Michael Held’s scientific contributions

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Publications (3)


Large scale optimization and the relaxation method
  • Article

January 1972

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11 Reads

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11 Citations

Michael Held

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Richard M. Karp

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Philip Wolfe

An algorithm is given for the maximization of any function of many variables which may be described as the envelope of a family of linear functions. It is shown how the large-scale problems of linear programming to which various decomposition schemes apply can be posed in this way. Computational experience with some of these problems is reviewed.



The traveling-salesman problem and minimum spanning trees: Part II

December 1971

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344 Reads

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1,099 Citations

Mathematical Programming

The relationship between the symmetric traveling-salesman problem and the minimum spanning tree problem yields a sharp lower bound on the cost of an optimum tour. An efficient iterative method for approximating this bound closely from below is presented. A branch-and-bound procedure based upon these considerations has easily produced proven optimum solutions to all traveling-salesman problems presented to it, ranging in size up to sixty-four cities. The bounds used are so sharp that the search trees are minuscule compared to those normally encountered in combinatorial problems of this type.

Citations (3)


... The following section contains a brief discussion of subgradient relaxation methods for solving the dual problem which have been studied by Held and Karp [23] and Held, Karp and Wolfe [24]. ...

Reference:

Constructive duality for discrete optimization
Large scale optimization and the relaxation method
  • Citing Conference Paper
  • January 1972

... En effet, certains problèmes peuvent contenir des sous-problèmes connus avec des algorithmes efficaces pour les résoudre. Une des techniques utiles dans ces situations est de recourir à la relaxation lagrangienne [Ree93], [HKW72]. Soit le programme linéaire en nombre entiers suivant, ...

Large scale optimization and the relaxation method
  • Citing Article
  • January 1972

... As shown in Fig. 2, the layout information in matrix form can be converted into a solution vector and solved as a TSP. Karp (1971) proved that the TSP is an NP-hard problem, and therefore, the problem addressed in this paper can also be understood as an NP-hard problem [38,39]. Figure 2 shows an example of converting layout information into solution vector. ...

The traveling-salesman problem and minimum spanning trees: Part II
  • Citing Article
  • December 1971

Mathematical Programming