Worst-case global optimization of black-box functions through Kriging and relaxation

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.

[1]  Scott D. Sudhoff,et al.  Evolutionary Algorithms for Minimax Problems in Robust Design , 2009, IEEE Transactions on Evolutionary Computation.

[2]  Julien Marzat,et al.  Automatic tuning via Kriging-based optimization of methods for fault detection and isolation , 2010, 2010 Conference on Control and Fault-Tolerant Systems (SysTol).

[3]  P. Pardalos,et al.  Minimax and applications , 1995 .

[4]  Thomas J. Santner,et al.  The Design and Analysis of Computer Experiments , 2003, Springer Series in Statistics.

[5]  Thomas J. Santner,et al.  Sequential design of computer experiments to minimize integrated response functions , 2000 .

[6]  Dumitru Dumitrescu,et al.  A new evolutionary approach to minimax problems , 2011, 2011 IEEE Congress of Evolutionary Computation (CEC).

[7]  Michael James Sasena,et al.  Flexibility and efficiency enhancements for constrained global design optimization with kriging approximations. , 2002 .

[8]  Julien Marzat,et al.  Learning Viewpoint Planning in Active Recognition on a Small Sampling Budget: A Kriging Approach , 2010, 2010 Ninth International Conference on Machine Learning and Applications.

[9]  M. D. McKay,et al.  A comparison of three methods for selecting values of input variables in the analysis of output from a computer code , 2000 .

[10]  Jack P. C. Kleijnen,et al.  Robust Optimization in Simulation: Taguchi and Response Surface Methodology , 2008 .

[11]  J. W. Helton,et al.  Worst case analysis in the frequency domain: The H ∞ approach to control , 1985 .

[12]  Michèle Basseville,et al.  Detection of abrupt changes: theory and application , 1993 .

[13]  G. Matheron.  Principles of geostatistics , 1963 .

[14]  T. Başar,et al.  Dynamic Noncooperative Game Theory , 1982 .

[15]  C. D. Perttunen,et al.  Lipschitzian optimization without the Lipschitz constant , 1993 .

[16]  S. Scholtes,et al.  Nondifferentiable and two-level mathematical programming , 1997 .

[17]  Farrokh Mistree,et al.  A procedure for robust design: Minimizing variations caused by noise factors and control factors , 1996 .

[18]  Arkadi Nemirovski,et al.  Robust Convex Optimization , 1998, Math. Oper. Res..

[19]  Kwon-Hee Lee,et al.  Th World Congresses of Structural and Multidisciplinary Optimization a Global Robust Optimization Using the Kriging Based Approximation Model , 2022 .

[20]  Berç Rustem,et al.  An Algorithm for the Global Optimization of a Class of Continuous Minimax Problems , 2009 .

[21]  Timothy W. Simpson,et al.  Metamodels for Computer-based Engineering Design: Survey and recommendations , 2001, Engineering with Computers.

[22]  Qingfu Zhang,et al.  A surrogate-assisted evolutionary algorithm for minimax optimization , 2010, IEEE Congress on Evolutionary Computation.

[23]  Andy J. Keane,et al.  Engineering Design via Surrogate Modelling - A Practical Guide , 2008 .

[24]  David J. C. MacKay,et al.  Information Theory, Inference, and Learning Algorithms , 2004, IEEE Transactions on Information Theory.

[25]  Donald R. Jones,et al.  A Taxonomy of Global Optimization Methods Based on Response Surfaces , 2001, J. Glob. Optim..

[26]  Raphael T. Haftka,et al.  Surrogate-based Analysis and Optimization , 2005 .

[27]  Eric Walter,et al.  Bayesian optimization for parameter identification on a small simulation budget , 2009 .

[28]  William J. Welch,et al.  Computer experiments and global optimization , 1997 .

[29]  A. Willsky,et al.  Analytical redundancy and the design of robust failure detection systems , 1984 .

[30]  Theodore T. Allen,et al.  Design and analysis of variable fidelity experimentation applied to engine valve heat treatment process design , 2005 .

[31]  H. Luthi.  Algorithms for Worst-Case Design and Applications to Risk Management:Algorithms for Worst-Case Design and Applications to Risk Management , 2002 .

[32]  E. Aiyoshi,et al.  Necessary conditions for min-max problems and algorithms by a relaxation procedure , 1980 .

[33]  P. Frank,et al.  Survey of robust residual generation and evaluation methods in observer-based fault detection systems , 1997 .

[34]  Donald R. Jones,et al.  Efficient Global Optimization of Expensive Black-Box Functions , 1998, J. Glob. Optim..

[35]  Berç Rustem.  Algorithms for nonlinear programming and multiple-objective decisions , 1998, Wiley-Interscience series in discrete mathematics and optimization.

[36]  Joseba Quevedo,et al.  Introduction to the DAMADICS actuator FDI benchmark study , 2006 .

[37]  E. Vázquez,et al.  Convergence properties of the expected improvement algorithm with fixed mean and covariance functions , 2007, 0712.3744.

[38]  Eric Walter,et al.  An informational approach to the global optimization of expensive-to-evaluate functions , 2006, J. Glob. Optim..

[39]  Berç Rustem,et al.  A global optimization algorithm for generalized semi-infinite, continuous minimax with coupled constraints and bi-level problems , 2009, J. Glob. Optim..

[40]  Bernhard Sendhoff,et al.  Robust Optimization - A Comprehensive Survey , 2007 .

[41]  D. Salmon,et al.  Minimax Controller Design , 1968 .

[42]  李幼升,et al.  Ph , 1989 .

[43]  William I. Notz,et al.  Sequential adaptive designs in computer experiments for response surface model fit , 2008 .

[44]  Tarunraj Singh,et al.  Minimax design of vibration absorbers for linear damped systems , 2011 .

[45]  N. Zheng,et al.  Global Optimization of Stochastic Black-Box Systems via Sequential Kriging Meta-Models , 2006, J. Glob. Optim..

[46]  William I. Notz,et al.  DESIGNING COMPUTER EXPERIMENTS TO DETERMINE ROBUST CONTROL VARIABLES , 2004 .

[47]  Carl E. Rasmussen,et al.  Gaussian processes for machine learning , 2005, Adaptive computation and machine learning.

[48]  Patrice Marcotte,et al.  An overview of bilevel optimization , 2007, Ann. Oper. Res..

[49]  Richard J. Beckman,et al.  A Comparison of Three Methods for Selecting Values of Input Variables in the Analysis of Output From a Computer Code , 2000, Technometrics.