Handbook of Global Optimization pp 87-113 | Cite as

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

## Abstract

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.

## Keywords

Global Optimization Local Optimization Global Optimization Problem Space Filling Curve Local Smoothing## Preview

Unable to display preview. Download preview PDF.

## References

- 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 - Beasley, J.E. (1998). Heuristic algorithms for the unconstrained binary quadratic programming problem. http://mscmga.ms.ic.ac.uk/jeb/bgp.pdf.Google Scholar
- Behrman, W. (1998).
*An efficient gradient flow method for unconstrained optimization*. PhD thesis, Scientific Computing and ComputationalGoogle Scholar - 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 - Boman, E.G. (1999). Infeasibility and negative curvature in optimization. PhD thesis, Scientific Computing and Computational Mathematics Department,
*Stanford University*, Stanford, California.Google Scholar - 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 - Werra, D. and Gay, Y. (1994). Chromatic scheduling and frequency assignment.
*Discrete Applied Mathematics*, 49: 165–174.MathSciNetMATHCrossRefGoogle Scholar - 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 - Dembo, R.S. and Steihaug, T. (1983). Truncated-Newton algorithms for large-scale unconstrained optimization.
*Mathematical Programming*, 26: 190–212.MathSciNetMATHCrossRefGoogle Scholar - Dorstenstein, T. (2001). Constructive and exchange algorithms for the frequency assignment problem. PhD thesis, Management Science and Engineering Department, Stanford University, Stanford,
*California*. http://www.geocities.com/dorstenstein/research.html.Google Scholar - Fiacco, A.V. and McCormick, G.P. (1968). Nonlinear Programming: Sequential Unconstrained Minimization Techniques.
*John Wiley and Sons*, New York and Toronto.Google Scholar - 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 - 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 - 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 - Gill, P.E. and Murray, W. (1974). Newton-type methods for unconstrained and linearly constrained optimization.
*Mathematical Programming*, 7: 311–350.MathSciNetMATHCrossRefGoogle Scholar - Gill, P.E., Murray, W., and Wright, M. (1981).
*Practical Optimization*. Academic Press, London, U.K.MATHGoogle Scholar - Glover, F., Alidaee, B., Rego, C., and Kochenberger, G. (2000). One-pass heuristics for large scale unconstrained binary quadratic problems. http://hces.bus.olemiss.edu/reports/hces0900.pdf.Google Scholar
- Goldstein, A.A. and Price, J.F. (1971). On descent from local minima.
*Mathematics of Computation*, 25: 569–574.MathSciNetMATHCrossRefGoogle Scholar - Lawler, E.L. and Bell, M.D. (1966). A method for solving discrete optimization problems.
*Operations Research*, 14: 1098–1112.CrossRefGoogle Scholar - 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 - 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 - Moré, J.J. and Wu, Z. (1997). Global continuation for distance geometry problems.
*SIAM Journal on Optimization*, 7: 814–836.MathSciNetMATHCrossRefGoogle Scholar - Nash, S.G. and Sofer, A. (1995).
*Linear and Nonlinear Programming*. McGraw-Hill, New York, New York.Google Scholar - Schulze, M.A. (2001). Active contours (snakes): A demonstration using Java.http://www.markschulze.net/snakes/.Google Scholar
- Sha, L. (1989). A
*Macrocell Placement Algorithm Using Mathematical Programming Techniques*. PhD thesis, Electrical Engineering Department, Stanford University, Stanford, California.Google Scholar - Shang, Y. (1997).
*Global search methods for solving nonlinear optimization problems*. PhD thesis, Computer Science Department, University of Illinois at Urbana-Champaign.Google Scholar - 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 - 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 - 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 - Zhang, J.J. (1999).
*Computing camera heading: A study*. PhD thesis, Scientific Computing and Computational Mathematics Department, Stanford University. Stanford, California.Google Scholar