Abstract
In this paper, Lipschitz univariate constrained global optimization problems where both the objective function and constraints can be multiextremal are considered. The constrained problem is reduced to a discontinuous unconstrained problem by the index scheme without introducing additional parameters or variables. A Branch-and-Bound method that does not use derivatives for solving the reduced problem is proposed. The method either determines the infeasibility of the original problem or finds lower and upper bounds for the global solution. Not all the constraints are evaluated during every iteration of the algorithm, providing a significant acceleration of the search. Convergence conditions of the new method are established. Extensive numerical experiments are presented.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Archetti, F. and Schoen, F. (1984), A survey on the global optimization problems: general theory and computational approaches, Annals of Operations Research 1, 87-110.
Bertsekas, D.P. (1996), Constrained Optimization and Lagrange Multiplier Methods, Athena Scientific, Belmont, MA.
Bertsekas, D.P. (1999), Nonlinear Programming, Second Edition, Athena Scientific, Belmont, MA.
Bomze, I.M., Csendes, T., Horst, R. and Pardalos, P.M. (1997) Developments in Global Optimization, Kluwer Academic Publishers, Dordrecht.
Breiman, L. and Cutler, A. (1993), A deterministic algorithm for global optimization, Math. Programming 58, 179-199.
Brooks, S.H. (1958), Discussion of random methods for locating surface maxima, Operation Research 6, 244-251.
Calvin, J. and Žilinskas, A. (1999), On the convergence of the P-algorithm for one-dimensional global optimization of smooth functions, JOTA 102, 479-495.
Evtushenko, Yu.G., Potapov, M.A. and Korotkich, V.V. (1992), Numerical methods for global optimization, Recent Advances in Global Optimization, ed. by C.A. Floudas and P.M. Pardalos, Princeton University Press, Princeton, NJ.
Famularo, D., Sergeyev Ya.D., and Pugliese, P. (2001), Test Problems for Lipschitz Univariate Global Optimization with Multiextremal Constraints, Stochastic and Global Optimization, eds. G. Dzemyda, V. Saltenis and A. Žilinskas, Kluwer Academic Publishers, Dordrecht, to appear.
Floudas, C.A. and. Pardalos, P.M (1996), State of the Art in Global Optimization, Kluwer Academic Publishers, Dordrecht.
Hansen, P. and Jaumard, B. (1995), Lipshitz optimization. In: Horst, R., and Pardalos, P.M. (Eds.). Handbook of Global Optimization, pp. 407-493, Kluwer Academic Publishers, Dordrecht.
Horst, R. and Pardalos, P.M. (1995), Handbook of Global Optimization, Kluwer Academic Publishers, Dordrecht.
Horst, R. and Tuy, H. (1996), Global Optimization-Deterministic Approaches, Springer-Verlag, Berlin, Third edition.
Lamar, B.W. (1999), A method for converting a class of univariate functions into d.c. functions, J. of Global Optimization, 15, 55-71.
Locatelli, M. and Schoen, F. (1995), An adaptive stochastic global optimisation algorithm for onedimensional functions, Annals of Operations Research 58, 263-278.
Locatelli, M. and Schoen, F. (1999), Random Linkage: a family of acceptance/rejection algorithms for global optimisation, Math. Programming 85, 379-396.
Lucidi, S. (1994), On the role of continuously differentiable exact penalty functions in constrained global optimization, J. of Global Optimization 5, 49-68.
MacLagan, D., Sturge, T., and Baritompa, W.P. (1996), Equivalent Methods for Global Optimization, State of the Art in Global Optimization, eds. C.A. Floudas, P.M. Pardalos, Kluwer Academic Publishers, Dordrecht, 201-212.
Mladineo, R. (1992), Convergence rates of a global optimization algorithm, Math. Programming 54, 223-232.
Nocedal, J. and S.J. Wright (1999), Numerical Optimization (Springer Series in Operations Research), Springer, Berlin.
Pardalos, P.M. and J.B. Rosen (1990), Eds., Computational Methods in Global Optimization, Annals of Operations Research, 25.
Patwardhan, A.A., Karim, M.N. and Shah, R. (1987), Controller tuning by a least-squares method, AIChE J. 33, 1735-1737.
Pijavskii, S.A. (1972), An Algorithm for Finding the Absolute Extremum of a Function, USSR Comput. Math. and Math. Physics 12, 57-67.
Pintér, J.D. (1996), Global Optimization in Action, Kluwer Academic Publisher, 1996.
Ralston, P.A.S., Watson, K.R., Patwardhan, A.A. and Deshpande, P.B. (1985), A computer algorithm for optimized control, Industrial and Engineering Chemistry, Product Research and Development 24, 1132.
Sergeyev, Ya.D. (1998), Global one-dimensional optimization using smooth auxiliary functions, Mathematical Programming 81, 127-146.
Sergeyev, Ya.D. (1999), On convergence of Divide the Best global optimization algorithms, Optimization 44, 303-325.
Sergeyev, Ya.D., Daponte, P., Grimaldi, D. and Molinaro, A. (1999), Two methods for solving optimization problems arising in electronic measurements and electrical engineering, SIAM J. Optimization 10, 1-21.
Sergeyev, Ya.D. and Markin, D.L. (1995), An algorithm for solving global optimization problems with nonlinear constraints, J. of Global Optimization 7, 407-419.
Strongin R.G. (1978), Numerical Methods on Multiextremal Problems, Nauka, Moscow, (In Russian).
Strongin, R.G. (1984). Numerical methods for multiextremal nonlinear programming problems with nonconvex constraints. In: Demyanov, V.F., and Pallaschke, D. (Eds.) Lecture Notes in Economics and Mathematical Systems 255, 278-282. Proceedings 1984. Springer. IIASA, Laxenburg/Austria.
Strongin, R.G. and Markin, D.L. (1986), Minimization of multiextremal functions with nonconvex constraints, Cybernetics 22, 486-493.
Strongin, R. G. and Sergeyev, Ya. D. (2000), Global Optimization with Non-Convex Constraints: Sequential and Parallel Algorithms, Kluwer Academic Publishers, Dordrecht.
Sun, X.L. and Li, D. (1999), Value-estimation function method for constrained global optimization, JOTA 102, 385-409.
Törn, A. and Žilinskas, A. (1989), Global Optimization, Springer-Verlag, Lecture Notes in Computer Science, 350.
Wang, X. and Chang, T.S. (1996), An improved univariate global optimization algorithm with improved linear bounding functions, J. of Global Optimization 8, 393-411.
Zhigljavsky, A.A. (1991), Theory of Global Random Search, Kluwer Academic Publishers, Dordrecht.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Sergeyev, Y.D., Famularo, D. & Pugliese, P. Index branch-and-bound algorithm for Lipschitz univariate global optimization with multiextremal constraints. Journal of Global Optimization 21, 317–341 (2001). https://doi.org/10.1023/A:1012391611462
Issue Date:
DOI: https://doi.org/10.1023/A:1012391611462