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  • Articles  (73,837)
  • Springer  (38,748)
  • National Academy of Sciences  (35,089)
  • 2000-2004  (27,346)
  • 1995-1999  (33,764)
  • 1975-1979  (12,727)
  • Natural Sciences in General  (47,991)
  • Computer Science  (25,987)
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  • Articles  (73,837)
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  • 1
    Electronic Resource
    Electronic Resource
    Explicit duality for convex homogeneous programs (1976)
    Springer
    Mathematical programming 10 (1976), S. 176-191 
    Details
    ISSN: 1436-4646
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract When all the functions that define a convex program are positively homogeneous, then a dual convex program can be constructed which is defined in terms of the primal data only (the primal variables do not appear). Furthermore, the dualizing process, carried out on the dual program, yields the primal. Several well-known examples of convex programs with explicit duals are shown to be special cases.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01580666
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  • 2
    Electronic Resource
    Electronic Resource
    Optimized relative step size random searches (1976)
    Springer
    Mathematical programming 10 (1976), S. 230-244 
    Details
    ISSN: 1436-4646
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract New results concerning the family of random searches as proposed by Rastrigin are presented. In particular, the random search with reversals and two optimized relative step size random searches are investigated. Random searches with reversals are found to be substantially better than their counterparts. A new principle of updating the step size for this family of searches is proposed.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01580669
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  • 3
    Electronic Resource
    Electronic Resource
    On generalizing shapley's index theory to labelled pseudomanifolds (1976)
    Springer
    Mathematical programming 10 (1976), S. 245-262 
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    ISSN: 1436-4646
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract For a bimatrix game one may visualize two bounded polyhedronsX andY, one for each player. OnX × Y one may visualize, as a graphG, the set of almost-complementary points (see text). $$\bar G$$ consists of an even number of nodes, one for each complementary point (one for the origin, others corresponding to extreme points which are equilibrium points); arcs (extreme point paths of almost complementary points); and possibly loops (paths with no equilibrium points). Shapley has shown that one may assign indices (+) and (−) to nodes, and directions called (+) and (−) to arcs or loops in such a way that, leaving a (+) node one moves always in a (+) direction, terminating at a (−) node. Indices and directions for a point are determined knowing only the point. In this paper, these concepts are generalized to labelled pseudomanifolds. An integer labelling of the vertices identifies theG-set of almost-completely labelled simplexes. It is shown that in order for theG-set of any labelling to be directed as above it is necessary and sufficient that the pseudomanifold be orientable. Realized examples for situations of current interest are also developed.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01580670
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  • 4
    Electronic Resource
    Electronic Resource
    On triangulations for computing fixed points (1976)
    Springer
    Mathematical programming 10 (1976), S. 322-346 
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    ISSN: 1436-4646
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract This paper introduces theoretical measures of efficiency for triangulations used in computing fixed points. The measures, for both regular triangulations ofR n and those with continuous refinement of grid size, are based on an average count of the number of simplices met by straight line segments. The computation of these measures is facilitated by a description of the facets as well as the vertices of a triangulation. We give a simple description of a modification of Eaves' and Saigal's K3 and compute the measures of efficiency of three regular triangulations ofR n and two triangulations of (0, 1] ×R n with continuous refinement of grid size, including the modified K3.
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    URL: http://dx.doi.org/10.1007/BF01580679
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  • 5
    Electronic Resource
    Electronic Resource
    Integer programming approaches to the travelling salesman problem (1976)
    Springer
    Mathematical programming 10 (1976), S. 367-378 
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    ISSN: 1436-4646
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract The availability of an LP routine where we can add constraints and reoptimize, makes it possible to adopt an integer programming approach to the travelling-salesman problem. Starting with some of the constraints that define the problem we use either a branching process or a cutting planes routine to eliminate fractional solutions. We then test the resulting integer solution against feasibility and if necessary we generate the violated constraints and reoptimize until a “genuine” feasible solution is achieved. Usually only a small number of the omitted constraints is generated. The generality of the method and the modest solution times achieved leads us to believe that such an LP approach to other combinatorial problems deserves further consideration.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01580682
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  • 6
    Electronic Resource
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    Mathematical programming study 4 (1976)
    Springer
    Mathematical programming 10 (1976), S. 409-412 
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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/BF01580687
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  • 7
    Electronic Resource
    Electronic Resource
    Convergence conditions for restarted conjugate gradient methods with inaccurate line searches (1976)
    Springer
    Mathematical programming 10 (1976), S. 32-51 
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    ISSN: 1436-4646
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract Convergence properties of restarted conjugate gradient methods are investigated for the case where the usual requirement that an exact line search be performed at each iteration is relaxed. The objective function is assumed to have continuous second derivatives and the eigenvalues of the Hessian are assumed to be bounded above and below by positive constants. It is further assumed that a Lipschitz condition on the second derivatives is satisfied at the location of the minimum. A class of descent methods is described which exhibitn-step quadratic convergence when restarted even though errors are permitted in the line search. It is then shown that two conjugate gradient methods belong to this class.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01580652
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  • 8
    Electronic Resource
    Electronic Resource
    Feasible directions algorithms for optimization problems with equality and inequality constraints (1976)
    Springer
    Mathematical programming 11 (1976), S. 67-80 
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    ISSN: 1436-4646
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract In this paper two algorithms, of the feasible-directions and dual feasible-directions type, are presented for optimization problems with equality and inequality constraints. An associated problem, having only inequality constraints, is defined, and shown to be equivalent to the original problem if a certain parameter is sufficiently large. The algorithms solve the associated problem, but incorporate a method for automatically increasing this parameter in order to ensure global convergence to a solution to the original problem. Any feasible directions algorithm can be similarly modified to enable it to handle equality constraints.
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    URL: http://dx.doi.org/10.1007/BF01580371
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  • 9
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    Superlinearly convergent variable metric algorithms for general nonlinear programming problems (1976)
    Springer
    Mathematical programming 11 (1976), S. 263-282 
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    ISSN: 1436-4646
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract In this paper sufficient conditions for local and superlinear convergence to a Kuhn—Tucker point are established for a class of algorithms which may be broadly defined and comprise a quadratic programming algorithm for repeated solution of a subproblem and a variable metric update to develop the Hessian in the subproblem. In particular the DFP update and an update attributed to Powell are shown to provide a superlinear convergent subclass of algorithms provided a start is made sufficiently close to the solution and the initial Hessian in the subproblem is sufficiently close to the Hessian of the Lagrangian at this point.
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    URL: http://dx.doi.org/10.1007/BF01580395
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  • 10
    Electronic Resource
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    The Mathematical Programming Society (1976)
    Springer
    Mathematical programming 11 (1976), S. 301-301 
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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/BF01580399
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  • 11
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    Bilinear programming: An exact algorithm (1977)
    Springer
    Mathematical programming 12 (1977), S. 173-194 
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    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.
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    URL: http://dx.doi.org/10.1007/BF01593787
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  • 12
    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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  • 13
    Electronic Resource
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    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.
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    URL: http://dx.doi.org/10.1007/BF01593779
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  • 14
    Electronic Resource
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    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
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    URL: http://dx.doi.org/10.1007/BF01593784
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  • 15
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    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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  • 16
    Electronic Resource
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    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
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    URL: http://dx.doi.org/10.1007/BF01593793
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  • 17
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    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
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    URL: http://dx.doi.org/10.1007/BF01593794
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  • 18
    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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  • 19
    Electronic Resource
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    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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  • 20
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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
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    URL: http://dx.doi.org/10.1007/BF01593809
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  • 21
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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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  • 22
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    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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  • 23
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    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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  • 24
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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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  • 25
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    Tenth International Symposium on Mathematical Programming (1979)
    Springer
    Mathematical programming 16 (1979), S. 261-261 
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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/BF01582113
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  • 26
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    On a class of least-element complementarity problems (1979)
    Springer
    Mathematical programming 16 (1979), S. 111-126 
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    ISSN: 1436-4646
    Keywords: Least Element ; Linear Complementarity ; Quadratic Programs ; Special Structure ; Applications ; Computational Experience
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract The present paper studies the linear complementarity problem of finding vectorsx andy inR + n such thatc + Dx + y ≧ 0,b − x ≧ 0 andx T (c + Dx + y) = y T (b − x) = 0 whereD is aZ-matrix andb 〉 0. Complementarity problems of this nature arise, for example, from the minimization of certain quadratic functions subject to upper and lower bounds on the variables. Two least-element characterizations of solutions to the above linear complementarity problem are established first. Next, a new and direct method to solve this class of problems, which depends on the idea of “least-element solution” is presented. Finally, applications and computational experience with its implementation are discussed.
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  • 27
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    Editorial policy (1979)
    Springer
    Mathematical programming 16 (1979), S. 139-139 
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    Source: Springer Online Journal Archives 1860-2000
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  • 28
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    An algorithm for a piecewise linear model of trade and production with negative prices and bankruptcy (1979)
    Springer
    Mathematical programming 16 (1979), S. 190-209 
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    ISSN: 1436-4646
    Keywords: Algorithm ; Piecewise Linear ; Complementary Pivot ; Equilibrium Model ; Trade ; Production Prices ; Linear Programming ; Fixed Point Methods
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract The general equilibrium model is approximated as a piecewise linear convex model and solved from the point of view of welfare economics using linear programming and fixed point methods.
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    URL: http://dx.doi.org/10.1007/BF01582108
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  • 29
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    Referees 1978 (1979)
    Springer
    Mathematical programming 16 (1979), S. 260-260 
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  • 30
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    The Mathematical Programming Society (1979)
    Springer
    Mathematical programming 16 (1979), S. 263-263 
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    URL: http://dx.doi.org/10.1007/BF01582115
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  • 31
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    On the solution of some (parametric) linear complementarity problems with applications to portfolio selection, structural engineering and actuarial graduation (1979)
    Springer
    Mathematical programming 16 (1979), S. 325-347 
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    ISSN: 1436-4646
    Keywords: Complementarity Problem ; Computational Results ; Structural Engineering ; Applications ; Portfolio Selection ; Actuarial Graduation
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract In this paper we discuss three applications of a class of (parametric) linear complementarity problems arising independently from such diverse areas as portfolio selection, structural engineering and actuarial graduation. After explaining how the complementarity problems emerge in these applications, we perform some analytical comparisons (based on operation counts and storage requirements) of several existing algorithms for solving this class of complementarity problems. We shall also present computational results to support the analytical comparisons. Finally, we deduce some conclusions about the general performance of these algorithms.
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  • 32
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    The 8th IFIP Conference on optimization techniques (1976)
    Springer
    Mathematical programming 11 (1976), S. 199-199 
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  • 33
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    Editor's note (1976)
    Springer
    Mathematical programming 11 (1976), S. 203-203 
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  • 34
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    An existence theorem for solutions tof(x) = 0 (1976)
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    Mathematical programming 11 (1976), S. 252-262 
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    Topics: Computer Science , Mathematics
    Notes: Abstract InR 1, if a continuous function has opposite signs at the endpoints of an interval, then the function has a zero in the interval. If the function has a nonvanishing derivative at a zero, then there is an interval such that the function has opposite signs at the endpoints. In this paper each of these results is extended toR n .
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    Fractional programming without differentiability (1976)
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    Notes: Abstract The notion of quasi-differentiability is examined and related to fractional programming. Necessary and sufficient conditions are given and various other properties of quasi-differentiable functions are discussed. Differentiability is not assumed.
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    Projections in minimax algebra (1976)
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    Mathematical programming 10 (1976), S. 111-123 
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    Notes: Abstract An axiomatic theory of linear operators can be constructed for abstract spaces defined over (R, ⊕, ⊗), that is over the (extended) real numbersR with the binary operationsx ⊕ y = max (x,y) andx ⊗ y = x + y. Many of the features of conventional linear operator theory can be reproduced in this theory, although the proof techniques are quite different. Specialisation of the theory to spaces ofn-tuples provides techniques for analysing a number of well-known operational research problems, whilst specialisation to function spaces provides a natural formal framework for certain familiar problems of approximation, optimisation and duality.
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    Council of the Mathematical Programming Society (1976)
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    Mathematical programming 10 (1976), S. 137-137 
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    Editorial policy (1976)
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    Mathematical programming 10 (1976), S. 144-144 
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    A set of geometric programming test problems and their solutions (1976)
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    Mathematical programming 10 (1976), S. 192-213 
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    Notes: Abstract This paper attempts to provide a set of standard test examples for researchers working in the area of geometric programming and general nonlinear, continuous, nonconvex programming algorithms. The examples consist partly of applications of nonlinear programming that have appeared in the literature and partly of original geometric programming applications. Solutions to all the problems are provided as well as the starting points from which these solutions were computed. Other computationally important aspects such as tolerances and degree of accuracy with which these problems were solved, are also included.
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    Analytical solution of A dynamic transaction flow problem (1976)
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    Mathematical programming 10 (1976), S. 214-229 
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    Notes: Abstract We analyze a mathematical programming model of a fractional flow process which consists ofn sectors and all possible time-dependent streams of flow between, into and out of the sectors. Assuming specific constraints on flow, least cost policies are determined for control of the system transactions involved over a given finite time horizon and over an infinite horizon. Several applications of the model are presented and discussed.
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    Quadratic duality over convex cone domains (1976)
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    Mathematical programming 10 (1976), S. 277-283 
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    On A characterization of optimality in convex programming (1976)
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    Mathematical programming 11 (1976), S. 81-88 
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    Notes: Abstract Necessary and sufficient conditions for optimality are given, for convex programming problems, without constraint qualification, in terms of a single mathematical program, which can be chosen to be bilinear.
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    A note on polyhedral sets having A least element (1976)
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    Mathematical programming 11 (1976), S. 94-96 
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    Finding the nearest point in A polytope (1976)
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    Mathematical programming 11 (1976), S. 128-149 
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    Notes: Abstract A terminating algorithm is developed for the problem of finding the point of smallest Euclidean norm in the convex hull of a given finite point set in Euclideann-space, or equivalently for finding an “optimal” hyperplane separating a given point from a given finite point set. Its efficiency and accuracy are investigated, and its extension to the separation of two sets and other convex programming problems described.
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    Transversal matroid intersections and related packings (1976)
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    Mathematical programming 11 (1976), S. 164-176 
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    Notes: Abstract Network flow techniques are used to show that a conjecture of Fulkerson on the blocking polyhedron associated with the intersection of two matroids is true for transversal matroids. This leads to some general integral packing results for partial transversals, bipartite edge matchings, and subpermutation matrices. A computational example for bipartite edge matchings is included.
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    Mathematical programming 11 (1976), S. 201-201 
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    A dual algorithm for the one-machine scheduling problem (1976)
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    Mathematical programming 11 (1976), S. 229-251 
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    Notes: Abstract A branch and bound algorithm is presented for the problem of schedulingn jobs on a single machine to minimize tardiness. The algorithm uses a dual problem to obtain a good feasible solution and an extremely sharp lower bound on the optimal objective value. To derive the dual problem we regard the single machine as imposing a constraint for each time period. A dual variable is associated with each of these constraints and used to form a Lagrangian problem in which the dualized constraints appear in the objective function. A lower bound is obtained by solving the Lagrangian problem with fixed multiplier values. The major theoretical result of the paper is an algorithm which solves the Lagrangian problem in a number of steps proportional to the product ofn 2 and the average job processing time. The search for multiplier values which maximize the lower bound leads to the formulation and optimization of the dual problem. The bounds obtained are so sharp that very little enumeration or computer time is required to solve even large problems. Computational experience with 20-, 30-, and 50-job problems is presented.
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    Bottleneck linear programming (1976)
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    Mathematical programming 11 (1976), S. 291-298 
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    Notes: Abstract The Bottleneck Linear Programming problem (BLP) is to maximizex 0 = max j c j ,x j 〉 0 subject toAx = b, x ⩾ 0. A relationship between the BLP and a problem solvable by a “greedy” algorithm is established. Two algorithms for the BLP are developed and computational experience is reported.
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    An algorithm for determining all extreme points of a convex polytope (1977)
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    Mathematical programming 12 (1977), S. 81-96 
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    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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    A monotone complementarity problem with feasible solutions but no complementary solutions (1977)
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    Mathematical programming 12 (1977), S. 131-132 
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    On some properties of L1 estimators (1977)
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    Mathematical programming 12 (1977), S. 139-140 
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    The Committee on Algorithms of the mathematical programming society: (1977)
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    Mathematical programming 12 (1977), S. 151-153 
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    The complementary unboundedness of dual feasible solution sets in convex programming (1977)
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    Mathematical programming 12 (1977), S. 392-405 
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    Keywords: Complementary Unboundedness ; Dual Feasible Solution Sets ; Convex Programming ; Geometric Programming ; Fenchel Duality ; Rockafellar Duality ; Ordinary Duality ; Quadratic Programming ; Optimal Location ; Traffic Equilibria
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    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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    Finite horizon approximations of infinite horizon linear programs (1977)
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    Mathematical programming 12 (1977), S. 1-17 
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    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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    Isotone solutions of parametric linear complementarity problems (1977)
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    Mathematical programming 12 (1977), S. 48-59 
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    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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    Optimality condition: Constraint regularization (1977)
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    Mathematical programming 13 (1977), S. 38-48 
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    Keywords: Constraint regularization ; Determinant cone ; Kuhn—Tucker constraint-qualification ; Lagrange regularity ; Optimality condition ; x 0-redundant constraints
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    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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    Ergodicity and symmetric mathematical programs (1977)
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    Mathematical programming 13 (1977), S. 81-87 
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    Keywords: Symmetric mathematical programs ; Ergodicity ; S-concavity and majorization ; Stochastic matrices ; Cyclic symmetry
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    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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    Call for papers: Conference on Combinatorial Optimization (1979)
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    Mathematical programming 16 (1979), S. 136-136 
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    A stable method for solving certain constrained least squares problems (1979)
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    Mathematical programming 16 (1979), S. 141-158 
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    Keywords: Constrained Least Squares Problems ; Positive Semidefinite Quadratic Programming ; Orthogonal Factorization ; Reorthogonalization
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    Notes: Abstract This paper presents a feasible descent algorithm for solving certain constrained least squares problems. These problems are specially structured quadratic programming problems with positive semidefinite Hessian matrices that are allowed to be singular. The algorithm generates a finite sequence of subproblems that are solved using the numerically stable technique of orthogonal factorization with reorthogonalization and Given's transformation updating.
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    The number of solutions of a class of linear complementarity problems (1979)
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    Mathematical programming 17 (1979), S. 104-105 
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    Keywords: Complementarity Problem ; Number of Solutions
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    Some remarks on condensation methods for geometric programs (1979)
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    Mathematical programming 17 (1979), S. 109-113 
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    Keywords: Geometric Programs ; Kuhn—Tucker Conditions ; Condensation Methods
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    Notes: Abstract The purpose of this paper is to present some points of clarification of a recently presented algorithm for geometric programs [7]. While presenting the clarification, we are able to identify the behavior of condensation type algorithms for generalized geometric programs.
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    The Mathematical Programming Society (1979)
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    Mathematical programming 17 (1979), S. 121-121 
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    A multiplier method with automatic limitation of penalty growth (1979)
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    Mathematical programming 17 (1979), S. 140-155 
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    Keywords: Multiplier Methods ; Adaptive Penalization ; Nonlinear Programming ; Quadratic Convergence
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    Notes: Abstract This paper presents a multiplier method for solving optimization problems with equality and inequality constraints. The method realizes all the good features that were foreseen by R. Fletcher for this type of algorithm in the past, but which suffers from none of the drawbacks of the earlier attempts.
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    Mathematical programming 17 (1979), S. 401-401 
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    Goal programming sensitivity analysis using interval penalty weights (1979)
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    Mathematical programming 17 (1979), S. 16-31 
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    Keywords: Goal Programming ; Sensitivity Analysis ; Vector-Maximum Algorithms ; Multiple Objective Linear Programming ; Multiple Criteria Decision Making
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    Notes: Abstract This paper presents an application of the vector-maximum research [4–8] to the sensitivity analysis of goal programming problems as several of the criterion function penalty weights are simultaneously and independently varied. A generalized goal programming capability is presented and a six-stage analytic procedure is described. The problem is generalized in the sense that the regular goal programming penalty weights can be expanded to intervals if desired. The solution procedure is new in that it depends upon an algorithm for the vector-maximum problem, “criterion cone” contraction procedures, and “filtering” techniques. Together they are able to generate and process all extreme points on the portion of the surface of the goal programming “augmented” feasible region corresponding to the interval penalty weights specified. In effect, the procedure and adapted algorithm of this paper delivers to goal programming an operational power of sensitivity analysis not previously available to users. A numerical example is provided in order to illustrate the computerized application of the total goal programming procedure outlined.
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    Mathematical programming 14 (1978), S. 127-127 
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    Algorithms for nonlinear constraints that use lagrangian functions (1978)
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    Mathematical programming 14 (1978), S. 224-248 
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    Keywords: Augmented Lagrangian ; Lagrangian Function ; Nonlinear Constraints ; Nonlinear Programming ; Optimization Algorithm
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    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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    The Mathematical Programming Society (1978)
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    Mathematical programming 14 (1978), S. 262-262 
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    The reformulation of two mixed integer programming problems (1978)
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    Mathematical programming 14 (1978), S. 325-331 
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    Keywords: Integer Programming ; Formulation ; Models
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    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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    Numerically stable methods for quadratic programming (1978)
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    Mathematical programming 14 (1978), S. 349-372 
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    Keywords: Indefinite Quadratic Programming ; Matrix Factorizations ; Numerical Software
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    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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    A mathematical programming system for preference and compatibility maximized menu planning and scheduling (1978)
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    Mathematical programming 15 (1978), S. 63-76 
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    Keywords: Menu Planning ; Separation Scheduling ; Menu Scheduling ; Decomposition ; Non-linear Programming ; Binary Knapsack Problem ; Lagrangian Relaxation ; Transportation Problem ; Branch and Bound
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    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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    On equivalent representations of certain multicommodity networks as single commodity flow problems (1978)
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    Mathematical programming 15 (1978), S. 92-99 
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    Keywords: Network Flows ; Multicommodity Networks
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    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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    Formation of the international association for statistical computing (1978)
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    Mathematical programming 15 (1978), S. 119-120 
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    A parametric linear complementarity problem involving derivatives (1978)
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    Mathematical programming 15 (1978), S. 146-154 
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    Keywords: Linear Complementarity Problem ; Principal Pivoting Algorithm
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    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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    A subgradient algorithm for certain minimax and minisum problems (1978)
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    Mathematical programming 15 (1978), S. 130-145 
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    Keywords: Minimax Problems ; Minisum Problems ; Nondifferentiable Optimization ; Subgradients ; Location Problems ; Linear Approximation Problems
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    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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    On the symmetric travelling salesman problem II: Lifting theorems and facets (1979)
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    Mathematical programming 16 (1979), S. 281-302 
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    Keywords: Linear Inequalities ; Convex Polytopes ; Facets ; Lifting Theorems ; Travelling Salesman Problem
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    Notes: Abstract Four lifting theorems are derived for the symmetric travelling salesman polytope. They provide constructions and state conditions under which a linear inequality which defines a facet of then-city travelling salesman polytope retains its facetial property for the (n + m)-city travelling salesman polytope, wherem ≥ 1 is an arbitrary integer. In particular, they permit a proof that all subtour-elimination as well as comb inequalities define facets of the convex hull of tours of then-city travelling salesman problem, wheren is an arbitrary integer.
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    The Mathematical Programming Society (1979)
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    Mathematical programming 16 (1979), S. 388-388 
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    A construction method in parametric programming (1979)
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    Mathematical programming 16 (1979), S. 21-36 
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    Keywords: Parametric Linear Programming ; Construction Method ; Degeneracy ; Primal Lexicographic Method
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    Notes: Abstract We consider a linear programming problem, with two parameters in the objective function, and present an algorithm for finding the decomposition of the parameter space into maximal polyhedral areas in which particular basic solutions are optimal. Special attention is paid to fill up areas of degenerate solutions.
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    A computational study of active set strategies in nonlinear programming with linear constraints (1979)
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    Mathematical programming 16 (1979), S. 81-97 
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    Keywords: Nonlinear Programming ; Linear Constraints ; Computational Experience
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    Notes: Abstract An essential part of many iterative methods for linearly constrained nonlinear programming problems is a procedure for determining those inequality constraints which will be “active” (that is, satisfied as equalities) at each iteration. We discuss experiments in which we used several strategies for identifying active constraints in conjunction with two well-known algorithms for linearly constrained optimization. The results indicate that in most cases a strategy which keeps the number of constraints in the active set as small as possible is computationally most efficient.
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    A direct dual method for the mixed plant location problem with some side constraints (1979)
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    Mathematical programming 17 (1979), S. 198-228 
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    Keywords: Plant Location ; Dual Method ; Orthogonality Conditions ; Enumeration ; Computational Results
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    Notes: Abstract The mixed plant location problem (mixed in the sense of allowing capacitated as well as uncapacitated plants) is a difficult and important mixed integer problem. We give a direct dual method, consisting of several phases (each of which appears essential for some data), to resolve a strong relaxed form of the problem with additional constraints over the integer variables (user specified, or derived from the data themselves). When all features of the algorithm are employed, there appears to be no difficulty with problems of 100 plants, even in an inefficient computer implementation. The primal solutions which we derive from the orthogonality conditions and a simple greedy heuristic are almost always much better than those we obtain from a standard relaxed problem in the Lagrangean sense. With an enumeration code in an efficient implementation we would expect to be capable of resolving very large problems (of perhaps up to 500 or 1000 plants) to within practically well acceptable tolerances.
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    Mathematical programming study 10 (1979)
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    Mathematical programming 17 (1979), S. 115-118 
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    Optimization: Techniques and Applications (1979)
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    Mathematical programming 17 (1979), S. 122-122 
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    On perfect graphs and polyhedra with (0, 1)-valued extreme points (1979)
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    Mathematical programming 17 (1979), S. 239-242 
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    Keywords: Anti-Blocking ; Perfect Graphs ; Polyhedra
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    Notes: Abstract LetC andA be (0, 1)-valued matrices with no zero columns. Fulkerson has shown that the extreme points of {x: Cx ≤ 1,x ≥ 0} are given by the rows ofA and their projections and the extreme points of {x: Ax ≤ 1,x ≥ 0} are given by the rows ofC and their projections if and only if the maximal rows ofC andA are the incidence vectors of maximal cliques and anticliques, respectively, of a perfect graph. This theorem is discussed and a new proof is given for the “only if” implication.
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    Exact penalty functions in nonlinear programming (1979)
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    Mathematical programming 17 (1979), S. 251-269 
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    Keywords: Nonlinear Programming ; Penalty Functions ; Exact Penalty Functions ; Constraint Qualification ; Second Order Optimality Conditions
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    Notes: Abstract It is shown that the existence of a strict local minimum satisfying the constraint qualification of [16] or McCormick's [12] second order sufficient optimality condition implies the existence of a class of exact local penalty functions (that is ones with a finite value of the penalty parameter) for a nonlinear programming problem. A lower bound to the penalty parameter is given by a norm of the optimal Lagrange multipliers which is dual to the norm used in the penalty function.
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    Computational complexity of LCPs associated with positive definite symmetric matrices (1979)
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    Mathematical programming 17 (1979), S. 335-344 
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    Keywords: Linear Complementarity Problem ; Complementary Cones ; Complementary Pivot Methods ; Computational Complexity ; Exponential Growth
    Source: Springer Online Journal Archives 1860-2000
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    Notes: Abstract Murty in a recent paper has shown that the computational effort required to solve a linear complementarity problem (LCP), by either of the two well known complementary pivot methods is not bounded above by a polynomial in the size of the problem. In that paper, by constructing a class of LCPs—one of ordern forn ≥ 2—he has shown that to solve the problem of ordern, either of the two methods goes through 2 n pivot steps before termination. However that paper leaves it as an open question to show whether or not the same property holds if the matrix,M, in the LCP is positive definite and symmetric. The class of LCPs in whichM is positive definite and symmetric is of particular interest because of the special structure of the problems, and also because they appear in many practical applications. In this paper, we study the computational growth of each of the two methods to solve the LCP, (q, M), whenM is positive definite and symmetric and obtain similar results.
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    Call for papers: Second Symposium on Mathematical Programming with Data Perturbations (1979)
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    Mathematical programming 17 (1979), S. 398-399 
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    The Mathematical Programming Society (1979)
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    Mathematical programming 17 (1979), S. 402-402 
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    The computation of Lagrange-multiplier estimates for constrained minimization (1979)
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    Mathematical programming 17 (1979), S. 32-60 
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    Keywords: Lagrange Multipliers ; Linearly-Constrained Optimization ; Augmented Lagrangian Functions ; Projected Lagrangian Methods ; Quadratic Sub Problems
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    Notes: Abstract Almost all efficient algorithms for constrained optimization require the repeated computation of Lagrange-multiplier estimates. In this paper we consider the difficulties in providing accurate estimates and what tests can be made in order to check the validity of the estimates obtained. A variety of formulae for the estimation of Lagrange multipliers are derived and their respective merits discussed. Finally the role of Lagrange multipliers within optimization algorithms is discussed and in addition to other results, it is shown that some algorithms are particularly sensitive to errors in the estimates.
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    Bicentennial conference on Mathematical Programming (1976)
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    Mathematical programming 10 (1976), S. 284-284 
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    Editorial policy (1976)
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    Mathematical programming 10 (1976), S. 286-286 
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    An algorithm for nonconvex programming problems (1976)
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    Mathematical programming 10 (1976), S. 312-321 
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    Notes: Abstract Branch and bound approaches for nonconvex programming problems had been given in [1] and [4]. Crucial for both are the use of rectangular partitions, convex envelopes and separable nonconvex portions of the objective function and constraints. We want to propose a similar algorithm which solves a sequence of problems in each of which the objective function is convex or even linear. The main difference between this approach and previous approaches is the use of general compact partitions instead of rectangular ones and a different refining rule such that the algorithm does not rely on the concept of convex envelopes and handles non-separable functions. First we describe a general algorithm and prove a convergence theorem under suitable regularity assumptions. Then we give as example an algorithm for concave minimization problems.
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    Quasi balanced matrices — An addendum (1976)
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    Mathematical programming 10 (1976), S. 405-407 
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    The Mathematical Programming Society (1976)
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    Mathematical programming 10 (1976), S. 413-413 
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    Optimal conditioning of self-scaling variable Metric algorithms (1976)
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    Mathematical programming 10 (1976), S. 70-90 
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    Notes: Abstract Variable Metric Methods are “Newton—Raphson-like” algorithms for unconstrained minimization in which the inverse Hessian is replaced by an approximation, inferred from previous gradients and updated at each iteration. During the past decade various approaches have been used to derive general classes of such algorithms having the common properties of being Conjugate Directions methods and having “quadratic termination”. Observed differences in actual performance of such methods motivated recent attempts to identify variable metric algorithms having additional properties that may be significant in practical situations (e.g. nonquadratic functions, inaccurate linesearch, etc.). The SSVM algorithms, introduced by this first author, are such methods that among their other properties, they automatically compensate for poor scaling of the objective function. This paper presents some new theoretical results identifying a subclass of SSVM algorithms that have the additional property of minimizing a sharp bound on the condition number of the inverse Hessian approximation at each iteration. Reducing this condition number is important for decreasing the roundoff error. The theoretical properties of this subclass are explored and two of its special cases are tested numerically in comparison with other SSVM algorithms.
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    A cutting plane algorithm for solving bilinear programs (1976)
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    Mathematical programming 11 (1976), S. 14-27 
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    Notes: Abstract This paper addresses itself to a special class of nonconvex quadratic program referred to as a bilinear program in the literature. We will propose here a cutting plane algorithm to solve this class of problems. The algorithm is along the lines of H. Tui and K. Ritter, but it differs in its exploitation of the special structure of the problem. Though the algorithm is not guaranteed at this stage of the research to converge to a global optimum, the preliminary results of numerical experiments are encouraging.
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    Approximate cutting planes in nonlinear programming (1976)
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    Mathematical programming 11 (1976), S. 194-198 
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    Notes: Abstract An approximation criterion is presented and a convergence proof is given that allows adaptation of the cutting plane algorithm when it is impractical to compute the exact cutting planes. A source application is briefly described and a class of functions is presented for which such approximate cutting planes can be conveniently computed.
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    The Mathematical Programming Society (1976)
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    Mathematical programming 11 (1976), S. 200-200 
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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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