A Multi-Layer Line Search Method to Improve the Initialization of Optimization Algorithms (Preprint submitted to Optimization Online)

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We introduce a novel metaheuristic methodology to improve the initialization of a given deterministic or stochastic optimization algorithm. Our objective is to improve the performance of the considered algorithm, called core optimization algorithm, by reducing its number of cost function evaluations, by increasing its success rate and by boosting the precision of its results. In our approach, the core optimization is considered as a suboptimization problem for a multi-layer line search method. The approach is presented and implemented for various particular core optimization algorithms: Steepest Descent, Heavy-Ball, Genetic Algorithm, Differential Evolution and Controlled Random Search. We validate our methodology by considering a set of low and high dimensional benchmark problems (i.e., problems of dimension between 2 and 1000). The results are compared to those obtained with the core optimization algorithms alone and with two additional global optimization methods (Direct Tabu Search and Continuous Greedy Randomized Adaptive Search). These latter also aim at improving the initial condition for the core algorithms. The numerical results seem to indicate that our approach improves the performances of the core optimization algorithms and allows to generate algorithms more efficient than the other optimization methods studied here. A Matlab optimization package called ”Global Optimization Platform” (GOP), implementing the algorithms presented here, has been developed and can be downloaded at:
Attouch, H., Goudou, X., and Redont, P. (2000). The heavy ball with friction method, i. the continuous dynamical system: Global exploration of the local minima of a real-valued function by asymptotic analysis of a dissipative dynamical system. Communications in Contemporary Mathematics, 02(01):1–34. Carrasco, M., Ivorra, B., and Ramos, A. M. (2012). A variance-expected compliance model for structural optimization. Journal of Optimization Theory and Applications, 152(1):136–151. Carrasco, M., Ivorra, B., and Ramos, A. M. (2015). Stochastic topology design optimization for continuous elastic materials. Computer Methods in Applied Mechanics and Engineering. Debiane, L., Ivorra, B., Mohammadi, B., Nicoud, F., Poinsot, T., Ern, A., and Pitsch, H. (2006). A low-complexity global optimization algorithm for temperature and pollution control in flames with complex chemistry. International Journal of Computational Fluid Dynamics, 20(2):93–98. Floudas, C. and Pardalos, P. (1999). Handbook of test problems in local and global optimization. Nonconvex optimization and its applications. Kluwer Academic Publishers. Gardeux, V., Chelouah, R., Siarry, P., and Glover, F. (2009). Unidimensional search for solving continuous high-dimensional optimization problems. In Intelligent Systems Design and Applications, 2009. ISDA ’09. Ninth International Conference on, pages 1096–1101. Gardeux, V., Chelouah, R., Siarry, P., and Glover, F. (2011). Em323: a line search based algorithm for solving high-dimensional continuous non-linear optimization problems. Soft Computing, 15(11):2275–2285. Glover, F. (2010). The 3-2-3, stratified split and nested interval line search algorithms. Research report, OptTek Systems, Boulder, CO. Goldberg, D. E. (1989). Genetic Algorithms in Search, Optimization and Machine Learning. Addison-Wesley Longman Publishing Co., Inc., Boston, MA, USA, 1st edition. Gomez, S., Ivorra, B., and Ramos, A. M. (2011). Optimization of a pumping ship trajectory to clean oil contamination in the open sea. Mathematical and Computer Modelling, 54(12):477 – 489. Gomez, S. and Levy, A. (1982). The tunnelling method for solving the constrained global optimization problem with several non-connected feasible regions. In Hennart, J., editor, Numerical Analysis, volume 909 of Lecture Notes in Mathematics, pages 34–47. Springer Berlin Heidelberg. Gonalves, J. F., de Magalhes Mendes, J. J., and Resende, M. G. C. (2002). A hybrid genetic algorithm for the job shop scheduling problem. European Journal of Operational Research, 167:2005. Grosan, C. and Abraham, A. (2007). Hybrid line search for multiobjective optimization. In Perrott, R., Chapman, B., Subhlok, J., de Mello, R., and Yang, L., editors, High Performance Computing and Communications, volume 4782 of Lecture Notes in Computer Science, pages 62–73. Springer Berlin Heidelberg. Hale, J. (2009). Ordinary Differential Equations. Dover Books on Mathematics Series. Dover Publications. Hedar, A.-R. and Fukushima, M. (2006). Tabu search directed by direct search methods for nonlinear global optimization. European Journal of Operational Research, 170(2):329 – 349. Hendrix, E., Ortigosa, P., and Garca, I. (2001). On success rates for controlled random search. Journal of Global Optimization, 21(3):239–263. Hirsch, M., Pardalos, P., and Resende, M. (2010). Speeding up continuous {GRASP}. European Journal of Operational Research, 205(3):507 – 521. Isebe, D., Azerad, P., Bouchette, F., Ivorra, B., and Mohammadi, B. (2008). Shape optimization of geotextile tubes for sandy beach protection. International Journal for Numerical Methods in Engineering, 74(8):1262–1277. Ivorra, B. (2006). Optimisation globale semi-deterministe et applications industrielles. ANRT-Grenoble, Reference: 06/MON2/0061. Ivorra, B., Hertzog, D. E., Mohammadi, B., and Santiago, J. G. (2006). Semideterministic and genetic algorithms for global optimization of microfluidic protein-folding devices. International Journal for Numerical Methods in Engineering, 66(2):319–333. Ivorra, B., Mohammadi, B., and Ramos, A. M. (2009). Optimization strategies in credit portfolio management. Journal of Global Optimization, 43(2-3):415–427. Ivorra, B., Mohammadi, B., and Ramos, A. M. (2014). Design of code division multiple access filters based on sampled fiber bragg grating by using global optimization algorithms. Optimization and Engineering, 15(3):677–695. Ivorra, B., Ramos, A. M., and Mohammadi, B. (2007). Semideterministic global optimization method: Application to a control problem of the burgers equation. Journal of Optimization Theory and Applications, 135(3):549–561. Ivorra, B., Redondo, J. L., Santiago, J. G., Ortigosa, P. M., and Ramos, A. M. (2013). Two- and three-dimensional modeling and optimization applied to the design of a fast hydrodynamic focusing microfluidic mixer for protein folding. Physics of Fluids (1994-present), 25(3):–. Lamghari, A. and Dimitrakopoulos, R. (2012). A diversified tabu search approach for the open-pit mine production scheduling problem with metal uncertainty. European Journal of Operational Research, 222(3):642 –652. Levy, A. and Gomez., S. (1985). The tunneling method applied to global optimization. In Numerical Optimization 1984: Proceedings of the SIAM Conference on Numerical Optimization, Boulder, Colorado, June 12-14, 1984. Proceedings in Applied Mathematics Series. SIAM. Li, X., Engelbrecht, A., and Epitropakis, M. G. (2013). Benchmark functions for cec’2013 special session and competition on niching methods for multimodal function optimization. 2013 IEEE Congress on Evolutionary Computation. Luenberger, D. and Ye, Y. (2008). Linear and Nonlinear Programming. International Series in Operations Research & Management Science. Springer. Mart, R., Campos, V., Resende, M. G., and Duarte, A. (2015). Multiobjective grasp with path relinking. European Journal of Operational Research, 240(1):54 – 71. Mohammadi, B. and Saïac, J. (2003). Pratique de la simulation numérique. Conception: Industrie et Technologies. Dunod. Muyl, F., Dumas, L., and Herbert, V. (2004). Hybrid method for aerodynamic shape optimization in automotive industry. Computers and Fluids, 33(5). Polyak, B. (2007). Newtons method and its use in optimization. European Journal of Operational Research, 181(3):1086 – 1096. Price, K., Storn, R. M., and Lampinen, J. A. (2005). Differential Evolution: A Practical Approach to Global Optimization (Natural Computing Series). Springer-Verlag New York, Inc., Secaucus, NJ, USA. Price, W. (1983). Global optimization by controlled random search. Journal of Optimization Theory and Applications, 40(3):333–348. Redondo, J. L., Fernández, J., García, I., and Ortigosa, P. M. (2009). Solving the multiple competitive facilities location and design problem on the plane. Evol. Comput., 17(1):21–53. Rocha, M. and Neves, J. (1999). Preventing premature convergence to local optima in genetic algorithms via random offspring generation. In Imam, I., Kodratoff, Y., El-Dessouki, A., and Ali, M., editors, Multiple Approaches to Intelligent Systems, volume 1611 of Lecture Notes in Computer Science, pages 127–136. Springer Berlin Heidelberg. Storn, R. and Price, K. (1997). Differential evolution a simple and efficient heuristic for global optimization over continuous spaces. Journal of Global Optimization, 11(4):341–359. Verhulst, F. (1996). Nonlinear Differential Equations and Dynamical Systems. Hochschultext / Universitext. Springer Berlin Heidelberg. Vieira, D. A. G. and Lisboa, A. C. (2014). Line search methods with guaranteed asymptotical convergence to an improving local optimum of multimodal functions. European Journal of Operational Research, 235(1):38 –46.