Abstract
A deterministic global optimization approach is proposed for nonconvex constrained nonlinear programming problems. Partitioning of the variables, along with the introduction of transformation variables, if necessary, converts the original problem into primal and relaxed dual subproblems that provide valid upper and lower bounds respectively on the global optimum. Theoretical properties are presented which allow for a rigorous solution of the relaxed dual problem. Proofs of ∈-finite convergence and ∈-global optimality are provided. The approach is shown to be particularly suited to (a) quadratic programming problems, (b) quadratically constrained problems, and (c) unconstrained and constrained optimization of polynomial and rational polynomial functions. The theoretical approach is illustrated through a few example problems. Finally, some further developments in the approach are briefly discussed.
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Dixon, L. C. W., andSzego, G. P., Editors,Toward Global Optimization, North-Holland, Amsterdam, Holland, 1975.
Dixon, L. C. W., andSzego, G. P., Editors,Toward Global Optimization 2, North-Holland, Amsterdam, Holland, 1978.
Archetti, F., andSchoen, F.,A Survey on the Global Minimization Problem: General Theory and Computational Approaches, Annals of Operations Research, Vol. 1, pp. 87–110, 1984.
Pardalos, P. M., andRosen, J. B.,Methods for Global Concave Minimization: A Bibliographic Survey, SIAM Review, Vol. 28, pp. 367–379, 1986.
Pardalos, P. M., andRosen, J. B.,Constrained Global Optimization: Algorithms and Applications, Lecture Notes in Computer Science, Springer-Verlag, Berlin, Germany, Vol. 268, 1987.
Törn, A., andZilinskas, A.,Global Optimization, Lecture Notes in Computer Science, Springer-Verlag, Berlin, Germany, Vol. 350, 1989.
Ratschek, H., andRokne, J.,New Computer Methods for Global Optimization, Halsted Press, Chichester, Great Britain, 1988.
Mockus, J.,Bayesian Approach to Global Optimization: Theory and Applications, Kluwer Academic Publishers, Amsterdam, Holland, 1989.
Horst, R., andTuy, H.,Global Optimization: Deterministic Approaches, Springer-Verlag, Berlin, Germany, 1990.
Floudas, C. A., andPardalos, P. M.,A Collection of Test Problems for Constrained Global Optimization Algorithms, Lecture Notes in Computer Science, Springer-Verlag, Berlin, Germany, Vol. 455, 1990.
Floudas, C. A., andPardalos, P. M.,Recent Advances in Global Optimization, Princeton University Press, Princeton, New Jersey, 1992.
Piyavskii, S. A.,An Algorithm for Finding the Absolute Extremum of a Function, USSR Computational Mathematics and Mathematical Physics, Vol. 12, pp. 57–67, 1972.
Pardalos, P. M., Glick, J. H., andRosen, J. B.,Global Minimization of Indefinite Quadratic Problems, Computing, Vol. 39, pp. 281–291, 1987.
Al-Khayyal, F. A.,Jointly Constrained Bilinear Programs and Related Problems: An Overview, Computers in Mathematical Applications, Vol. 19, pp. 53–62, 1990.
Hansen, P., Jaumard, B., andLu, S. H.,An Analytical Approach to Global Optimization, Mathematical Programming, Vol. 52, pp. 227–241, 1991.
Tuy, H., Thieu, T. V., andThai, N. Q.,A Conical Algorithm for Globally Minimizing a Concave Function over a Closed Convex Set, Mathematics of Operations Research, Vol. 10, pp. 498–514, 1985.
Tuy, H.,Global Minimum of a Difference of Two Convex Functions, Mathematical Programming Study, Vol. 30, pp. 150–182, 1987.
Tuy, H.,Convex Programs with an Additional Reverse Convex Constraint, Journal of Optimization Theory and Applications, Vol. 52, pp. 463–472, 1987.
Tuy, H.,On Outer Approximation Methods for Solving Concave Minimization Problems, Acta Mathematica Vietnamica, Vol. 8, 1983.
Horst, R., Thoai, N. V., andDe Vries, J.,A New Simplicial Cover Technique in Constrained Global Optimization, Journal of Global Optimization, Vol. 2, pp. 1–19, 1992.
Shor, N. Z.,Dual Quadratic Estimates in Polynomial and Boolean Programming, Annals of Operations Research, Vol. 25, pp. 163–168, 1990.
Floudas, C. A., Aggarwal, A., andCiric, A. R.,Global Optimum Search for Nonconvex NLP and MINLP Problems, Computers and Chemical Engineering, Vol. 13, pp. 1117–1132, 1989.
Aggarwal, A., andFloudas, C. A.,A Decomposition Approach for Global Optimum Search in QP, NLP, and MINLP Problems, Annals of Operations Research, Vol. 25, pp. 119–146, 1990.
Sherali, H., andTuncbilek, C. H.,A Global Optimization Algorithm for Polynomial Programming Problems Using a Reformulation-Linearization Technique, Journal of Global Optimization, Vol. 2, pp. 101–112, 1992.
Hansen, E. R.,Global Optimization Using Interval Analysis: The Multidimensional Case, Numerische Mathematik, Vol. 34, pp. 247–270, 1980.
Floudas, C. A., andVisweswaran, V.,A Global Optimization Algorithm for Certain Classes of Nonconvex NLPs, Part 1: Theory, Computers and Chemical Engineering, Vol. 14, pp. 1397–1417, 1990.
Visweswaran, V., andFloudas, C. A.,A Global Optimization Algorithm for Certain Classes of Nonconvex NLPs, Part 2: Application of Theory and Test Problems, Computers and Chemical Engineering, Vol. 14, pp. 1419–1434. 1990.
Geoffrion, A. M.,Generalized Benders Decomposition, Journal of Optimization Theory and Applications, Vol. 10, pp. 237–260, 1972.
Wolsey, L. A.,A Resource Decomposition Algorithm for General Mathematical Programs, Mathematical Programming Study, Vol. 14, pp. 244–257, 1981.
Flippo, O. E.,Stability, Duality and Decomposition in General Mathematical Programming, PhD Thesis, Erasmus University, Rotterdam, Holland, 1990.
Hansen, P., andJaumard, B.,Reduction of Indefinite Quadratic Programs to Bilinear Programs, Journal of Global Optimization, Vol. 2, pp. 41–60, 1992.
Bazaraa, M. S., andShetty, C. M.,Nonlinear Programming: Theory and Algorithms, John Wiley and Sons, New York, New York, 1979.
Visweswaran, V., andFloudas, C. A.,New Properties and Computational Improvement of the GOP Algorithm for Problems with Quadratic Objective Function and Constraints, Journal of Global Optimization, Vol. 3, No. 3, 1993.
Meyer, R.,The Validity of a Family of Optimization Methods, SIAM Journal on Control, Vol. 8, pp. 41–54, 1970.
Berge, C.,Topological Spaces, Macmillan Company, New York, New York, 1963.
Visweswaran, V., andFloudas, C. A.,Unconstrained and Constrained Global Optimization of Polynomial Functions in One Variable, Journal of Global Optimization, Vol. 2, pp. 73–100, 1992.
Floudas, C. A., Jaumard, B., andVisweswaran, V.,Decomposition Techniques for Global Optimization (to appear).
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Communicated by R. Sargent
The authors gratefully acknowledge financial support from National Science Foundation Presidential Young Investigator Award CBT-88-57013. The authors are also grateful to Drs. F. A. Al-Khayyal, B. Jaumard, P. M. Pardalos, and H. D. Sherali for helpful comments on an earlier draft of this paper.
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Floudas, C.A., Visweswaran, V. Primal-relaxed dual global optimization approach. J Optim Theory Appl 78, 187–225 (1993). https://doi.org/10.1007/BF00939667
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DOI: https://doi.org/10.1007/BF00939667