A Derivative-Free Algorithm for Constrained Global Optimization Based on Exact Penalty Functions

详细信息    查看全文

• 作者：Gianni Di Pillo (1)
  Stefano Lucidi (1)
  Francesco Rinaldi (2)
  
  1. Dipartimento di Ingegneria Informatica ; Automatica e Gestionale 鈥淎. Ruberti鈥? 鈥淪apienza鈥?Universit脿 di Roma ; Via Ariosto 25 ; 00185 ; Rome ; Italy
  2. Dipartimento di Matematica ; Universit脿 di Padova ; Via Trieste 63 ; 35121 ; Padova ; Italy
  
• 关键词：Nonlinear optimization ; Global optimization ; Exact penalty functions ; Derivative ; free minimization ; DIRECT algorithm
• 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：862-882
• 全文大小：692 KB
• 参考文献：1. Floudas, C.A. (1999) Deterministic Global Optimization: Theory, Methods and Application. Kluwer Academic, Dordrecht
  2. Horst, R., Pardalos, P.M., Thoai, N.V. (1995) Introduction to Global Optimization. Kluwer Academic, Dordrecht
  3. Horst, R., Tuy, H. (1996) Global Optimization: Deterministic Approaches. Springer, Berlin CrossRef
  4. Jones, D.R., Perttunen, C.D., Stuckman, B.E. (1993) Lipschitzian optimization without the Lipschitz constant. J. Optim. Theory Appl. 79: pp. 157-181 CrossRef
  5. Jones, D.R. The direct global optimization algorithm. In: Floudas, C., Pardalos, P. eds. (2009) Encyclopedia of Optimization. Springer, New York, pp. 725-735 CrossRef
  6. Pinter, J.D. (1996) Global Optimization in Action. Kluwer Academic, Dordrecht CrossRef
  7. Tawarmalani, M., Sahinidis, N.V. (2002) Convexification and Global Optimization in Continuous and Mixed-Integer Nonlinear Programming. Kluwer Academic, Dordrecht CrossRef
  8. Neumaier, A., Shcherbina, O., Huyer, W., Vink贸, T. (2005) A comparison of complete global optimization solvers. Math. Program. 103: pp. 335-356 CrossRef
  9. Birgin, E.G., Floudas, C.A., Martinez, J.M. (2010) Global minimization using an augmented Lagrangian method with variable lower-level constraints. Math. Program. 125: pp. 139-162 CrossRef
  10. Luo, H.Z., Sun, X.L., Li, D. (2007) On the convergence of augmented Lagrangian methods for constrained global optimization. SIAM J. Optim. 18: pp. 1209-1230 CrossRef
  11. Wang, C.Y., Li, D. (2009) Unified theory of augmented Lagrangian methods for constrained global optimization. J. Glob. Optim. 44: pp. 433-458 CrossRef
  12. Cassioli, A., Schoen, F. (2013) Global optimization of expensive black box problems with a known lower bound. J. Glob. Optim. 57: pp. 177-190 CrossRef
  13. Pillo, G., Lucidi, S., Rinaldi, F. (2012) An approach to constrained global optimization based on exact penalty functions. J. Glob. Optim. 54: pp. 251-260 CrossRef
  14. Huang, L.R., Ng, K.F. (1994) Second-order necessary and sufficient conditions in nonsmooth optimization. Math. Program. 66: pp. 379-402 CrossRef
  15. Liuzzi, G., Lucidi, S., Piccialli, V., Sotgiu, A. (2004) A magnetic resonance device designed via global optimization techniques. Math. Program. 101: pp. 339-364 CrossRef
  16. Neumaier, A., Sam-Haroud, D., Vu, X.H., Nguyen, T.V. Benchmarking global optimization and constraint satisfaction codes. In: Bliek, C., Jermann, C., Neumaier, A. eds. (2003) Global Optimization and Constraint Satisfaction. Springer, Berlin, pp. 211-222
  17. Liuzzi, G., Lucidi, S., Piccialli, V.: Exploiting derivative-free local searches in DIRECT-type algorithms for global optimization. IASI Technical Report, (2013)
  18. Fasano, G., Liuzzi, G., Lucidi, S., Rinaldi, F.: A linesearch-based derivative-free approach for nonsmooth optimization. IASI Technical Report (2013)
  19. Gablonsky, J.M.: DIRECT version 2.0, user guide. Technical Report, North Carolina State University, (2001)
  
• 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

Constrained global optimization problems can be tackled by using exact penalty approaches. In a preceding paper, we proposed an exact penalty algorithm for constrained problems which combines an unconstrained global minimization technique for minimizing a non-differentiable exact penalty function for given values of the penalty parameter, and an automatic updating of the penalty parameter that occurs only a finite number of times. However, in the updating of the penalty parameter, the method requires the evaluation of the derivatives of the problem functions. In this work, we show that an efficient updating can be implemented also without using the problem derivatives, in this way making the approach suitable for globally solving constrained problems where the derivatives are not available. In the algorithm, any efficient derivative-free unconstrained global minimization technique can be used. In particular, we adopt an improved version of the DIRECT algorithm. In addition, to improve the performances, the approach is enriched by resorting to derivative-free local searches, in a multistart framework. In this context, we prove that, under suitable assumptions, for every global minimum point there exists a neighborhood of attraction for the local search. An extensive numerical experience is reported.