Abstract
A powerful approach to solve a combinatorial optimization problem is to embed the discrete space of admissible points in a continuous space and apply classical optimization techniques such as gradient descent to the embedded problem. The method of recasting certain combinatorial optimization problems as linear programming problems and solving them by some interior solution technique, such as the Karmarkar algorithm [l], can be viewed as an application of this idea. In [2] and [3], some gradient flows arising from geometrical matching problems in computer vision are investigated. The continuous space used in that approach is the special orthogonal group SO(n), the group of orthogonal matrices with determinant 1. A more classical approach is to embed the permutation matrices in the space of doubly stochastic matrices. The benefits, in this case, derive from the well known characterization of Birkhoff: the permutation matrices are the extreme points of the set of doubly stochastic matrices and the set of doubly stochastic matrices is the convex hull of the set of permutation matrices [4]. Of course, SO(n) is not a polytope and has no vertices. It is, however, a group containing the set of permutation matrices with determinant 1 as a subset and even subgroup. It is also a compact manifold without boundary and gradient flows on it always have at least one stable stationary point. Instead of doubly stochastic matrices it is orthostochastic matrices, that is, matrices of the form (θ 2 ij ) with (θ ij ) orthogonal, which play the main role.
Keywords
- Assignment Problem
- Travel Salesman Problem
- Combinatorial Optimization Problem
- Permutation Matrix
- Gradient Flow
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.
This work was supported in part by the US Army Research Office under grant DAAL03-86-K-0171 and DARPA grant AFOSR-89-0506.
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References
Karmarkar, N, A New Polynomial-Time Algorithm for Linear Programming, Combinatorica 4(4) (1984), 373–395.
Brockett, R. W., Dynamical Systems that Sort Lists and Solve Linear Programming Problems, Proc. 27th IEEE Conference on Decision and Control, (1988) 799–803.
Brockett, R. W., A Geometrical Matching Problem, J. of Linear Algebra and Its Applications 122 (1989), 761–777.
Marshall, A. W., and I. Olkin, Inequalities: Theory of Majorization and Its Applications, Academic Press, Orlando, Florida, (1979).
Spivey, W. A., and R. M. Thrall, Linear Optimization, Holt, Rinehart and Winston, New York, NY, (1970).
Burkard, R. E., “Traveling Salesman and Assignment Problems: A Survey,” in Discrete Optimization I, North-Holland, Amsterdam, (1979).
Papadimitriou, C. H., and K. Steiglitz, Combinatorial Optimization: Algorithms and Complexity, Prentice-Hall, Englewood Cliffs, New Jersey, (1982).
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© 1991 Springer Science+Business Media New York
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Brockett, R.W., Wong, W.S. (1991). A Gradient Flow for the Assignment Problem. In: New Trends in Systems Theory. Progress in Systems and Control Theory, vol 7. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0439-8_20
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DOI: https://doi.org/10.1007/978-1-4612-0439-8_20
Publisher Name: Birkhäuser, Boston, MA
Print ISBN: 978-1-4612-6760-7
Online ISBN: 978-1-4612-0439-8
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