A Bayesian approach to constrained single- and multi-objective optimization

This article addresses the problem of derivative-free (single- or multi-objective) optimization subject to multiple inequality constraints. Both the objective and constraint functions are assumed to be smooth, non-linear and expensive to evaluate. As a consequence, the number of evaluations that can be used to carry out the optimization is very limited, as in complex industrial design optimization problems. The method we propose to overcome this difficulty has its roots in both the Bayesian and the multi-objective optimization literatures. More specifically, an extended domination rule is used to handle objectives and constraints in a unified way, and a corresponding expected hyper-volume improvement sampling criterion is proposed. This new criterion is naturally adapted to the search of a feasible point when none is available, and reduces to existing Bayesian sampling criteria—the classical Expected Improvement (EI) criterion and some of its constrained/multi-objective extensions—as soon as at least one feasible point is available. The calculation and optimization of the criterion are performed using Sequential Monte Carlo techniques. In particular, an algorithm similar to the subset simulation method, which is well known in the field of structural reliability, is used to estimate the criterion. The method, which we call BMOO (for Bayesian Multi-Objective Optimization), is compared to state-of-the-art algorithms for single- and multi-objective constrained optimization.

[1]  Sébastien Le Digabel,et al.  Modeling an Augmented Lagrangian for Blackbox Constrained Optimization , 2014, Technometrics.

[2]  Eric Moulines,et al.  Comparison of resampling schemes for particle filtering , 2005, ISPA 2005. Proceedings of the 4th International Symposium on Image and Signal Processing and Analysis, 2005..

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

[4]  Kalyanmoy Deb,et al.  A fast and elitist multiobjective genetic algorithm: NSGA-II , 2002, IEEE Trans. Evol. Comput..

[5]  Michael A. Gelbart,et al.  Constrained Bayesian Optimization and Applications , 2015 .

[6]  Thomas J. Santner,et al.  Sequential Design of Computer Experiments for Constrained Optimization , 2010 .

[7]  Qingfu Zhang,et al.  Expensive Multiobjective Optimization by MOEA/D With Gaussian Process Model , 2010, IEEE Transactions on Evolutionary Computation.

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

[9]  M. Sasena,et al.  Exploration of Metamodeling Sampling Criteria for Constrained Global Optimization , 2002 .

[10]  J. Mockus.  Bayesian Approach to Global Optimization: Theory and Applications , 1989 .

[11]  Ling Li,et al.  Sequential design of computer experiments for the estimation of a probability of failure , 2010, Statistics and Computing.

[12]  Victor Picheny,et al.  A Stepwise uncertainty reduction approach to constrained global optimization , 2014, AISTATS.

[13]  Andy J. Keane,et al.  Statistical Improvement Criteria for Use in Multiobjective Design Optimization , 2006 .

[14]  C. Andrieu,et al.  The pseudo-marginal approach for efficient Monte Carlo computations , 2009, 0903.5480.

[15]  Ling Li,et al.  Sequential Design of Experiments to Estimate a Probability of Failure. , 2012 .

[16]  F. Archetti,et al.  A probabilistic algorithm for global optimization , 1979 .

[17]  Shigeru Obayashi,et al.  Optimization of Combustion Chamber for Diesel Engine Using Kriging Model , 2006 .

[18]  A. Oyama,et al.  New Constraint-Handling Method for Multi-Objective and Multi-Constraint Evolutionary Optimization , 2007 .

[19]  Hoon Kim,et al.  Monte Carlo Statistical Methods , 2000, Technometrics.

[20]  R. Regis.  Constrained optimization by radial basis function interpolation for high-dimensional expensive black-box problems with infeasible initial points , 2014 .

[21]  Wolfgang Ponweiser,et al.  On Expected-Improvement Criteria for Model-based Multi-objective Optimization , 2010, PPSN.

[22]  Donald R. Jones,et al.  Global versus local search in constrained optimization of computer models , 1998 .

[23]  Ling Li,et al.  Bayesian Subset Simulation: a kriging-based subset simulation algorithm for the estimation of small probabilities of failure , 2012, 1207.1963.

[24]  Julien Bect,et al.  A new integral loss function for Bayesian optimization , 2014, ArXiv.

[25]  Harold J. Kushner,et al.  A New Method of Locating the Maximum Point of an Arbitrary Multipeak Curve in the Presence of Noise , 1964 .

[26]  Shigeru Obayashi,et al.  Updating Kriging Surrogate Models Based on the Hypervolume Indicator in Multi-Objective Optimization , 2013 .

[27]  Yaochu Jin,et al.  Surrogate-assisted evolutionary computation: Recent advances and future challenges , 2011, Swarm Evol. Comput..

[28]  K. C. Seow,et al.  MULTIOBJECTIVE DESIGN OPTIMIZATION BY AN EVOLUTIONARY ALGORITHM , 2001 .

[29]  Jerome Sacks,et al.  Choosing the Sample Size of a Computer Experiment: A Practical Guide , 2009, Technometrics.

[30]  Nicola Beume,et al.  S-Metric Calculation by Considering Dominated Hypervolume as Klee's Measure Problem , 2009, Evolutionary Computation.

[31]  D. Ginsbourger,et al.  Towards Gaussian Process-based Optimization with Finite Time Horizon , 2010 .

[32]  Sonja Kuhnt,et al.  Design and analysis of computer experiments , 2010 .

[33]  Michael L. Stein,et al.  Interpolation of spatial data , 1999 .

[34]  Michael T. M. Emmerich,et al.  Faster Exact Algorithms for Computing Expected Hypervolume Improvement , 2015, EMO.

[35]  P. Moral,et al.  Sequential Monte Carlo samplers , 2002, cond-mat/0212648.

[36]  P. Toint,et al.  A globally convergent augmented Lagrangian algorithm for optimization with general constraints and simple bounds , 1991 .

[37]  Victor Picheny,et al.  Multiobjective optimization using Gaussian process emulators via stepwise uncertainty reduction , 2013, Statistics and Computing.

[38]  Daniel Hern'andez-Lobato,et al.  Predictive Entropy Search for Multi-objective Bayesian Optimization with Constraints , 2016, Neurocomputing.

[39]  Roger Woodard,et al.  Interpolation of Spatial Data: Some Theory for Kriging , 1999, Technometrics.

[40]  M. Emmerich,et al.  The computation of the expected improvement in dominated hypervolume of Pareto front approximations , 2008 .

[41]  Joshua D. Knowles,et al.  Multiobjective Optimization on a Budget of 250 Evaluations , 2005, EMO.

[42]  Christian P. Robert,et al.  Monte Carlo Statistical Methods (Springer Texts in Statistics) , 2005 .

[43]  Matthew W. Hoffman,et al.  Predictive Entropy Search for Bayesian Optimization with Unknown Constraints , 2015, ICML.

[44]  J. Beck,et al.  Estimation of Small Failure Probabilities in High Dimensions by Subset Simulation , 2001 .

[45]  Thomas J. Santner,et al.  Multiobjective Optimization of Expensive Black-box Functions via Expected Maximin Improvement , 2022 .

[46]  Carl E. Rasmussen,et al.  Warped Gaussian Processes , 2003, NIPS.

[47]  Dianne T. Bautista.  A sequential design for approximating the Pareto front using the expected Pareto improvement function , 2009 .

[48]  Joshua D. Knowles,et al.  ParEGO: a hybrid algorithm with on-line landscape approximation for expensive multiobjective optimization problems , 2006, IEEE Transactions on Evolutionary Computation.

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

[50]  James M. Parr,et al.  Infill sampling criteria for surrogate-based optimization with constraint handling , 2012 .

[51]  Shigeru Obayashi,et al.  Efficient global optimization (EGO) for multi-objective problem and data mining , 2005, 2005 IEEE Congress on Evolutionary Computation.

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

[53]  Peter J. Fleming,et al.  Multiobjective optimization and multiple constraint handling with evolutionary algorithms. I. A unified formulation , 1998, IEEE Trans. Syst. Man Cybern. Part A.

[54]  Eckart Zitzler,et al.  HypE: An Algorithm for Fast Hypervolume-Based Many-Objective Optimization , 2011, Evolutionary Computation.

[55]  D. Cox,et al.  An Analysis of Transformations , 1964 .

[56]  Neil D. Lawrence,et al.  Deep Gaussian Processes , 2012, AISTATS.

[57]  M. Powell.  A Direct Search Optimization Method That Models the Objective and Constraint Functions by Linear Interpolation , 1994 .

[58]  Romain Benassi,et al.  Nouvel algorithme d'optimisation bayésien utilisant une approche Monte-Carlo séquentielle. , 2013 .

[59]  Michael T. M. Emmerich,et al.  Faster Computation of Expected Hypervolume Improvement , 2014, ArXiv.

[60]  Tom Dhaene,et al.  Fast calculation of multiobjective probability of improvement and expected improvement criteria for Pareto optimization , 2014, J. Glob. Optim..

[61]  Khaled Rasheed,et al.  Constrained Multi-objective Optimization Using Steady State Genetic Algorithms , 2003, GECCO.

[62]  Tim Hesterberg,et al.  Monte Carlo Strategies in Scientific Computing , 2002, Technometrics.

[63]  Andy J. Keane,et al.  Non-stationary kriging for design optimization , 2012 .

[64]  Michael T. M. Emmerich,et al.  Single- and multiobjective evolutionary optimization assisted by Gaussian random field metamodels , 2006, IEEE Transactions on Evolutionary Computation.

[65]  Bernd Bischl,et al.  Model-Based Multi-objective Optimization: Taxonomy, Multi-Point Proposal, Toolbox and Benchmark , 2015, EMO.

[66]  Pierre Del Moral,et al.  Sequential Monte Carlo for rare event estimation , 2012, Stat. Comput..

[67]  Gareth O. Roberts,et al.  Examples of Adaptive MCMC , 2009 .

[68]  Marco Laumanns,et al.  SPEA2: Improving the Strength Pareto Evolutionary Algorithm For Multiobjective Optimization , 2002 .

[69]  Christophe Andrieu,et al.  A tutorial on adaptive MCMC , 2008, Stat. Comput..

[70]  Marco Laumanns,et al.  SPEA2: Improving the strength pareto evolutionary algorithm , 2001 .

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

[72]  Wolfgang Ponweiser,et al.  Multiobjective Optimization on a Limited Budget of Evaluations Using Model-Assisted -Metric Selection , 2008, PPSN.

[73]  Robert B. Gramacy,et al.  Optimization Under Unknown Constraints , 2010, 1004.4027.

[74]  J. Mockus,et al.  The Bayesian approach to global optimization , 1989 .

[75]  Jasper Snoek,et al.  Bayesian Optimization with Unknown Constraints , 2014, UAI.

[76]  Julien Bect,et al.  Bayesian Optimization Using Sequential Monte Carlo , 2012, LION.

[77]  Victor Picheny,et al.  Fast Parallel Kriging-Based Stepwise Uncertainty Reduction With Application to the Identification of an Excursion Set , 2014, Technometrics.