Evolutionary bilevel optimization

Many practical optimization problems should better be posed as bilevel optimization problems in which there are two levels of optimization tasks. A solution at the upper level is feasible if the corresponding lower level variable vector is optimal for the lower level optimization problem. Consider, for example, an inverted pendulum problem for which the motion of the platform relates to the upper level optimization problem of performing the balancing task in a time-optimal manner. For a given motion of the platform, whether the pendulum can be balanced at all becomes a lower level optimization problem of maximizing stability margin. Such nested optimization problems are commonly found in transportation, engineering design, game playing and business models. They are also known as Stackelberg games in the operations research community. These problems are too complex to be solved using classical optimization methods simply due to the "nestedness" of one optimization task into another. Evolutionary Algorithms (EAs) provide some amenable ways to solve such problems due to their flexibility and ability to handle constrained search spaces efficiently. Clearly, EAs have an edge in solving such difficult yet practically important problems. In the recent past, there has been a surge in research activities towards solving bilevel optimization problems. In this tutorial, we will introduce principles of bilevel optimization for single and multiple objectives, and discuss the difficulties in solving such problems in general. With a brief survey of the existing literature, we will present a few viable evolutionary algorithms for both single and multi-objective EAs for bilevel optimization. Our recent studies on bilevel test problems and some application studies will be discussed. Finally, a number of immediate and future research ideas on bilevel optimization will also be highlighted.

[1]  G. Eichfelder Solving Nonlinear Multiobjective Bilevel Optimization Problems with Coupled Upper Level Constraints , 2007 .

[2]  R. Conrad,et al.  Resource taxation with heterogeneous quality and endogenous reserves , 1981 .

[3]  Kalyanmoy Deb,et al.  Multi-objective Stackelberg game between a regulating authority and a mining company: A case study in environmental economics , 2013, 2013 IEEE Congress on Evolutionary Computation.

[4]  B. Mordukhovich,et al.  New necessary optimality conditions in optimistic bilevel programming , 2007 .

[5]  Lucio Bianco,et al.  A Bilevel flow model for HazMat transportation network design , 2008 .

[6]  Ankur Sinha,et al.  Bilevel Multi-objective Optimization Problem Solving Using Progressively Interactive EMO , 2011, EMO.

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

[8]  Yuping Wang,et al.  An evolutionary algorithm for solving nonlinear bilevel programming based on a new constraint-handling scheme , 2005, IEEE Transactions on Systems, Man, and Cybernetics, Part C (Applications and Reviews).

[9]  Jane J. Ye,et al.  Optimality conditions for bilevel programming problems , 1995 .

[10]  Jonathan F. Bard,et al.  An explicit solution to the multi-level programming problem , 1982, Comput. Oper. Res..

[11]  K. Deb,et al.  An Evolutionary Approach for Bilevel Multi-objective Problems , 2009 .

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

[13]  Tharam S. Dillon,et al.  Decentralized multi-objective bilevel decision making with fuzzy demands , 2007, Knowl. Based Syst..

[14]  Rama Rao Pakala A Study on Applications of Stackelberg Game Strategies in Concurrent Design Models , 1994 .

[15]  Jonathan F. Bard,et al.  Practical Bilevel Optimization: Algorithms and Applications (Nonconvex Optimization and Its Applications) , 2006 .

[16]  A. Frantsev,et al.  Finding Optimal Strategies in Multi-Period Stackelberg Games Using an Evolutionary Framework , 2012 .

[17]  Kalyanmoy Deb,et al.  A bilevel optimization approach to automated parameter tuning , 2014, GECCO.

[18]  R. Conrad Output taxes and the quantity-quality trade-off in the mining firm , 1981 .

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

[20]  Kalyanmoy Deb,et al.  Unconstrained scalable test problems for single-objective bilevel optimization , 2012, 2012 IEEE Congress on Evolutionary Computation.

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

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

[23]  Sanaz Mostaghim,et al.  Bilevel Optimization of Multi-Component Chemical Systems Using Particle Swarm Optimization , 2006, 2006 IEEE International Conference on Evolutionary Computation.

[24]  J. Herskovits,et al.  Contact shape optimization: a bilevel programming approach , 2000 .

[25]  Kalyanmoy Deb,et al.  An Efficient and Accurate Solution Methodology for Bilevel Multi-Objective Programming Problems Using a Hybrid Evolutionary-Local-Search Algorithm , 2010, Evolutionary Computation.

[26]  Martine Labbé,et al.  A parallel between two classes of pricing problems in transportation and marketing , 2010 .

[27]  J. Helliwell Effects of taxes and royalties on copper mining investment in British Columbia , 1978 .

[28]  Yuping Wang,et al.  A New Evolutionary Algorithm for a Class of Nonlinear Bilevel Programming Problems and Its Global Convergence , 2011, INFORMS J. Comput..

[29]  Kalyanmoy Deb,et al.  Constructing test problems for bilevel evolutionary multi-objective optimization , 2009, 2009 IEEE Congress on Evolutionary Computation.

[30]  Kalyanmoy Deb,et al.  Towards Understanding Evolutionary Bilevel Multi-Objective Optimization Algorithm , 2009 .

[31]  L. N. Vicente,et al.  Multicriteria Approach to Bilevel Optimization , 2006 .

[32]  Eitaro Aiyoshi,et al.  HIERARCHICAL DECENTRALIZED SYSTEM AND ITS NEW SOLUTION BY A BARRIER METHOD. , 1980 .

[33]  Kalyanmoy Deb,et al.  Solving Bilevel Multi-Objective Optimization Problems Using Evolutionary Algorithms , 2009, EMO.

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

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