Derivative-Free Methods for Mixed-Integer Constrained Optimization Problems

详细信息    查看全文

• 作者：Giampaolo Liuzzi (1)
  Stefano Lucidi (2)
  Francesco Rinaldi (3)
  
  1. Istituto di Analisi dei Sistemi ed Informatica (IASI) 鈥淎.Ruberti鈥? CNR ; Viale Manzoni 30 ; 00185聽 ; Rome ; Italy
  2. Dipartimento di Informatica e Sistemistica 鈥淎. Ruberti鈥? 鈥淪apienza鈥?Universit脿 di Roma ; Via Ariosto 25 ; 00185聽 ; Rome ; Italy
  3. Dipartimento di Matematica ; Universit脿 di Padova ; Via Trieste 63 ; 35121 ; 聽Padua ; Italy
  
• 关键词：Mixed integer nonlinear programming ; Derivative ; free optimization ; Nonlinear constrained optimization ; 90C11 ; 90C30 ; 90C56
• 刊名：Journal of Optimization Theory and Applications
• 出版年：2015
• 出版时间：March 2015
• 年：2015
• 卷：164
• 期：3
• 页码：933-965
• 全文大小：1,081 KB
• 参考文献：1. Conn, A, Scheinberg, K, Vicente, L (2009) Introduction to Derivative-Free Optimization. MOS/SIAM Series on Optimization. SIAM, Philadelphia, MA CrossRef
  2. Abramson, M, Audet, C, Chrissis, J, Walston, J (2009) Mesh adaptive direct search algorithms for mixed variable optimization. Optim. Lett. 3: pp. 35-47 CrossRef
  3. Audet, C, Dennis, J (2006) Mesh adaptive direct search algorithms for constrained optimization. SIAM J. Optim. 17: pp. 188-217 CrossRef
  4. Torczon, V (1997) On the convergence of pattern search algorithms. SIAM J. Optim. 7: pp. 1-25 CrossRef
  5. Audet, C, Dennis, J (2003) Analysis of generalized pattern searches. SIAM J. Optim. 13: pp. 889-903 CrossRef
  6. Kolda, T, Lewis, R, Torczon, V (2003) Optimization by direct search: new perspectives on some classical and modern methods. SIAM Rev. 45: pp. 385-482 CrossRef
  7. Audet, C, Dennis, J (2001) Pattern search algorithms for mixed variable programming. SIAM J. Optim. 11: pp. 573-594 CrossRef
  8. Kokkolaras, M, Audet, C, Dennis, J (2001) Mixed variable optimization of the number and composition of heat intercepts in a thermal insulation system. Optim. Eng. 2: pp. 5-29 CrossRef
  9. Abramson, M.: Pattern search filter algorithms for mixed variable general constrained optimization problems. Ph.D. thesis, Department of Computational and Applied Mathematics, Rice University (2002)
  10. Abramson, M, Audet, C, Dennis, J (2007) Filter pattern search algorithms for mixed variable constrained optimization problems. Pac. J. Optim. 3: pp. 477-500
  11. Audet, C, Dennis, J (2004) A pattern search filter method for nonlinear programming without derivatives. SIAM J. Optim. 14: pp. 980-1010 CrossRef
  12. Vicente, L (2009) Implicitly and densely discrete black-box optimization problems. Optim. Lett. 3: pp. 475-482 CrossRef
  13. Lucidi, S, Piccialli, V, Sciandrone, M (2005) An algorithm model for mixed variable programming. SIAM J. Optim. 14: pp. 1057-1084 CrossRef
  14. Lucidi, S, Sciandrone, M, Tseng, P (2002) Objective-derivative-free methods for constrained optimization. Math. Program. 92: pp. 37-59 CrossRef
  15. Garc铆a-Palomares, U, Costa-Montenegro, E, Asorey-Cacheda, R, Gonzalez-Castano, FJ (2012) Adapting derivative free optimization methods to engineering models with discrete variables. Optim. Eng. 13: pp. 579-594
  16. Garc铆a-Palomares, U, Rodr铆guez, J (2002) New sequential and parallel derivative-free algorithms for unconstrained optimization. SIAM J. Optim. 13: pp. 79-96 CrossRef
  17. M眉ller, J, Shoemaker, C, Pich茅, R (2013) So-mi: a surrogate model algorithm for computationally expensive nonlinear mixed-integer black-box global optimization problems. Comput. Oper. Res. 40: pp. 1383-1400 CrossRef
  18. Laguna, M., Gort谩zar, F., Gallego, M., Duarte, A., Mart铆, R.: A black-box scatter search for optimization problems with integer variables. J. Glob. Optim. (2013). doi:10.1007/s10898-013-0061-2
  19. Liuzzi, G, Lucidi, S, Rinaldi, F (2012) Derivative-free methods for bound constrained mixed-integer optimization. Comput. Optim. Appl. 53: pp. 505-526 CrossRef
  20. Liuzzi, G, Lucidi, S, Sciandrone, M (2010) Sequential penalty derivative-free methods for nonlinear constrained optimization. SIAM J. Optim. 20: pp. 2814-2835 CrossRef
  21. Lucidi, S, Sciandrone, M (2002) A derivative-free algorithm for bound constrained optimization. Comput. Optim. Appl. 21: pp. 119-142 CrossRef
  22. Vicente, L (2013) Worst case complexity of direct search. EURO J. Comput. Optim. 1: pp. 143-153 CrossRef
  23. Nesterov, Y (2004) Introductory Lectures on Convex Optimization. Kluwer Academic Publishers, Dordrecht CrossRef
  24. Lin, CJ, Lucidi, S, Palagi, L, Risi, A, Sciandrone, M (2009) A decomposition algorithm model for singly linearly constrained problems subject to lower and upper bounds. J. Optim. Theory Appl. 141: pp. 107-126 CrossRef
  25. Bertsekas, D (1999) Nonlinear Programming. Athena Scientific, Belmont, MA
  26. Digabel, S (2011) Algorithm 909: Nomad: nonlinear optimization with the mads algorithm. ACM Trans. Math. Softw. 37: pp. 1-15 CrossRef
  27. Hock, W, Schittkowski, K (1981) Test Examples for Nonlinear Programming Codes. Lecture notes in Economics and Mathematical Systems. Springer, Berlin CrossRef
  28. Hock, W, Schittkowski, K (1987) More Test Examples for Nonlinear Programming Codes. Lecture notes in Economics and Mathematical Systems. Springer, Berlin
  29. Mor茅, J, Wild, S (2009) Benchmarking derivative-free optimization algorithms. SIAM J. Optim. 20: pp. 172-191 CrossRef
  30. Lucidi, S, Parasiliti, F, Rinaldi, F, Villani, M (2012) Finite element based multi-objective design optimization procedure of interior permanent magnet synchronous motors for wide constant-power region operation. IEEE Trans. Ind. Electron. 59: pp. 2503-2514 CrossRef
  
• 刊物主题：Calculus of Variations and Optimal Control; Optimization; Optimization; Theory of Computation; Applications of Mathematics; Engineering, general; Operations Research/Decision Theory;
• 出版者：Springer US
• ISSN：1573-2878

文摘

Methods which do not use any derivative information are becoming popular among researchers, since they allow to solve many real-world engineering problems. Such problems are frequently characterized by the presence of discrete variables, which can further complicate the optimization process. In this paper, we propose derivative-free algorithms for solving continuously differentiable Mixed Integer NonLinear Programming problems with general nonlinear constraints and explicit handling of bound constraints on the problem variables. We use an exterior penalty approach to handle the general nonlinear constraints and a local search approach to take into account the presence of discrete variables. We show that the proposed algorithms globally converge to points satisfying different necessary optimality conditions. We report a computational experience and a comparison with a well-known derivative-free optimization software package, i.e., NOMAD, on a set of test problems. Furthermore, we employ the proposed methods and NOMAD to solve a real problem concerning the optimal design of an industrial electric motor. This allows to show that the method converging to the better extended stationary points obtains the best solution also from an applicative point of view.