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  • 1
    Electronic Resource
    Electronic Resource
    Springer
    Mathematical programming 35 (1986), S. 110-119 
    ISSN: 1436-4646
    Keywords: Convex programming ; Frank-Wolfe algorithm
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract We give a detailed proof, under slightly weaker conditions on the objective function, that a modified Frank-Wolfe algorithm based on Wolfe's ‘away step’ strategy can achieve geometric convergence, provided a strict complementarity assumption holds.
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  • 2
    Electronic Resource
    Electronic Resource
    Springer
    Mathematical programming 35 (1986), S. 125-125 
    ISSN: 1436-4646
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
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  • 3
    Electronic Resource
    Electronic Resource
    Springer
    Mathematical programming 35 (1986), S. 140-172 
    ISSN: 1436-4646
    Keywords: Probabilistic analysis ; self-dual simplex ; spherical symmetry
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract In this paper we analyze the average number of steps performed by the self-dual simplex algorithm for linear programming, under the probabilistic model of spherical symmetry. The model was proposed by Smale. Consider a problem ofn variables withm constraints. Smale established that for every number of constraintsm, there is a constantc(m) such that the number of pivot steps of the self-dual algorithm,ρ(m, n), is less thanc(m)(lnn) m(m+1) . We improve upon this estimate by showing thatρ(m, n) is bounded by a function ofm only. The symmetry of the function inm andn implies thatρ(m, n) is in fact bounded by a function of the smaller ofm andn.
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  • 4
    Electronic Resource
    Electronic Resource
    Springer
    Mathematical programming 35 (1986), S. 17-31 
    ISSN: 1436-4646
    Keywords: Linear complementarity problem ; K-matrix ; Q 0-matrix ; finite characterization ; Q-matrix
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract The class of realn × n matricesM, known asK-matrices, for which the linear complementarity problemw − Mz = q, w ≥ 0, z ≥ 0, w T z =0 has a solution wheneverw − Mz =q, w ≥ 0, z ≥ 0 has a solution is characterized for dimensionsn 〈4. The characterization is finite and ‘practical’. Several necessary conditions, sufficient conditions, and counterexamples pertaining toK-matrices are also given. A finite characterization of completelyK-matrices (K-matrices all of whose principal submatrices are alsoK-matrices) is proved for dimensions 〈4.
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  • 5
    Electronic Resource
    Electronic Resource
    Springer
    Mathematical programming 34 (1986), S. 1-33 
    ISSN: 1436-4646
    Keywords: Polyhedra ; Chinese Postman ; Binary Group Problems ; Binary Matroids ; Blocking Clutters
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract A new proof of the characterization of the Chinese postman polyhedra is given. In developing this proof, a theorem of Gomory about homomorphic lifting of facets for group polyhedra is generalized to subproblems. Some results for the Chinese postman problem are generalized to binary group problems. In addition, a connection is made between Fulkerson's blocking polyhedra and a blocking pair of binary group problems. A connection is also developed between minors and lifting of facets for group problems.
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  • 6
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    Springer
    Mathematical programming 34 (1986), S. 48-61 
    ISSN: 1436-4646
    Keywords: Total Dual Integrality ; Hilbert Basis ; Polyhedra
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract Edmonds and Giles introduced the class of box totally dual integral polyhedra as a generalization of submodular flow polyhedra. In this paper a geometric characterization of these polyhedra is given. This geometric result is used to show that each TDI defining system for a box TDI polyhedron is in fact a box TDI system, that the class of box TDI polyhedra is in co-NP and is closed under taking projections and dominants, that the class of box perfect graphs is in co-NP, and a result of Edmonds and Giles which is related to the facets of box TDI polyhdera.
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  • 7
    Electronic Resource
    Electronic Resource
    Springer
    Mathematical programming 34 (1986), S. 72-83 
    ISSN: 1436-4646
    Keywords: Cutting Planes ; Disjunctive Programming
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract The duality between facets of the convex hull of disjunctive sets and the extreme points of reverse polars of these sets is utilized to establish simple rules for the derivation of all facet cuts for simple disjunctions, namely, elementary disjunctions in nonnegative variables. These rules generalize the cut generation procedure underlying polyhedral convexity cuts with negative edge extensions. The latter are also shown to possess some interesting properties with respect to a biextremal problem that maximizes the distance, from the origin, of the nearest point feasible to the cut. A computationally inexpensive procedure is given to generate facet cuts for simple disjunctions which are dominant with respect to any specified preemptive ordering of variables.
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  • 8
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    Electronic Resource
    Springer
    Mathematical programming 34 (1986), S. 111-124 
    ISSN: 1436-4646
    Keywords: Global Optimization ; Stochastic Optimization ; Annealing ; Metropolis Method ; NP ; Hill Climbing ; Local Improvement
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract The annealing algorithm is a stochastic optimization method which has attracted attention because of its success with certain difficult problems, including NP-hard combinatorial problems such as the travelling salesman, Steiner trees and others. There is an appealing physical analogy for its operation, but a more formal model seems desirable. In this paper we present such a model and prove that the algorithm converges with probability arbitrarily close to 1. We also show that there are cases where convergence takes exponentially long—that is, it is no better than a deterministic method. We study how the convergence rate is affected by the form of the problem. Finally we describe a version of the algorithm that terminates in polynomial time and allows a good deal of ‘practical’ confidence in the solution.
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  • 9
    Electronic Resource
    Electronic Resource
    Springer
    Mathematical programming 34 (1986), S. 125-141 
    ISSN: 1436-4646
    Keywords: Assignment Problem ; Dual Method ; Signature ; Linear Programming ; Simplex Method ; Pivoting ; Average Behavior
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract “Where there is abundance of mystery and confusion in every direction, the truth seldom remains hidden for long. It's a matter of having plenty of angles to go at it from. Only the utterly simple crimes - the simplex crimes, you may say - have the trick of remaining baffling.” - Sir John (from Michael Innes,The Open House (A Sir John Appleby Mystery), Penguin Books, 1974). A dual simplex method for the assignment problem leaves open to choice the activity (i,j) of rowi and columnj that is to be dropped in pivoting so long asx ij 〈 0. A choice (i,j) over columnsj having at least 3 basic activities that minimizesx ij is shown to converge in at most ( 2 n-1 ) pivots, and at most O(n 3) time, and it is argued that on average the number of pivots is at mostn logn.
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  • 10
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    Mathematical programming 34 (1986), S. 142-162 
    ISSN: 1436-4646
    Keywords: Network Design ; Bilevel Programming ; Variational Inequalities ; Stackelberg Games
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract Recently much attention has been focused on multilevel programming, a branch of mathematical programming that can be viewed either as a generalization of min-max problems or as a particular class of Stackelberg games with continuous variables. The network design problem with continuous decision variables representing link capacities can be cast into such a framework. We first give a formal description of the problem and then develop various suboptimal procedures to solve it. Worst-case behaviour results concerning the heuristics, as well as numerical results on a small network, are presented.
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