Modeling an augmented Lagrangian for improved blackbox constrained optimization

Constrained blackbox optimization is a difficult problem, with most approaches coming from the mathematical programming literature. The statistical literature is sparse, especially in addressing problems with nontrivial constraints. This situation is unfortunate because statistical methods have many attractive properties: global scope, handling noisy objectives, sensitivity analysis, and so forth. To narrow that gap, we propose a combination of response surface modeling, expected improvement, and the augmented Lagrangian numerical optimization framework. This hybrid approach allows the statistical model to think globally and the augmented Lagrangian to act locally. We focus on problems where the constraints are the primary bottleneck, requiring expensive simulation to evaluate and substantial modeling effort to map out. In that context, our hybridization presents a simple yet effective solution that allows existing objective-oriented statistical approaches, like those based on Gaussian process surrogates and expected improvement heuristics, to be applied to the constrained setting with minor modification. This work is motivated by a challenging, real-data benchmark problem from hydrology where, even with a simple linear objective function, learning a nontrivial valid region complicates the search for a global minimum.

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

[2]  Robert B. Gramacy,et al.  Particle Learning of Gaussian Process Models for Sequential Design and Optimization , 2009, 0909.5262.

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

[4]  Sébastien Le Digabel,et al.  Algorithm xxx : NOMAD : Nonlinear Optimization with the MADS algorithm , 2010 .

[5]  Stephen J. Wright,et al.  Numerical Optimization , 2018, Fundamental Statistical Inference.

[6]  Katya Scheinberg,et al.  Introduction to derivative-free optimization , 2010, Math. Comput..

[7]  Christine A. Shoemaker,et al.  Global Convergence of Radial Basis Function Trust-Region Algorithms for Derivative-Free Optimization , 2013, SIAM Rev..

[8]  Sébastien Le Digabel,et al.  Use of quadratic models with mesh-adaptive direct search for constrained black box optimization , 2011, Optim. Methods Softw..

[9]  A. O'Hagan,et al.  Bayesian inference for non‐stationary spatial covariance structure via spatial deformations , 2003 .

[10]  Victor Picheny,et al.  Quantile-Based Optimization of Noisy Computer Experiments With Tunable Precision , 2013, Technometrics.

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

[12]  Adam D. Bull,et al.  Convergence Rates of Efficient Global Optimization Algorithms , 2011, J. Mach. Learn. Res..

[13]  Robert B. Gramacy,et al.  Ja n 20 08 Bayesian Treed Gaussian Process Models with an Application to Computer Modeling , 2009 .

[14]  Nando de Freitas,et al.  A Tutorial on Bayesian Optimization of Expensive Cost Functions, with Application to Active User Modeling and Hierarchical Reinforcement Learning , 2010, ArXiv.

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

[16]  Cass T. Miller,et al.  Optimal design for problems involving flow and transport phenomena in subsurface systems , 2001 .

[17]  Phillip Boyle,et al.  Gaussian Processes for Regression and Optimisation , 2007 .

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

[19]  Charles Audet,et al.  A surrogate-model-based method for constrained optimization , 2000 .

[20]  Charles Audet,et al.  A MADS Algorithm with a Progressive Barrier for Derivative-Free Nonlinear Programming , 2007 .

[21]  Derek J. Pike,et al.  Empirical Model‐building and Response Surfaces. , 1988 .

[22]  Hans van Duijne,et al.  Chapter 7.1:Prevention and Reduction of Groundwater Pollution at Contaminated Megasites: Integrated Management Strategy and its Application on Megasite Cases , 2007 .

[23]  Dimitri P. Bertsekas,et al.  Constrained Optimization and Lagrange Multiplier Methods , 1982 .

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

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

[26]  Charles Audet,et al.  Reducing the Number of Function Evaluations in Mesh Adaptive Direct Search Algorithms , 2012, SIAM J. Optim..

[27]  Herbert K. H. Lee,et al.  Bayesian Guided Pattern Search for Robust Local Optimization , 2009, Technometrics.

[28]  Stefan M. Wild,et al.  BENEFITS OF DEEPER ANALYSIS IN SIMULATION- BASED GROUNDWATER OPTIMIZATION PROBLEMS , 2012 .

[29]  Robert Hooke,et al.  `` Direct Search'' Solution of Numerical and Statistical Problems , 1961, JACM.

[30]  Robert B. Gramacy,et al.  Enhancing Parallel Pattern Search Optimization with a Gaussian Process Oracle , 2006 .

[31]  James R. Craig,et al.  Pump-and-treat optimization using analytic element method flow models , 2006 .

[32]  Christopher J Paciorek,et al.  Spatial modelling using a new class of nonstationary covariance functions , 2006, Environmetrics.

[33]  L. Shawn Matott,et al.  Application of MATLAB and Python optimizers to two case studies involving groundwater flow and contaminant transport modeling , 2011, Comput. Geosci..

[34]  Charles Audet,et al.  Comparison of derivative-free optimization methods for groundwater supply and hydraulic capture community problems , 2008 .

[35]  A. J. Booker,et al.  A rigorous framework for optimization of expensive functions by surrogates , 1998 .

[36]  Tamara G. Kolda,et al.  Optimization by Direct Search: New Perspectives on Some Classical and Modern Methods , 2003, SIAM Rev..

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

[38]  Charles Audet,et al.  Mesh Adaptive Direct Search Algorithms for Constrained Optimization , 2006, SIAM J. Optim..

[39]  David Volent Lindberg,et al.  Optimization Under Constraints by Applying an Asymmetric Entropy Measure , 2015 .

[40]  Sébastien Le Digabel,et al.  The mesh adaptive direct search algorithm with treed Gaussian process surrogates , 2011 .

[41]  Stanley H. Cohen,et al.  Design and Analysis , 2010 .