Foundations of Bilevel Programming (Nonconvex Optimization and Its Applications)
S. Dempe
Bilevel programming problems are hierarchical optimization  problems where the constraints of one problem (the so-called upper  level problem) are defined in part by a second parametric optimization  problem (the lower level problem). If the lower level problem has a  unique optimal solution for all parameter values, this problem is  equivalent to a one-level optimization problem having an implicitly  defined objective function. Special emphasize in the book is on  problems having non-unique lower level optimal solutions, the  optimistic (or weak) and the pessimistic (or strong) approaches are  discussed. The book starts with the required results in parametric  nonlinear optimization. This is followed by the main theoretical  results including necessary and sufficient optimality conditions and  solution algorithms for bilevel problems. Stationarity conditions can  be applied to the lower level problem to transform the optimistic  bilevel programming problem into a one-level problem. Properties of  the resulting problem are highlighted and its relation to the bilevel  problem is investigated. Stability properties, numerical complexity,  and problems having additional integrality conditions on the variables  are also discussed.  Audience: Applied mathematicians and economists working in  optimization, operations research, and economic modelling. Students  interested in optimization will also find this book useful.