A surrogate-assisted metaheuristic for bilevel optimization

A Bilevel Optimization Problem (BOP) is related to two optimization problems in a hierarchical structure. A BOP is solved when an optimum of the upper level problem is found, subject to the optimal response of the respective lower level problem. This paper presents a metaheuristic method assisted by a kernel interpolation numerical technique to approximate optimal solutions of a BOP. Two surrogate methods approximate upper and lower level objective functions on solutions obtained by a population-based algorithm adapted to save upper level objective function evaluations. Some theoretical properties about kernel interpolation are used to study global convergence in some particular BOPs. The empirical results of this approach are analyzed when representative test functions for bilevel optimization are solved. The overall performance provided by the proposal is competitive.

[1]  Nello Cristianini,et al.  Kernel Methods for Pattern Analysis , 2003, ICTAI.

[2]  Bernhard Schölkopf,et al.  Comparing support vector machines with Gaussian kernels to radial basis function classifiers , 1997, IEEE Trans. Signal Process..

[3]  R. Fletcher.  Practical Methods of Optimization , 1988 .

[4]  Helio J. C. Barbosa,et al.  Performance Evaluation of Local Surrogate Models in Bilevel Optimization , 2019, LOD.

[5]  Efrén Mezura-Montes,et al.  A New Evolutionary Optimization Method Based on Center of Mass , 2018, Decision Science in Action.

[6]  Kalyanmoy Deb,et al.  Using Karush-Kuhn-Tucker proximity measure for solving bilevel optimization problems , 2019, Swarm Evol. Comput..

[7]  Heinrich von Stackelberg.  Market Structure and Equilibrium , 2010 .

[8]  El-Ghazali Talbi,et al.  A Taxonomy of Metaheuristics for Bi-level Optimization , 2013 .

[9]  G. Gary Wang,et al.  Survey of modeling and optimization strategies to solve high-dimensional design problems with computationally-expensive black-box functions , 2010 .

[10]  Patrick Kofod Mogensen,et al.  Optim: A mathematical optimization package for Julia , 2018, J. Open Source Softw..

[11]  Alan Edelman,et al.  Julia: A Fresh Approach to Numerical Computing , 2014, SIAM Rev..

[12]  Helio J. C. Barbosa,et al.  Differential evolution for bilevel programming , 2013, 2013 IEEE Congress on Evolutionary Computation.

[13]  Xiaotie Deng,et al.  Complexity Issues in Bilevel Linear Programming , 1998 .

[14]  A. Keane,et al.  Evolutionary Optimization of Computationally Expensive Problems via Surrogate Modeling , 2003 .

[15]  Kalyanmoy Deb,et al.  Bilevel optimization based on iterative approximation of multiple mappings , 2020, J. Heuristics.

[16]  Yuping Wang,et al.  An Evolutionary Algorithm Based on a New Decomposition Scheme for Nonlinear Bilevel Programming Problems , 2010, Int. J. Commun. Netw. Syst. Sci..

[17]  El-Ghazali Talbi,et al.  Metaheuristics for Bi-level Optimization , 2013 .

[18]  Robert G. Jeroslow,et al.  The polynomial hierarchy and a simple model for competitive analysis , 1985, Math. Program..

[19]  Tapabrata Ray,et al.  An Enhanced Memetic Algorithm for Single-Objective Bilevel Optimization Problems , 2016, Evolutionary Computation.

[20]  Kalyanmoy Deb,et al.  An improved bilevel evolutionary algorithm based on Quadratic Approximations , 2014, 2014 IEEE Congress on Evolutionary Computation (CEC).

[21]  Jonathan F. Bard,et al.  Practical Bilevel Optimization: Algorithms and Applications , 1998 .

[22]  Adrian S. Lewis,et al.  Nonsmooth optimization via quasi-Newton methods , 2012, Mathematical Programming.

[23]  L. N. Vicente,et al.  Descent approaches for quadratic bilevel programming , 1994 .

[24]  Kalyanmoy Deb,et al.  A Review on Bilevel Optimization: From Classical to Evolutionary Approaches and Applications , 2017, IEEE Transactions on Evolutionary Computation.

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

[26]  Andrew R. Conn,et al.  Bilevel derivative-free optimization and its application to robust optimization , 2012, Optim. Methods Softw..

[27]  Tapabrata Ray,et al.  A Surrogate Assisted Approach for Single-Objective Bilevel Optimization , 2017, IEEE Transactions on Evolutionary Computation.

[28]  D K Smith,et al.  Numerical Optimization , 2001, J. Oper. Res. Soc..

[29]  Tapabrata Ray,et al.  Efficient Global Optimization for Solving Computationally Expensive Bilevel Optimization Problems , 2018, 2018 IEEE Congress on Evolutionary Computation (CEC).

[30]  Jorge Nocedal,et al.  On the limited memory BFGS method for large scale optimization , 1989, Math. Program..

[31]  E. Rowland.  Theory of Games and Economic Behavior , 1946, Nature.

[32]  Andrew Koh.  Solving transportation bi-level programs with Differential Evolution , 2007, 2007 IEEE Congress on Evolutionary Computation.

[33]  Efrén Mezura-Montes,et al.  A Metaheuristic for Bilevel Optimization Using Tykhonov Regularization and the Quasi-Newton Method , 2019, 2019 IEEE Congress on Evolutionary Computation (CEC).

[34]  Sonja Kuhnt,et al.  Kernel interpolation , 2011, Comput. Stat. Data Anal..

[35]  Xin Yao,et al.  Model-based evolutionary algorithms: a short survey , 2018, Complex & Intelligent Systems.

[36]  Kalyanmoy Deb,et al.  Test Problem Construction for Single-Objective Bilevel Optimization , 2014, Evolutionary Computation.

[37]  Martine Labbé,et al.  A Bilevel Model for Toll Optimization on a Multicommodity Transportation Network , 2000, Transp. Sci..

[38]  Kalyanmoy Deb,et al.  Transportation policy formulation as a multi-objective bilevel optimization problem , 2015, 2015 IEEE Congress on Evolutionary Computation (CEC).

[39]  Pierre Hansen,et al.  New Branch-and-Bound Rules for Linear Bilevel Programming , 1989, SIAM J. Sci. Comput..

[40]  A. Copeland.  Book Review: Theory of games and economic behavior , 1945 .

[41]  Paul H. Calamai,et al.  Bilevel and multilevel programming: A bibliography review , 1994, J. Glob. Optim..

[42]  Nataliya I. Kalashnykova,et al.  Comparison of algorithms for solving a bi-level toll setting problem , 2010 .

[43]  Anthony Widjaja,et al.  Learning with Kernels: Support Vector Machines, Regularization, Optimization, and Beyond , 2003, IEEE Transactions on Neural Networks.

[44]  Stephan Dempe,et al.  Foundations of Bilevel Programming , 2002 .

[45]  Kalyanmoy Deb,et al.  Efficient Evolutionary Algorithm for Single-Objective Bilevel Optimization , 2013, ArXiv.

[46]  Kim N. Dirks,et al.  A bilevel multi-objective road pricing model for economic, environmental and health sustainability , 2014 .