Abstract
We both propose and test an implicit strategy that is based on changing the search space from points to directions, which in combination with the Differential Evolution (DE) algorithm, is easily implemented for solving boundary optimization of a generic continuous function. In particular, we see that the DE method can be efficiently implemented to find solutions on the boundary of a convex and bounded feasible set resulting when the constraints are bounds on the variables, linear inequalities and quadratic convex inequalities. The computational results are performed on different classes of boundary minimization problems. The proposed technique is compared with the Generalized Differential Evolution method.
Similar content being viewed by others
References
Ali M.M., Fatti L.P.: A differential free point generation scheme in the differential evolution algorithm. J. Glob. Optim. 35, 551–572 (2006)
Benson H.P.: Concave minimization: theory, applications and algorithms. In: Horst, R., Pardalos, P.M. (eds) Handbook of Global Optimization, pp. 43–148. Kluwer, Dordrecht (1995)
Bomze I.M., De Klerk E.: Solving standard quadratic optimization problems via linear, semidefinite and copositive programming. J. Glob. Optim. 24, 163–185 (2002)
Brest, J., Sepesy Maucec, M.: Population size reduction for the differential evolution algorithm. Appl. Intell. doi:10.1007/s10489-007-0091-x, Springer, LLC (2007)
Cai Z., Wang Y.: A multiobjective optimization-based evolutionary algorithm for constrained optimization. IEEE Trans. Evol. Comput. 10, 658–675 (2006)
Churilov L., Ralph D., Sniedovich M.: A note on composite concave quadratic programming. Oper. Res. Lett. 23, 163–169 (1998)
Coello C.A.C.: Theoretical and numerical constraint-handling techniques used with evolutionary algorithms: a survey of the state if the art. Comput. Methods Appl. Mech. Eng. 191, 1245–1287 (2002)
Cox, S.E., Haftka, R.T., Baker, C., Grossman, B., Mason, W.H., Watson, L.T.: Global multidisciplinary optimization of a high speed civil transport. In: Proceedings of the Aerospace Numerical Simulation Symposium ’99, Tokyo, pp. 23–28. June (1999)
Deb K.: An efficient constraint handling method for genetic algorithms. Comput. Methods Appl. Mech. Eng. 186, 311–338 (2000)
Fang K.T., Li R.Z.: Some methods for generating both NT-net and the uniform distribution on a Stiefel manifold and their applications. Comput. Stat. Data Anal. 24, 29–46 (1997)
Fang K.T., Wang Y.: Number Theoretic Methods in Statistics. Chapman & Hall, London (1994)
Floudas C.A., Pardalos P.M. et al.: Handbook of Test Problems in Local and Global Optimization. Kluwer Academic Publ, Dordrecht (1999)
Jeyakumar V., Rubinov A.M., Wu Z.Y.: Sufficient global optimality conditions for nonconvex quadratic minimization problems with box constraints. J. Global Optim. 36, 471–481 (2006)
Kaelo P., Ali M.M.: Some variants of the controlled random search algorithm for global optimization. J.O.T.A. 130, 253–264 (2006)
Kreinovich V., Nguyen H.T., Wu B.: On line algorithms for computing mean and variance of interval data and their use in intelligent systems. Inf. Sci. 177, 3228–3238 (2007)
Lampinen, J. (2001) Multi-Constrained Nonlinear Optimization by the Differential Evolution Algorithm. Thechnical report, Lappeenranta University of Technology, Department of Information Technology
Lampinen, J.: A Constraint Handling Approach for the Differential Evolution Algorithm. In: Proceedings of the 2002 Congress on Evolutionary Computation (CEC2002), pp. 1468–1473
Leguizamon, G., Coello Coello, C.A.: Boundary search for constrained numerical optimization problems in ACO algorithms. In: Dorigo, M., et al. (eds.) Proceedings of ANTS 2006, Lecture Notes in Computer Science No. 4150, pp. 108–119. Springer, Berlin (2006)
Locatelli M., Thoai N.V.: Finite exact branch and bound algorithms for concave minimization over polytopes. J. Glob. Optim. 18, 107–128 (2000)
Maranas C.D., Floudas C.A., Pardalos P.M.: New results in the packing of equal circles in a square. Discret. Math. 142, 287–293 (1995)
Mezura-Montes, E., Velazquez-Reyes, J., Coello Coello, C.A.: Modified differential evolution for constrained optimization. In: Proceedings of Congress on Evolutionary Computation (CEC), pp. 25–32 (2006)
Mezura-Montes E., Miranda-Varela M.E., del Carmen Gomez-Ramon R.: Differential evolution in constrained numerical optimization: an empirical study. Inf. Sci. 180, 4223–4262 (2010)
Mezura-Montes, E., Coello Coello, C.A., Tun-Morales, E.I.: Simple feasibility rules and differential evolution for constrained optimization. In: Proceeding of IMICAI 2004, Lecture Notes in Artificial Intelligence, 2972, pp. 707–716. Springer, Berlin (2004)
Mladenovic N., Drazic M., Kovacevic-Vujcic V., Cangalovic M.: General variable neighborhood search for the continuous optimization. Eur. J. Oper. Res. 191, 753–770 (2008)
Moshirvaziri K.: Construction of test problems for concave minimization under linear and nonlinear constraints. J.O.T.A. 98, 83–108 (1998)
Price K.: An introduction to differential evolution. In: Corne, D., Dorigo, M., Glover, F. (eds) New Ideas in Optimization, pp. 79–108. McGraw Hill, New York (1999)
Rosen J.B., Pardalos P.M.: Global minimization of large scale constrained concave quadratic problems by separable programming. Math. Program. 34, 163–174 (1986)
Rubinov A.M., Andramonov M.Yu.: Minimizing increasing star-shaped functions based on abstract convexity. J. Glob. Optim. 15, 19–39 (1999)
Salman A., Engelbrecht A.P., Omran M.G.H.: Empirical analysis of self adaptive differential evolution. Eur. J. Oper. Res. 183, 785–804 (2007)
Shectman J.P., Sahinidis N.V.: A finite algorithm for global minimization of separable concave programs. J. Glob. Optim. 12, 1–36 (1998)
Spadoni, M., Stefanini, L.: Handling box, linear and quadratic-convex constraints for boundary optimization with differential evolution algorithms. In: Proceedings of 2009 Ninth International Conference on Intelligent Systems Design and Applications (ISDA), pp. 7–12. Pisa (2009)
Storn, R., Price, K.: Differential Evolution: A Simple and Efficient Heuristic for Global Optimization Over Continuous Spaces. ICSI technical report TR-95-012, Berkeley University, 1995. Also, J. Glob. Optim. 11, 341–359 (1997)
Storn R.: System design by constraint adaptation and differential evolution. IEEE Trans. Evol. Comput. 3, 22–34 (1999)
Storn, R.: Differential Evolution for MATLAB, International Computer Science Institute (ICSI), Berkeley, CA 94704. http://http.icsi.berkeley.edu/~storn (1997)
Sun X.L., Li J.L.: A branch-and-bound based method for solving monotone optimization problems. J. Glob. Optim. 35, 367–385 (2006)
Takahama, T., Sakai, S.: Constrained optimization by the ɛ-constrained differential evolution with gradient-based mutation and feasible elites. In: Proceedings of 2006 IEEE Congress on Evolutionary Computation, pp. 308–315. Vancouver, Canada (2006)
Tuy H.: Convex Analysis and Global Optimization. Kluwer, Dordrecht (1998)
Tuy H.: Monotonic Optimization: problems and solution approaches. SIAM J. Optim. 11, 464–494 (2000)
Zaharie, D.: Critical Values for the Control Parameters of Differential Evolution algorithms. In: Proceeding of Mendel 2002, 8th International Conference on Soft Computing (2002)
Zhang M., Luo W., Wang X.: Differential evolution with dynamic stochastic selection for constrained optimization. Inf. Sci. 178, 3043–3074 (2008)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Spadoni, M., Stefanini, L. A Differential Evolution algorithm to deal with box, linear and quadratic-convex constraints for boundary optimization. J Glob Optim 52, 171–192 (2012). https://doi.org/10.1007/s10898-011-9695-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10898-011-9695-0