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Solution Methods for Bilevel Optimization Andrey Tin A. Tin@soton. ac. uk School of Mathematics Solution Methods for Bilevel Optimization Andrey Tin A. [email protected] ac. uk School of Mathematics Supervisors: Dr Alain B. Zemkoho, Professor Jörg Fliege

Overview Ø Definition of a bilevel problem and its general form Ø Optimality (KKT-type) Overview Ø Definition of a bilevel problem and its general form Ø Optimality (KKT-type) conditions Ø Reformulation of a general bilevel problem Ø Iterative (descent direction) methods Ø Numerical results

Stackelberg Game (Bilevel problem) § Players: the Leader and the Follower § The Leader Stackelberg Game (Bilevel problem) § Players: the Leader and the Follower § The Leader is first to make a decision § Follower reacts optimally to Leader’s decision § The payoff for the Leader depends on the follower’s reaction

Example § Taxation of a factory § Leader – government § Objectives: maximize profit Example § Taxation of a factory § Leader – government § Objectives: maximize profit and minimize pollution § Follower – factory owner § Objectives: maximize profit

General structure of a Bilevel problem General structure of a Bilevel problem

Important Sets Important Sets

General linear Bilevel problem General linear Bilevel problem

Solution methods §Vertex enumeration in the context of Simplex method §Kuhn-Tucker approach §Penalty approach Solution methods §Vertex enumeration in the context of Simplex method §Kuhn-Tucker approach §Penalty approach §Extract gradient information from a lower objective function to compute directional derivatives of an upper objective function

Concept of KKT conditions Concept of KKT conditions

Value function reformulation Value function reformulation

KKT for value function reformulation KKT for value function reformulation

Assumptions Assumptions

KKT-type optimality conditions for Bilevel KKT-type optimality conditions for Bilevel

Further Assumptions (for simpler version) Further Assumptions (for simpler version)

Simpler version of KKT-type conditions Simpler version of KKT-type conditions

NCP-Functions NCP-Functions

Problems with differentiability § Fischer-Burmeister is not differentiable at 0 Problems with differentiability § Fischer-Burmeister is not differentiable at 0

Simpler version with perturbed Fischer. Burmeister NCP functions Simpler version with perturbed Fischer. Burmeister NCP functions

Iterative methods Iterative methods

Newton method Newton method

Pseudo inverse Pseudo inverse

Gauss-Newton method Gauss-Newton method

Singular Value Decomposition (SVD) Singular Value Decomposition (SVD)

SVD for wrong direction SVD for wrong direction

SVD for right direction SVD for right direction

Levenberg-Marquardt method Levenberg-Marquardt method

Numerical results Numerical results

Plans for further work Plans for further work

Plans for further work 6. Construct the own code for Levenberg-Marquardt method in the Plans for further work 6. Construct the own code for Levenberg-Marquardt method in the context of solving bilevel problems within defined reformulation. 7. Search for good starting point techniques for our problem. 8. Do the numerical calculations for the harder reformulation defined. 9. Code Newton method with pseudo-inverse. 10. Solve the problem assuming strict complementarity 11. Look at other solution methods.

Thank you! Questions? Thank you! Questions?

References References

References References