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Measure concentration in optimization

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Abstract

We discuss some consequences of the measure concentration phenomenon for optimization and computational problems. Topics include average case analysis in optimization, efficient approximate counting, computation of mixed discriminants and permanents, and semidefinite relaxation in quadratic programming.

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References

  1. A.I. Barvinok, Problems of distance geometry and convex properties of quadratic maps,Discrete & Computational Geometry, 13 (2) (1995) 189–202.

    Article  MATH  MathSciNet  Google Scholar 

  2. A.I. Barvinok, Integral geometry of higher-dimensional polytopes and the average case in combinatorial optimization, in:Proceedings of the 36th Annual Symposium on Foundations of Computer Science (FOCS’95) (IEEE Computer Society Press, 1995) 275–283.

  3. A.I. Barvinok, Two algorithmic results for the Traveling Salesman Problem,Mathematics of Operations Research 21 (1996) 65–84.

    Article  MATH  MathSciNet  Google Scholar 

  4. A.I. Barvinok, Computing mixed discriminants, mixed volumes and permanents,Discrete and Computational Geometry, to appear.

  5. A.I. Barvinok, Approximate counting via random optimization,Random Structures & Algorithms, to appear.

  6. A.I. Barvinok and R. Robb, On the mean radius of permutation polytopes, Submitted.

  7. C. Borell, The Brunn-Minkowski inequality in Gauss space,Inventiones Mathematicae 30 (2) (1975) 207–216.

    Article  MATH  MathSciNet  Google Scholar 

  8. V.A. Emelichev, M.M. Kovalev and M.K. Kravtsov,Polytopes, Graphs and Optimization (Cambridge University Press, New York, 1984).

    Google Scholar 

  9. M.X. Goemans and D.P. Williamson, Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming.J. Assoc. Comput. Mach. 42 (6) (1995) 1115–1145.

    MATH  MathSciNet  Google Scholar 

  10. O. Goldreich, Probabilistic proof systems, in:Proceedings of the International Congress of Mathematicians, Zürich, Switzerland, 1994, Vol. 2 (Birkhäuser, Basel, 1995) 1395–1406.

    Google Scholar 

  11. D.Yu. Grigoriev and M. Karpinski, The matching problem for bipartite graphs with polynomially bounded permanents is in NC,Proceedings of the 28th Annual IEEE Symposium Foundations of Computer Science (IEEE Computer Society Press, Washington, DC, 1987) 162–172.

    Google Scholar 

  12. L.H. Harper, Optimal numberings and isoperimetric problems on graphs,Journal of Combinatorial Theory 1 (1966) 385–393.

    MATH  MathSciNet  Google Scholar 

  13. M. Jerrum and A. Sinclair, The Markov chain Monte Carlo method: An approach to approximate counting and integration, in: D.S. Hochbaum, ed.,Approximation Algorithms for NP-hard Problems (PWS Publishing, Boston, MA, 1996) 482–520.

    Google Scholar 

  14. E.L. Lawler, J.K. Lenstra, A.H.G. Rinnoy Kan and D.B. Shmoys, eds.,The Traveling Salesman Problem. A Guided Tour of Combinatorial Optimization (Wiley-Interscience, Chichester, 1990).

    Google Scholar 

  15. K. Leichtweiß, Convexity and Differential Geometry, in: P.M. Gruber and J.M. Wills, eds.,Handbook of Convex Geometry, Vol. B (North-Holland, Amsterdam, 1993) Chapter 4.1, 1045–1080.

    Google Scholar 

  16. L. Lovász and M.D. Plummer,Matching Theory (North-Holland, Amsterdam and Akadémiai Kiadó, Budapest, 1986).

    MATH  Google Scholar 

  17. A. Megretskii, Personal communication, 1995.

  18. V.D. Milman and G. Schechtman,Asymptotic Theory of Finite Dimensional Normed Spaces, Lecture Notes in Mathematics, Vol. 1200 (Springer, Berlin, 1986).

    Google Scholar 

  19. Yu. Nesterov and A. Nemirovskii,Interior-Point Polynomial Algorithms in Convex Programming, SIAM Studies in Applied Mathematics, Vol. 3 (SIAM, Philadelphia, PA, 1994).

    Google Scholar 

  20. C.H. Papadimitriou and K. Steiglitz,Combinatorial Optimization: Algorithms and Complexity (Prentice-Hall, Englewood Cliffs, NJ, 1982).

    MATH  Google Scholar 

  21. G. Pisier,The Volume of Convex Bodies and Banach Space Geometry, Cambridge Tracts in Mathematics, Vol. 94 (Cambridge University Press, Cambridge, 1989).

    Google Scholar 

  22. L.E. Rasmussen, Approximating the permanent: A simple approach,Random Structures & Algorithms 5 (1994) 349–361.

    MATH  MathSciNet  Google Scholar 

  23. M.J. Steele,Probability Theory and Combinatorial Optimization, CBMS-NSF Regional Conference Series in Applied Mathematics, Vol. 69 (SIAM, Philadelphia, PA, 1997).

    Google Scholar 

  24. M. Talagrand, Concentration of measure and isoperimetric inequalities in product spaces,Institut des Hautes Études Scientifiques. Publications Mathématiques 81 (1995) 73–205.

    MATH  MathSciNet  Google Scholar 

Download references

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Authors and Affiliations

  1. Department of Mathematics, University of Michigan, 48109-1109, Ann Arbor, MI, USA

    Alexander Barvinok

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  1. Alexander Barvinok
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Barvinok, A. Measure concentration in optimization. Mathematical Programming 79, 33–53 (1997). https://doi.org/10.1007/BF02614310

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  • DOI: https://doi.org/10.1007/BF02614310

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