Abstract
In this paper we propose a new simplicial partition-based deterministic algorithm for global optimization of Lipschitz-continuous functions without requiring any knowledge of the Lipschitz constant. Our algorithm is motivated by the well-known Direct algorithm which evaluates the objective function on a set of points that tries to cover the most promising subregions of the feasible region. Almost all previous modifications of Direct algorithm use hyper-rectangular partitions. However, other types of partitions may be more suitable for some optimization problems. Simplicial partitions may be preferable when the initial feasible region is either already a simplex or may be covered by one or a manageable number of simplices. Therefore in this paper we propose and investigate simplicial versions of the partition-based algorithm. In the case of simplicial partitions, definition of potentially optimal subregion cannot be the same as in the rectangular version. In this paper we propose and investigate two definitions of potentially optimal simplices: one involves function values at the vertices of the simplex and another uses function value at the centroid of the simplex. We use experimental investigation to compare performance of the algorithms with different definitions of potentially optimal partitions. The experimental investigation shows, that proposed simplicial algorithm gives very competitive results to Direct algorithm using standard test problems and performs particularly well when the search space and the numbers of local and global optimizers may be reduced by taking into account symmetries of the objective function.
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Baker, C.A., Watson, L.T., Grossman, B., Mason, W.H., Haftka, R.T.: Parallel global aircraft configuration design space exploration. In: Tentner, A. (ed.) High Performance Computing Symposium 2000, pp. 54–66. Soc. for Computer Simulation Internat (2000)
Bartholomew-Biggs, M.C., Parkhurst, S.C., Wilson, S.P.: Using DIRECT to solve an aircraft routing problem. Comput. Optim. Appl. 21(3), 311–323 (2002). doi:10.1023/A:1013729320435
Björkman, M., Holmström, K.: Global optimization using the direct algorithm in Matlab. Adv. Model. Optim. 1(2), 17–37 (1999)
Carter, R.G., Gablonsky, J.M., Patrick, A., Kelley, C.T., Eslinger, O.J.: Algorithms for noisy problems in gas transmission pipeline optimization. Optim. Eng. 2(2), 139–157 (2001). doi:10.1023/A:1013123110266
Casado, L., Hendrix, E., García, I.: Infeasibility spheres for finding robust solutions of blending problems with quadratic constraints. J. Glob. Optim. 39(4), 577–593 (2007). doi:10.1007/s10898-007-9157-x
Chiter, L.: DIRECT algorithm: a new definition of potentially optimal hyperrectangles. Appl. Math. Comput. 179(2), 742–749 (2006). doi:10.1016/j.amc.2005.11.127
Cox, S.E., Haftka, R.T., Baker, C.A., Grossman, B., Mason, W.H., Watson, L.T.: A comparison of global optimization methods for the design of a high-speed civil transport. J. Glob. Optim. 21(4), 415–432 (2001). doi:10.1023/A:1012782825166
Dixon, L., Szegö, C.: The global optimisation problem: an introduction. In: Dixon, L., Szegö, G. (eds.) Towards Global Optimization, vol. 2, pp. 1–15. North-Holland Publishing Company, Amsterdam (1978)
Finkel, D.E., Kelley, C.T.: Additive scaling and the DIRECT algorithm. J. Glob. Optim. 36(4), 597–608 (2006). doi:10.1007/s10898-006-9029-9
Gablonsky, J.M., Kelley, C.T.: A locally-biased form of the DIRECT algorithm. J. Glob. Optim. 21(1), 27–37 (2001). doi:10.1023/A:1017930332101
Gorodetsky, S.: Paraboloid triangulation methods in solving multiextremal optimization problems with constraints for a class of functions with Lipschitz directional derivatives. Vestnik of Lobachevsky State University of Nizhni Novgorod 1(1), 144–155 (2012)
Grbić, R., Nyarko, E.K., Scitovski, R.: A modification of the direct method for Lipschitz global optimization for a symmetric function. J. Glob. Optim. 1–20 (2012). doi:10.1007/s10898-012-0020-3
He, J., Watson, L.T., Ramakrishnan, N., Shaffer, C.A., Verstak, A., Jiang, J., Bae, K., Tranter, W.H.: Dynamic data structures for a DIRECT search algorithm. Comput. Optim. Appl. 23(1), 5–25 (2002). doi:10.1023/A:1019992822938
Horst, R.: On generalized bisection of n-simplices. Math. Comput. 66(218), 691–698 (1997)
Horst, R.: Bisecton by global optimization revisited. J. Optim. Theory Appl. 144(3), 501–510 (2010)
Horst, R., Pardalos, P.M., Thoai, N.V.: Introduction to Global Optimization. Nonconvex Optimization and Its Application. Kluwer Academic Publishers, Boston (1995)
Horst, R., Tuy, H.: Global Optimization: Deterministic Approaches. Springer, Berlin (1996)
Jennrich, R.I., Sampson, P.F.: Application of stepwise regression to non-linear estimation. Technometrics 10(1), 63–72 (1968). doi:10.1080/00401706.1968.10490535
Jones, D.R., Perttunen, C.D., Stuckman, B.E.: Lipschitzian optimization without the Lipschitz constant. J. Optim. Theory Appl. 79(1), 157–181 (1993). doi:10.1007/BF00941892
Křivý, I., Tvrdík, J., Krpec, R.: Stochastic algorithms in nonlinear regression. Comput. Stat. Data Anal. 33(3), 277–290 (2000). doi:10.1016/S0167-9473(99)00059-6
Kvasov, D.E., Sergeyev, Y.D.: A univariate global search working with a set of Lipschitz constants for the first derivative. Optim. Lett. 3(2), 303–318 (2009). doi:10.1007/s11590-008-0110-9
Kvasov, D.E., Sergeyev, Y.D.: Lipschitz gradients for global optimization in a one-point-based partitioning scheme. J. Comput. Appl. Math. 236(16), 4042–4054 (2012). doi:10.1016/j.cam.2012.02.020
Kvasov, D.E., Sergeyev, Y.D.: Univariate geometric Lipschitz global optimization algorithms. Numer. Algebra Control Optim. 2(1), 69–90 (2012). doi:10.3934/naco.2012.2.69
Lanczos, C.: Applied Analysis. Prentice Hall, Englewood Cliffs (1956)
Liuzzi, G., Lucidi, S., Piccialli, V.: A direct-based approach for large-scale global optimization problems. Comput. Optim. Appl. 45(2), 353–375 (2010)
Liuzzi, G., Lucidi, S., Piccialli, V.: A partition-based global optimization algorithm. J. Global Optim. 48(1), 113–128 (2010). doi:10.1007/s10898-009-9515-y
Mockus, J.: On the Pareto optimality in the context of Lipschitzian optimization. Informatica 22(4), 521–536 (2011)
Nast, M.: Subdivision of simplices relative to a cutting plane and finite concave minimization. J. Global Optim. 9(1), 65–93 (1996). doi:10.1007/BF00121751
Osborne, M.R.: Some aspects of nonlinear least squares calculations. In: Lootsma, F.A. (ed.) Numerical Methods for Nonlinear Optimization, pp. 171–189. Academic Press, New York (1972)
Paulavičius, R., Žilinskas, J.: Analysis of different norms and corresponding Lipschitz constants for global optimization in multidimensional case. Inf. Technol. Control 36(4), 383–387 (2007)
Paulavičius, R., Žilinskas, J.: Influence of Lipschitz bounds on the speed of global optimization. Technol. Econ. Dev. Econ. 18(1), 54–66 (2012). doi:10.3846/20294913.2012.661170
Paulavičius, R., Žilinskas, J., Grothey, A.: Investigation of selection strategies in branch and bound algorithm with simplicial partitions and combination of Lipschitz bounds. Optim. Lett. 4(2), 173–183 (2010). doi:10.1007/s11590-009-0156-3
Pintér, J.D.: Global Optimization in Action: Continuous and Lipschitz Optimization: Algorithms, Implementations and Applications. Springer, New York (1996)
di Serafino, D., Liuzzi, G., Piccialli, V., Riccio, F., Toraldo, G.: A modified DIviding RECTangles algorithm for a problem in astrophysics. J. Optim. Theory Appl. 151(1), 175–190 (2011). doi:10.1007/s10957-011-9856-9
Sergeyev, Y.D., Kvasov, D.E.: Global search based on efficient diagonal partitions and a set of Lipschitz constants. SIAM J. Optim. 16(3), 910–937 (2006). doi:10.1137/040621132
Sergeyev, Y.D., Kvasov, D.E.: Lipschitz global optimization. In: Cochran, J. (ed.) Wiley Encyclopedia of Operations Research and Management Science, vol. 4, pp. 2812–2828. Wiley, New York (2011)
Strongin, R.G., Sergeyev, Y.D.: Global Optimization with Non-Convex Constraints: Sequential and Parallel Algorithms. Kluwer Academic Publishers, Dordrecht (2000)
Todt, M.J.: The Computation of Fixed Points and Applications, Lecture Notes in Economics and Mathematical Systems, vol. 24 (1976)
Yao, Y.: Dynamic tunneling algorithm for global optimization. IEEE Trans. Syst. Man Cybern. 19(5), 1222–1230 (1989)
Zhigljavsky, A., Žilinskas, A.: Stochastic Global Optimization. Springer, New York (2008)
Žilinskas, A.: On strong homogeneity of two global optimization algorithms based on statistical models of multimodal objective functions. Appl. Math. Comput. 218(16), 8131–8136 (2012). doi:10.1016/j.amc.2011.07.051
Žilinskas, A., Žilinskas, J.: Global optimization based on a statistical model and simplicial partitioning. Comput. Math. Appl. 44(7), 957–967 (2002). doi:10.1016/S0898-1221(02)00206-7
Žilinskas, A., Žilinskas, J.: A hybrid global optimization algorithm for non-linear least squares regression. J. Global Optim. 56(2), 265–277 (2013). doi:10.1007/s10898-011-9840-9
Žilinskas, J.: Reducing of search space of multidimensional scaling problems with data exposing symmetries. Inf. Technol. Control 36(4), 377–382 (2007)
Žilinskas, J.: Branch and bound with simplicial partitions for global optimization. Math. Modell. Anal. 13(1), 145–159 (2008). doi:10.3846/1392-6292.2008.13.145-159
Acknowledgments
The authors would like to thank anonymous referees for their careful reading of the paper and insightful comments that helped us improve the paper. Postdoctoral fellowship of R. Paulavičius is being funded by European Union Structural Funds project “Postdoctoral Fellowship Implementation in Lithuania” within the framework of the Measure for Enhancing Mobility of Scholars and Other Researchers and the Promotion of Student Research (VP1-3.1-ŠMM-01) of the Program of Human Resources Development Action Plan.
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Paulavičius, R., Žilinskas, J. Simplicial Lipschitz optimization without the Lipschitz constant. J Glob Optim 59, 23–40 (2014). https://doi.org/10.1007/s10898-013-0089-3
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DOI: https://doi.org/10.1007/s10898-013-0089-3