Comparison of derivative-free optimization methods for groundwater supply and hydraulic capture community problems

Management decisions involving groundwater supply and remediation often rely on optimization techniques to determine an effective strategy. We introduce several derivative-free sampling methods for solving constrained optimization problems that have not yet been considered in this field, and we include a genetic algorithm for completeness. Two well-documented community problems are used for illustration purposes: a groundwater supply problem and a hydraulic capture problem. The community problems were found to be challenging applications due to the objective functions being nonsmooth, nonlinear, and having many local minima. Because the results were found to be sensitive to initial iterates for some methods, guidance is provided in selecting initial iterates for these problems that improve the likelihood of achieving significant reductions in the objective function to be minimized. In addition, we suggest some potentially fruitful areas for future research.

[1]  John H. Holland,et al.  Genetic Algorithms and the Optimal Allocation of Trials , 1973, SIAM J. Comput..

[2]  T. Kolda,et al.  Nonlinearly-constrained optimization using asynchronous parallel generating set search. , 2007 .

[3]  Charles Audet,et al.  Analysis of Generalized Pattern Searches , 2000, SIAM J. Optim..

[4]  Tamara G. Kolda,et al.  Revisiting Asynchronous Parallel Pattern Search for Nonlinear Optimization , 2005, SIAM J. Optim..

[5]  Kalyanmoy Deb,et al.  Simulated Binary Crossover for Continuous Search Space , 1995, Complex Syst..

[6]  C. T. Kelley,et al.  A Locally-Biased form of the DIRECT Algorithm , 2001, J. Glob. Optim..

[7]  Tamara G. Kolda,et al.  Asynchronous Parallel Generating Set Search for Linearly Constrained Optimization , 2008, SIAM J. Sci. Comput..

[8]  V. Torczon,et al.  RANK ORDERING AND POSITIVE BASES IN PATTERN SEARCH ALGORITHMS , 1996 .

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

[10]  D. Ahlfeld,et al.  Advective control of groundwater contaminant plumes: Model development and comparison to hydraulic control , 1999 .

[11]  Daniel E. Finkel,et al.  Global optimization with the direct algorithm , 2005 .

[12]  C. D. Gelatt,et al.  Optimization by Simulated Annealing , 1983, Science.

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

[14]  Carl Tim Kelley,et al.  Iterative methods for optimization , 1999, Frontiers in applied mathematics.

[15]  Ali S. Hadi,et al.  Finding Groups in Data: An Introduction to Chster Analysis , 1991 .

[16]  T. Tsuchiya,et al.  On the formulation and theory of the Newton interior-point method for nonlinear programming , 1996 .

[17]  Tamara G. Kolda,et al.  On the Convergence of Asynchronous Parallel Pattern Search , 2002, SIAM J. Optim..

[18]  R. Tapia,et al.  Numerical Comparisons of Path-Following Strategies for a Primal-Dual Interior-Point Method for Nonlinear Programming , 2002 .

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

[20]  Christine A. Shoemaker,et al.  Constrained Global Optimization of Expensive Black Box Functions Using Radial Basis Functions , 2005, J. Glob. Optim..

[21]  Evin J. Cramer,et al.  Effective Parallel Optimization of Complex Computer Simulations , 2004 .

[22]  Patrick M. Reed,et al.  Striking the Balance: Long-Term Groundwater Monitoring Design for Conflicting Objectives , 2004 .

[23]  Juan C. Meza,et al.  OPT++: An object-oriented class library for nonlinear optimization , 1994 .

[24]  J. M. Gablonsky.  Direct version 2.0 user guide , 2001 .

[25]  Cass T. Miller,et al.  Solution of a Well-Field Design Problem with Implicit Filtering , 2004 .

[26]  Cass T. Miller,et al.  Optimal design for problems involving flow and transport phenomena in saturated subsurface systems , 2002 .

[27]  Oskar von Stryk,et al.  A mixed-integer simulation-based optimization approach with surrogate functions in water resources management , 2008 .

[28]  Charles Audet,et al.  A Pattern Search Filter Method for Nonlinear Programming without Derivatives , 2001, SIAM J. Optim..

[29]  R. Tapia,et al.  On the Global Convergence of a Modified Augmented Lagrangian Linesearch Interior-Point Newton Method for Nonlinear Programming , 2002 .

[30]  D. Finkel,et al.  Convergence analysis of the direct algorithm , 2004 .

[31]  Arlen W. Harbaugh,et al.  A modular three-dimensional finite-difference ground-water flow model , 1984 .

[32]  Tamara G. Kolda,et al.  Asynchronous Parallel Pattern Search for Nonlinear Optimization , 2001, SIAM J. Sci. Comput..

[33]  D. McKinney,et al.  Genetic algorithm solution of groundwater management models , 1994 .

[34]  C. T. Kelley,et al.  Modifications of the direct algorithm , 2001 .

[35]  Tamara G. Kolda,et al.  Understanding Asynchronous Parallel Pattern Search , 2003 .

[36]  Christopher E. Kees,et al.  A hydraulic capture application for optimal remediation design , 2004 .

[37]  R. C. Peralta,et al.  Closure to discussion on Optimal in-situ bioremediation design by hybrid genetic algorithm-simulated annealing , 2005 .

[38]  C. Kelley,et al.  Yield optimization using a GaAs process simulator coupled to a physical device model , 1992 .

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

[40]  David E. Goldberg,et al.  Designing a competent simple genetic algorithm for search and optimization , 2000 .

[41]  C. T. Kelley,et al.  An Implicit Filtering Algorithm for Optimization of Functions with Many Local Minima , 1995, SIAM J. Optim..

[42]  Arlen W. Harbaugh,et al.  User's documentation for MODFLOW-96, an update to the U.S. Geological Survey modular finite-difference ground-water flow model , 1996 .

[43]  Robert A. Marryott,et al.  Optimal Groundwater Management: 1. Simulated Annealing , 1991 .

[44]  O. V. Stryk,et al.  DERIVATIVE-FREE OPTIMIZATION METHODS FOR HANDLING FIXED COSTS IN OPTIMAL GROUNDWATER REMEDIATION DESIGN , 2006 .

[45]  Kalyanmoy Deb,et al.  Controlled Elitist Non-dominated Sorting Genetic Algorithms for Better Convergence , 2001, EMO.

[46]  Kenneth Holmström,et al.  Global Optimization Using the DIRECT Algorithm in Matlab , 1999 .

[47]  Owen J. Eslinger,et al.  Algorithms for Noisy Problems in Gas Transmission Pipeline Optimization , 2001 .

[48]  Barbara S. Minsker,et al.  Which Groundwater Remediation Objective is Better: A Realistic One or a Simple One? , 2005 .

[49]  Kalyanmoy Deb,et al.  Self-Adaptive Genetic Algorithms with Simulated Binary Crossover , 2001, Evolutionary Computation.

[50]  David E. Goldberg,et al.  Genetic Algorithms in Search Optimization and Machine Learning , 1988 .

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

[52]  Tamara G. Kolda,et al.  Stationarity Results for Generating Set Search for Linearly Constrained Optimization , 2006, SIAM J. Optim..

[53]  Jeffrey Horn,et al.  Multi-objective optimal design of groundwater remediation systems: application of the niched Pareto genetic algorithm (NPGA) , 2002 .

[54]  K. Deb.  An Efficient Constraint Handling Method for Genetic Algorithms , 2000 .

[55]  Maria da Conceição Cunha,et al.  On solving aquifer management problems with simulated annealing algorithms , 1999 .

[56]  Owen J. Eslinger,et al.  IFFCO: implicit filtering for constrained optimization, version 2 , 1999 .

[57]  John E. Dennis,et al.  Numerical methods for unconstrained optimization and nonlinear equations , 1983, Prentice Hall series in computational mathematics.

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

[59]  Sven Leyffer,et al.  Nonlinear programming without a penalty function , 2002, Math. Program..

[60]  David P. Ahlfeld,et al.  Solution of the groundwater transport management problem by sequential relaxation , 1998 .

[61]  K. R. Fowler,et al.  Approaching the Groundwater Remediation Problem Using Multifidelity Optimization. , 2006 .

[62]  Paul P. Wang,et al.  MT3DMS: A Modular Three-Dimensional Multispecies Transport Model for Simulation of Advection, Dispersion, and Chemical Reactions of Contaminants in Groundwater Systems; Documentation and User's Guide , 1999 .

[63]  Lothar Thiele,et al.  Comparison of Multiobjective Evolutionary Algorithms: Empirical Results , 2000, Evolutionary Computation.

[64]  D. McKinney,et al.  Approximate Mixed‐Integer Nonlinear Programming Methods for Optimal Aquifer Remediation Design , 1995 .

[65]  Cass T. Miller,et al.  Solution of a Groundwater Control Problem with Implicit Filtering , 2002 .

[66]  John H. Holland,et al.  Adaptation in Natural and Artificial Systems: An Introductory Analysis with Applications to Biology, Control, and Artificial Intelligence , 1992 .

[67]  Efficient Parallel Implementation of Hybrid Optimization Approaches for Solving Groundwater Inverse Problems , 2005 .

[68]  Tamara G. Kolda,et al.  Algorithm 856: APPSPACK 4.0: asynchronous parallel pattern search for derivative-free optimization , 2006, TOMS.