Abstract
A new algorithm is proposed to deal with the worst-case optimization of black-box functions evaluated through costly computer simulations. The input variables of these computer experiments are assumed to be of two types. Control variables must be tuned while environmental variables have an undesirable effect, to which the design of the control variables should be robust. The algorithm to be proposed searches for a minimax solution, i.e., values of the control variables that minimize the maximum of the objective function with respect to the environmental variables. The problem is particularly difficult when the control and environmental variables live in continuous spaces. Combining a relaxation procedure with Kriging-based optimization makes it possible to deal with the continuity of the variables and the fact that no analytical expression of the objective function is available in most real-case problems. Numerical experiments are conducted to assess the accuracy and efficiency of the algorithm, both on analytical test functions with known results and on an engineering application.
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References
Santner T.J., Williams B.J., Notz W.: The Design and Analysis of Computer Experiments. Springer, Berlin, Heidelberg (2003)
Jones R.: A taxonomy of global optimization methods based on response surfaces. J. Glob. Optim. 21(4), 345–383 (2001)
Queipo N.V., Haftka R.T., Shyy W., Goel T., Vaidyanathan R., Tucker P.K.: Surrogate-based analysis and optimization. Prog. Aerosp. Sci. 41(1), 1–28 (2005)
Simpson T.W., Poplinski J.D., Koch P.N., Allen J.K.: Metamodels for computer-based engineering design: survey and recommendations. Eng. Comput. 17(2), 129–150 (2001)
McKay M.D., Beckman R.J., Conover W.J.: A comparison of three methods for selecting values of input variables in the analysis of output from a computer code. Technometrics 21(2), 239–245 (1979)
Matheron G.: Principles of geostatistics. Econ. Geol. 58(8), 1246–1266 (1963)
Rasmussen C.E., Williams C.K.I.: Gaussian Processes for Machine Learning. Springer, New York, NY (2006)
Jones D.R., Schonlau M.J., Welch W.J.: Efficient global optimization of expensive black-box functions. J. Glob. Optim. 13(4), 455–492 (1998)
Forrester A.I.J., Sobester A., Keane A.J.: Engineering Design via Surrogate Modelling: A Practical Guide. Wiley, Chichester (2008)
Huang D., Allen T.T., Notz W.I., Zeng N.: Global optimization of stochastic black-box systems via sequential Kriging meta-models. J. Glob. Optim. 34(3), 441–466 (2006)
Sasena, M.J.: Flexibility and efficiency enhancements for constrained global design optimization with Kriging approximations. Ph.D. thesis, University of Michigan, USA (2002)
Villemonteix J., Vazquez E., Walter E.: An informational approach to the global optimization of expensive-to-evaluate functions. J. Glob. Optim. 44(4), 509–534 (2009)
Vazquez E., Bect J.: Convergence properties of the expected improvement algorithm with fixed mean and covariance functions. J. Stat. Plan. Inference 140(11), 3088–3095 (2010)
Huang D., Allen T.T.: Design and analysis of variable fidelity experimentation applied to engine valve heat treatment process design. J. R. Stat. Soc. Ser. C Appl. Stat. 54(2), 443–463 (2005)
Villemonteix, J., Vazquez, E., Walter, E.: Bayesian optimization for parameter identification on a small simulation budget. In: Proceedings of the 15th IFAC Symposium on System Identification, SYSID 2009, Saint-Malo France (2009)
Marzat, J., Walter, E., Piet-Lahanier, H., Damongeot, F.: Automatic tuning via Kriging-based optimization of methods for fault detection and isolation. In: Proceedings of the IEEE Conference on Control and Fault-Tolerant Systems, SYSTOL 2010, Nice, France, pp. 505–510 (2010)
Defretin, J., Marzat, J., Piet-Lahanier, H.: Learning viewpoint planning in active recognition on a small sampling budget: a Kriging approach. In: Proceedings of the 9th IEEE International Conference on Machine Learning and Applications, ICMLA 2010, Washington, USA, pp. 169–174 (2010)
Beyer H.G., Sendhoff B.: Robust optimization—a comprehensive survey. Comput. Methods Appl. Mech. Eng. 196(33–34), 3190–3218 (2007)
Dellino G., Kleijnen J.P.C., Meloni C.: Robust optimization in simulation: Taguchi and response surface methodology. Int. J. Prod. Econ. 125(1), 52–59 (2010)
Chen W., Allen J.K., Tsui K.L., Mistree F.: A procedure for robust design: minimizing variations caused by noise factors and control factors. ASME J. Mech. Des. 118, 478–485 (1996)
Lee, K., Park, G., Joo, W.: A global robust optimization using the Kriging based approximation model. In: Proceedings of the 6th World Congresses of Structural and Multidisciplinary Optimization, Rio de Janeiro, Brazil (2005)
Williams B.J., Santner T.J., Notz W.I.: Sequential design of computer experiments to minimize integrated response functions. Statistica Sinica 10(4), 1133–1152 (2000)
Lehman J.S., Santner T.J., Notz W.I.: Designing computer experiments to determine robust control variables. Statistica Sinica 14(2), 571–590 (2004)
Lam, C.Q.: Sequential adaptive designs in computer experiments for response surface model fit. Ph.D. thesis, The Ohio State University (2008)
Cramer A.M., Sudhoff S.D., Zivi E.L.: Evolutionary algorithms for minimax problems in robust design. IEEE Trans. Evolut. Comput. 13(2), 444–453 (2009)
Lung, R.I., Dumitrescu, D.: A new evolutionary approach to minimax problems. In: Proceedings of the 2011 IEEE Congress on Evolutionary Computation, New Orleans, USA, pp. 1902–1905 (2011)
Zhou, A., Zhang, Q.: A surrogate-assisted evolutionary algorithm for minimax optimization. In: Proceedings of the 2010 IEEE Congress on Evolutionary Computation, Barcelona, Spain, pp. 1–7 (2010)
Shimizu K., Aiyoshi E.: Necessary conditions for min-max problems and algorithms by a relaxation procedure. IEEE Trans. Autom. Control 25(1), 62–66 (1980)
Rustem B., Howe M.: Algorithms for Worst-Case Design and Applications to Risk Management. Princeton University Press, Princeton, NJ (2002)
Brown B., Singh T.: Minimax design of vibration absorbers for linear damped systems. J. Sound Vib. 330(11), 2437–2448 (2011)
Salmon D.M.: Minimax controller design. IEEE Trans. Autom. Control 13(4), 369–376 (1968)
Helton J.: Worst case analysis in the frequency domain: the H ∞ approach to control. IEEE Trans. Autom. Control 30(12), 1154–1170 (1985)
Chow E.Y., Willsky A.S.: Analytical redundancy and the design of robust failure detection systems. IEEE Trans. Autom. Control 29, 603–614 (1984)
Frank P.M., Ding X.: Survey of robust residual generation and evaluation methods in observer-based fault detection systems. J. Process Control 7(6), 403–424 (1997)
Colson B., Marcotte P., Savard G.: An overview of bilevel optimization. Ann. Oper. Res. 153(1), 235–256 (2007)
Ben-Tal A., Nemirovski A.: Robust convex optimization. Math. Oper. Res. 23(4), 769–805 (1998)
Başar T., Olsder G.J.: Dynamic Noncooperative Game Theory. Society for Industrial Mathematics, New York, NY (1999)
Du D., Pardalos P.M.: Minimax and Applications. Kluwer, Norwell (1995)
Parpas P., Rustem B.: An algorithm for the global optimization of a class of continuous minimax problems. J. Optim. Theory Appl. 141(2), 461–473 (2009)
Tsoukalas A., Rustem B., Pistikopoulos E.N.: A global optimization algorithm for generalized semi-infinite, continuous minimax with coupled constraints and bi-level problems. J. Glob. Optim. 44(2), 235–250 (2009)
Rustem B.: Algorithms for Nonlinear Programming and Multiple Objective Decisions. Wiley, Chichester (1998)
Shimizu K., Ishizuka Y., Bard J.F.: Nondifferentiable and Two-level Mathematical Programming. Kluwer, Norwell (1997)
MacKay D.J.C.: Information Theory, Inference, and Learning Algorithms. Cambridge University Press, Cambridge, MA (2003)
Schonlau, M.: Computer experiments and global optimization. Ph.D. thesis, University of Waterloo, Canada (1997)
Jones D.R., Perttunen C.D., Stuckman B.E.: Lipschitzian optimization without the Lipschitz constant. J. Optim. Theory Appl. 79(1), 157–181 (1993)
Basseville M., Nikiforov I.V.: Detection of Abrupt Changes: Theory and Application. Prentice Hall, Englewood Cliffs, NJ (1993)
Bartyś M., Patton R.J., Syfert M., delas Heras S., Quevedo J.: Introduction to the DAMADICS actuator FDI benchmark study. Control Eng. Pract. 14(6), 577–596 (2006)
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Marzat, J., Walter, E. & Piet-Lahanier, H. Worst-case global optimization of black-box functions through Kriging and relaxation. J Glob Optim 55, 707–727 (2013). https://doi.org/10.1007/s10898-012-9899-y
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DOI: https://doi.org/10.1007/s10898-012-9899-y