## Abstract

The “relaxation” procedure introduced by Held and Karp for approximately solving a large linear programming problem related to the traveling-salesman problem is refined and studied experimentally on several classes of specially structured large-scale linear programming problems, and results on the use of the procedure for obtaining exact solutions are given. It is concluded that the method shows promise for large-scale linear programming

## Keywords

Exact Solution Mathematical Method Programming Problem Linear Programming Problem Subgradient Optimization
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Preview

Unable to display preview. Download preview PDF.

## References

- [1]J. Abadie and M. Sakarovitch, “Two methods of decomposition for linear programming”, in:
*Proceedings of the Princeton symposium on mathematical programming*Ed. H.W. Kuhn (Princeton University Press, Princeton, N.J., 1970) pp 1–23.Google Scholar - [2]S. Agmon, “The relaxation method for linear inequalities”,
*Canadian Journal of Mathematics*6 (1954) 382–392.Google Scholar - [3]D.P. Bertsekas and S.K. Mitter, “Steepest descent for optimization problems with nondifferentiable cost functionals”, in:
*Proceedings of the 5*^{th}*annual Princeton conference on information sciences and systems*, 1971.Google Scholar - [4]G.B. Dantzig, D.R. Fulkerson and S.M. Johnson, “Solution of a large-scale traveling-salesman problem”,
*Operations Research*2 (1954) 393–410.Google Scholar - [5]V.F. Dem'janov, “Seeking a minimax on a bounded set”,
*Soviet Mathematics Doklady*11 (1970) 517–521. [Translation of:*Doklady Akademii Nauk SSSR*191 (1970).]Google Scholar - [6]M.L. Fisher and J.F. Shapiro, “Constructive duality in integer programming”, Working Paper OR 008-72, Operations Research Center, Massachusetts Institute of Technology, Cambridge, Mass. (April, 1972).Google Scholar
- [7]L.R. Ford, Jr. and D.R. Fulkerson,
*Flows in networks*(Princeton University Press, Princeton, N.J., 1962).Google Scholar - [8]A.M. Geoffrion, “Elements of large-scale mathematical programming”,
*Management Science*16 (1970) 652–691.Google Scholar - [9]R.C. Grinold, “Steepest ascent for large-scale linear programs”,
*SIAM Review*14 (1972) 447–464.Google Scholar - [10]M. Held and R.M. Karp, “The traveling-salesman problem and minimum spanning trees”,
*Operations Research*18 (1970) 1138–1162.Google Scholar - [11]M. Held and R.M. Karp, “The traveling-salesman problem and minimum spanning trees: part II”,
*Mathematical Programming*1 (1971) 6–25.Google Scholar - [12]M. Held and R.M. Karp, “A dynamic programming approach to sequencing problems”,
*Journal of the Society for Industrial and Applied Mathematics*10 (1962) 196–210.Google Scholar - [13]L.L. Karg and G.L. Thompson, “A heuristic approach to solving traveling-salesman problems”,
*Management Science*10 (1964) 225–248.Google Scholar - [14]H.W. Kuhn and A.W. Tucker, “Nonlinear programming”, in:
*Proceedings of the second Berkeley symposium on mathematical statistics and probability*Ed. J. Neyman (University of California Press, Berkeley, Calif., 1951) pp. 481–492.Google Scholar - [15]L.S. Lasdon,
*Optimization theory for large systems*(Macmillan, London, 1970).Google Scholar - [16]R.E. Marsten and J.W. Blankenship, “Boxstep: a new strategy for Lagrangian decomposition”, Department of Industrial Engineering and Management Sciences, Northwestern University, Evanston, Ill. (March, 1973).Google Scholar
- [17]T. Motzkin and I.J. Schoenberg, “The relaxation method for linear inequalities”,
*Canadian Journal of Mathematics*6 (1954) 393–404.Google Scholar - [18]J. von Neumann, “A certain zero-sum two-person game equivalent to the optimal assignment problem”, in:
*Contributions to the theory of games*, Vol. II, Eds. H.W. Kuhn and A.W. Tucker, Annals of Mathematics Study No. 28 (Princeton University Press, Princeton, N.J., 1953).Google Scholar - [19]W. Oettli, “An iterative method, having linear rate of convergence, for solving a pair of dual linear programs”,
*Mathematical Programming*3 (1972) 302–311.Google Scholar - [20]B.T. Poljak, “A general method of solving extremum problems”, Soviet Mathematics Doklady 8 (1967) 593–597. [Translation of
*Doklady Akademii Nauk SSSR*174 (1967).]Google Scholar - [21]B.T. Poljak, “Minimization of unsmooth functionals”,
*U.S.S.R. Computational Mathematics and Mathematical Physics*14–29. [Translation of: Žurnal Vyčislitel'no\(\mathop i\limits^ \vee \) Matematiki i Matematičesko\(\mathop i\limits^ \vee \) Fiziki 9 (1969) 509–521.]Google Scholar - [22]R.T. Rockafellar,
*Convex analysis*(Princeton University Press, Princeton, N.J., 1970).Google Scholar - [23]N.Z. Shor, “On the structure of algorithms for the numerical solution of optimal planning and design problems”, Dissertation, Cybernetics Institute, Academy of Sciences U.S.S.R. (1964).Google Scholar
- [24]P. Wolfe, M. Held and R.M. Karp, “Large-scale optimization and the relaxation method”, in:
*Proceedings of the 25*^{th}*national ACM meeting*, Boston, Mass. (August 1972).Google Scholar

## Copyright information

© The Mathematical Programming Society 1974