric Methods ; Quasi-Newton Methods ; Optimal Conditioning ; Round-off Errors
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract A conjecture of Dixon relating to the behaviour of variable metric methods on functions with special symmetry is validated under suitable onditions. The relation between Huang's class and Oren's class is explored. Then the equivalence of Davidon's and Oren and Spedicato's approaches to optimal conditioning is demonstrated.
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    URL: http://dx.doi.org/10.1007/BF01609011
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