Abstract
The theoretical foundation of integral global optimization has become widely known and well accepted [4],[24],[25]. However, more effort is needed to demonstrate the effectiveness of the integral global optimization algorithms. In this work we detail the implementation of the integral global minimization algorithms. We describe how the integral global optimization method handles nonconvex unconstrained or box constrained, constrained or discrete minimization problems. We illustrate the flexibility and the efficiency of integral global optimization method by presenting the performance of algorithms on a collection of well known test problems in global optimization literature. We provide the software which solves these test problems and other minimization problems. The performance of the computations demonstrates that the integral global algorithms are not only extremely flexible and reliable but also very efficient.
Similar content being viewed by others
References
Arthanar, T. S. and Dodge, Y.,Mathematical Programming in Statistics, John Wiley and Sons, New York, (1981).
Aubin, J. P. and Ekeland, I.,Applied Nonlinear Analysis, Wiley-Interscience, New York, 1983.
Ballard, D. H., Jelinek, C. O. and Schinzinger, R., “An algorithm for the solution of constrained generalized polynomial programming problem”,Computer Journal,17 (1974), 261–266.
Chew, S. H. and Zheng, Q.,Integral Global Optimization, Lecture Notes in Economics and Mathematical Systems, No.298, Springer-Verlag, 1988.
Dickman, H. B. and Gilman, M. J., “Monte Carlo optimization”,Journal of Optimization Theory and its Applications,60 (1989), 149–157.
Dixon, L. and Szegö, G.,Towards Global Optimization, North Holland, Amsterdam, 1975.
Fiacco, A. V. and McCormick, G. P.,Nonlinear Programming: Sequential Unconstrained Minimization Techniques, John Wiley and Sons, New York, 1968.
Gomez, S. and Levy, A. V., “The tunneling method for solving the constrained global optimization problem with several non-connected feasible regions”, inNumerical Analysis, J. P. Hennart ed., Lecture Notes in Mathematics, Springer-Verlag,909, 1982, 34–47.
Hansen, E.,Global Optimization Using Interval Analysis, Marcel Dekker, New York, 1992.
Himmelblau, D. M.,Applied Nonlinear Programming, McGraw-Hill, New York, 1972.
Hock, W. and Schittkowski, K.,Test Examples for Nonlinear Programming Code, Lecture Notes in Economics and Mathematical Systems, No.187, Springer-Verlag, 1981.
Horst, R. and Tuy, H.,Global Optimization, Springer-Verlag, Berlin, 1990.
Lee, L. E., “Weight minimization of a spped reducer”, an ASME publication, 77-DET-163, 1977.
Levy, A. V. and Montalvo, A., “The tunneling algorithm for the global minimization of functions”,SIAM J. Sci. Sta. Comput.,6 (1985), 15–29.
Loh, H. T.,A Sequential Linearization Approach for Mixed-Discrete Nonlinear Design Optimization, Doctoral Dissertation, Department of Mechanical Engineering and Applied Mechanics, The University of Michigan, Ann Arbor, 1989.
Loh, H. T. and Papalambros, P. Y. “Sequential linearization approach for solving mixeddiscrete nonlinear design optimization”,Journal of Mechanical Design,113 (1991), 325–334.
Moor, R., Hansen, E. and Leclerc, A. “Rigorous methods for global optimization”, inRecent Advances in Global Optimization, C. A. Floudas and P. M. Pardalos eds., Princeton University Press, 1992, 321–342.
Muu, L. D. and Oettli, W., “Method for minimizing a convex-concave function over a convex set”,Journal of Optimization, Theorem and Applications,70 (1991), 377–384.
Pardalos, P. M. and Rosen, J. B.,Constrained Global Optimization: Algorithms and Applications, Lecture Notes in Computer Science,268, Springer-Verlag, 1987.
Pillo, G. Di. and Grippo, L., “Exact penalty functions in constrained optimization”,SIAM Journal Control and Optimization,28 (1989), 1333–1360.
Price, W. L., “Global optimization by controlled random search,Journal of Optimization Theory and Applications,40 (1983), 333–348.
Ritter, K., “A method for solving maximum-problems with a nonconcave quadratic objective function”,Z. Wahrscheinlichkeitstheorie verw. Geb. 4 (1966), 340–351.
Schittkowski, K.,More Test Examples for Nonlinear Programming Code, Lecture Note in Economics and Mathematical Systems, No.282, Springer-Verlag, New York, 1987.
Shi, S., Zheng, Q. and Zhuang, D., “Discontinuous robust mappings are approximatable”,Transaction AMS, 1995.
Shi, S., Zheng, Q. and Zhuang, D., “Set-valued mappings and approximatable mappings”,Journal of Mathematical Analysis and Applications,183 (1994), 706–726.
Tui, H. “Concave programming under linear constraints”,Dokl. Akad. Naul. SSSR 159 (1964), 32–35, (English).
Tu, R. and Zheng, Q., “Integral global optimization method in statistical applications”, to appear inComputers and Mathematics — with Applications.
Zheng, Q., “Strategies of changed domain for searching global extrema”,Numerical Computation and Applications of Computer 3:4 (1981), 257–261. (In Chinese).
Zheng, Q., “Robust Analysis and global optimization”,Annals of Operations Research,24 (1990), 273–286.
Zheng, Q., “Robust analysis and global minimization of a class of discontinuous functions (I)”,Acta Mathematicae Applicatae Sinica (English Series),6:3 (1990), 205–223.
Zheng, Q., “Robust Analysis and global optimization of a class of discontinuous functions (II)”,Acta Mathematicae Applicatae Sinica (English Series),6:4 (1990), 317–337.
Zheng, Q., “ Global minimization of constrained problems with discontinuous penalty functions”, to appear.
Zheng, Q., Jiang, B.C., and Zhuang, S.L., “A method for finding global extrema”,Acta Mathematicae Applicatae Sinica,2:1 (1978), 161–174. (In Chinese).
Zheng, Q. and Zhuang, D., “Integral global optimization of constrained problems in function space with discontinuous penalty functions”, inRecent Advances in Global Optimization, C. A. Floudas and P. M. Pardalos eds., 298–320, Princeton University Press, 1992.
Zheng, Q. and Zhuang, D., “Equi-robust set-valued mappings and the approximation of fixed points,” inProceedings of The Second International Conference on Fixed Point Theory and Applications, K. K. Tan ed., World Scientific Publishing, 1992, 346–361.
Zheng, Q. and Zhuang, D., “Finite dimensional approximation to solutions of minimization problems in function spaces”,Optimization,26 (1992), 33–50.
Zheng, Q. and Zhuang, D., “Integral global optimization and its Monte Carlo implementation”, inProceedings of Conference on Scientific and Engineering Computing, National Defence Industry Press, Beijing, 1993, 262–266.
P. B. Zwart, Nonlinear programming: counterexamples to global optimization algorithms by Ritter and Tui,Operations Research,21 (1973), 1260–1266.
Author information
Authors and Affiliations
Additional information
Research supported partially by NSERC grant and Mount St Vincent University research grant.
Rights and permissions
About this article
Cite this article
Zheng, Q., Zhuang, D. Integral global minimization: Algorithms, implementations and numerical tests. J Glob Optim 7, 421–454 (1995). https://doi.org/10.1007/BF01099651
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF01099651