Algorithms for Global Optimization and Discrete Problems Based on Methods for Local Optimization

  • Walter Murray
  • Kien-Ming Ng
Part of the Nonconvex Optimization and Its Applications book series (NOIA, volume 62)


One of the most challenging optimization problems is determining the minimizer of a nonlinear nonconvex problem in which there are some discrete variables. Any such problem may be transformed to that of finding a global optimum of a problem in continuous variables. However, such transformed problems have astronomically large numbers of local minimizers, making them harder to solve than typical global optimization problems. Despite this apparent disadvantage we show the approach is not hopeless.

Since the technique requires finding a global minimizer we review the approaches to solving such problems. Our interest is in problems for differentiable functions, and our focus is on algorithms that utilize the large body of work available on finding local minimizers of smooth functions. The method we advocate convexifies the problem and uses an homotopy approach to find the required solution. To illustrate how well the new algorithm performs we apply it to a hard frequency assignment problem, and to binary quadratic problems taken from the literature.


Global Optimization Local Optimization Global Optimization Problem Space Filling Curve Local Smoothing 
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.


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  1. Anderssen, R.S. and Bloomfield, P. (1975). Properties of the random search in global optimization. Journal of Optimization Theory and Applications, 16: 383–398.MathSciNetMATHCrossRefGoogle Scholar
  2. Beasley, J.E. (1998). Heuristic algorithms for the unconstrained binary quadratic programming problem. Scholar
  3. Behrman, W. (1998). An efficient gradient flow method for unconstrained optimization. PhD thesis, Scientific Computing and ComputationalGoogle Scholar
  4. Mathematics Department, Stanford University, Stanford, California. Betrò, B. and Rotondi, R. (1984). A bayesian algorithm for global opti-mization. Annals of Operations Research, 1: 111–128.CrossRefGoogle Scholar
  5. Boman, E.G. (1999). Infeasibility and negative curvature in optimization. PhD thesis, Scientific Computing and Computational Mathematics Department, Stanford University, Stanford, California.Google Scholar
  6. Branin, F.H. (1972). Widely convergent methods for finding multiple solutions of simultaneous nonlinear equations. IBM Journal of Research and Development, 16: 504–522.MathSciNetMATHCrossRefGoogle Scholar
  7. Werra, D. and Gay, Y. (1994). Chromatic scheduling and frequency assignment. Discrete Applied Mathematics, 49: 165–174.MathSciNetMATHCrossRefGoogle Scholar
  8. Gatto, A. (2000). A subspace method based on a differential equation approach to solve unconstrained optimization problems. PhD thesis, Management Science and Engineering Department, Stanford University, Stanford, California.Google Scholar
  9. Dembo, R.S. and Steihaug, T. (1983). Truncated-Newton algorithms for large-scale unconstrained optimization. Mathematical Programming, 26: 190–212.MathSciNetMATHCrossRefGoogle Scholar
  10. Dorstenstein, T. (2001). Constructive and exchange algorithms for the frequency assignment problem. PhD thesis, Management Science and Engineering Department, Stanford University, Stanford, California. Scholar
  11. Fiacco, A.V. and McCormick, G.P. (1968). Nonlinear Programming: Sequential Unconstrained Minimization Techniques. John Wiley and Sons, New York and Toronto.Google Scholar
  12. Forsgren, A., Gill, P.E., and Murray, W. (1995). Computing modified Newton directions using a partial Cholesky factorization. SIAM Journal on Scientific Computing, 16: 139–450.MathSciNetMATHCrossRefGoogle Scholar
  13. Garcia, C.B. and Gould, F.J. (1980). Relations between several path following algorithms and local arid global Newton methods. SIAM Review, 22: 263–274.MathSciNetMATHCrossRefGoogle Scholar
  14. Ge, R.P. and Qin, Y.F. (1987). A class of filled functions for finding global minimizers of a function of several variables. Journal of Optimization Theory and Applications, 54: 241–252.MathSciNetMATHCrossRefGoogle Scholar
  15. Gill, P.E. and Murray, W. (1974). Newton-type methods for unconstrained and linearly constrained optimization. Mathematical Programming, 7: 311–350.MathSciNetMATHCrossRefGoogle Scholar
  16. Gill, P.E., Murray, W., and Wright, M. (1981). Practical Optimization. Academic Press, London, U.K.MATHGoogle Scholar
  17. Glover, F., Alidaee, B., Rego, C., and Kochenberger, G. (2000). One-pass heuristics for large scale unconstrained binary quadratic problems. Scholar
  18. Goldstein, A.A. and Price, J.F. (1971). On descent from local minima. Mathematics of Computation, 25: 569–574.MathSciNetMATHCrossRefGoogle Scholar
  19. Lawler, E.L. and Bell, M.D. (1966). A method for solving discrete optimization problems. Operations Research, 14: 1098–1112.CrossRefGoogle Scholar
  20. Levy, A.V. and Gomez, S. (1985). The tunneling method applied to global optimization. In Boggs, P.T., editor, Numerical Optimization 1981,pages 213–244. R.Google Scholar
  21. Levy, A.V. and Montalvo, A. (1985). The tunneling algorithm for the global minimization of functions. SIAM Journal of Scientific and Statistical Computation, 6: 15–29.MathSciNetMATHCrossRefGoogle Scholar
  22. Moré, J.J. and Wu, Z. (1997). Global continuation for distance geometry problems. SIAM Journal on Optimization, 7: 814–836.MathSciNetMATHCrossRefGoogle Scholar
  23. Nash, S.G. and Sofer, A. (1995). Linear and Nonlinear Programming. McGraw-Hill, New York, New York.Google Scholar
  24. Schulze, M.A. (2001). Active contours (snakes): A demonstration using Java. Scholar
  25. Sha, L. (1989). A Macrocell Placement Algorithm Using Mathematical Programming Techniques. PhD thesis, Electrical Engineering Department, Stanford University, Stanford, California.Google Scholar
  26. Shang, Y. (1997). Global search methods for solving nonlinear optimization problems. PhD thesis, Computer Science Department, University of Illinois at Urbana-Champaign.Google Scholar
  27. Snyman, J.A. and Fatti, L.P. (1987). A multi-start global minimization algorithm with dynamic search trajectories. Journal of Optimization Theory and Applications, 54: 121–141.MathSciNetMATHCrossRefGoogle Scholar
  28. Törn, A.A. (1977). Cluster analysis using seed points and density-determined hyperspheres as an aid to global optimization. IEEE Transactions on Systems, Man and Cybernetics, 7: 610–616.MATHCrossRefGoogle Scholar
  29. Vilkov, A.V., Zhidkov, N.P., and Shchedrin, B.M. (1975). A method for finding the global minimum of a function of one variable. USSR Computational Mathematics and Mathematical Physics, 15 (4): 221–224.CrossRefGoogle Scholar
  30. Zhang, J.J. (1999). Computing camera heading: A study. PhD thesis, Scientific Computing and Computational Mathematics Department, Stanford University. Stanford, California.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2002

Authors and Affiliations

  • Walter Murray
    • 1
  • Kien-Ming Ng
    • 1
  1. 1.Systems Optimization Laboratory Department of Management Science and EngineeringStanford UniversityStanfordUSA

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