Test Problem Construction for Single-Objective Bilevel Optimization

In this paper, we propose a procedure for designing controlled test problems for single-objective bilevel optimization. The construction procedure is flexible and allows its user to control the different complexities that are to be included in the test problems independently of each other. In addition to properties that control the difficulty in convergence, the procedure also allows the user to introduce difficulties caused by interaction of the two levels. As a companion to the test problem construction framework, the paper presents a standard test suite of 12 problems, which includes eight unconstrained and four constrained problems. Most of the problems are scalable in terms of variables and constraints. To provide baseline results, we have solved the proposed test problems using a nested bilevel evolutionary algorithm. The results can be used for comparison, while evaluating the performance of any other bilevel optimization algorithm. The code related to the paper may be accessed from the website http://bilevel.org.

[1]  Paul H. Calamai,et al.  Generating quadratic bilevel programming test problems , 1994, TOMS.

[2]  Yafeng Yin,et al.  Genetic-Algorithms-Based Approach for Bilevel Programming Models , 2000 .

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

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

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

[6]  Athanasios Migdalas,et al.  Bilevel programming in traffic planning: Models, methods and challenge , 1995, J. Glob. Optim..

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

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

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

[10]  S. Dempe.  Annotated Bibliography on Bilevel Programming and Mathematical Programs with Equilibrium Constraints , 2003 .

[11]  Gao Zi-You,et al.  A bi-level programming model and solution algorithm for the location of logistics distribution centers , 2008 .

[12]  Yew-Soon Ong,et al.  Evolutionary multitasking in bi-level optimization , 2015 .

[13]  Aravind Srinivasan,et al.  A Population-Based, Parent Centric Procedure for Constrained Real-Parameter Optimization , 2006, 2006 IEEE International Conference on Evolutionary Computation.

[14]  Kalyanmoy Deb,et al.  A population-based, steady-state procedure for real-parameter optimization , 2005, 2005 IEEE Congress on Evolutionary Computation.

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

[16]  Kalyanmoy Deb,et al.  A Computationally Efficient Evolutionary Algorithm for Real-Parameter Optimization , 2002, Evolutionary Computation.

[17]  M. Dufwenberg.  Game theory. , 2011, Wiley interdisciplinary reviews. Cognitive science.

[18]  Michael Florian,et al.  Optimizing frequencies in a transit network: a nonlinear bi-level programming approach , 1995 .

[19]  Per Kristian Lehre,et al.  A Parameterised Complexity Analysis of Bi-level Optimisation with Evolutionary Algorithms , 2014, Evolutionary Computation.

[20]  Zhongping Wan,et al.  Genetic algorithm based on simplex method for solving linear-quadratic bilevel programming problem , 2008, Comput. Math. Appl..

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

[22]  E. Polak,et al.  An Outer Approximations Approach to Reliability-Based Optimal Design of Structures , 1998 .

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

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

[25]  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.

[26]  Jonathan F. Bard,et al.  Coordination of a multidivisional organization through two levels of management , 1983 .

[27]  Paul H. Calamai,et al.  Generating Linear and Linear-Quadratic Bilevel Programming Problems , 1993, SIAM J. Sci. Comput..

[28]  Mahyar A. Amouzegar,et al.  Test problem construction for linear bilevel programming problems , 1996, J. Glob. Optim..

[29]  Kalyanmoy Deb,et al.  Finding optimal strategies in a multi-period multi-leader-follower Stackelberg game using an evolutionary algorithm , 2013, Comput. Oper. Res..

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

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

[32]  J. Periaux,et al.  Multi-point optimization using GAs and Nash/Stackelberg games for high lift multi-airfoil design in aerodynamics , 2001, Proceedings of the 2001 Congress on Evolutionary Computation (IEEE Cat. No.01TH8546).

[33]  Hong Li,et al.  A Differential Evolution with Two Mutation Strategies and a Selection Based on an Improved Constraint-Handling Technique for Bilevel Programming Problems , 2014 .

[34]  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.

[35]  William R. Smith,et al.  Chemical Reaction Equilibrium Analysis: Theory and Algorithms , 1982 .