Foundations of bilevel programming
type: Item
instance of: scholarly article
title: Foundations of bilevel programming
zbMATH Open document ID: 1038.90097
published in: Nonconvex Optimization and Its Applications
publication date: 2002-08-26T00:00:00Z
review text: Bilevel programming (BP in brief) deals with optimization problems of the form \[ \begin{aligned} (P) \text{``min''}\quad & f(x,y)\\ \text{s.t.}\quad & g_i(x,y) \leq 0,\;i\in I\\ \quad & x\in\Psi(y),\;y\in\mathbb{R}^m,\end{aligned} \] where \(| I |<\infty\) and \(\Psi(y)\subset\mathbb{R}^n\) denotes the optimal set of a certain optimization problem, \((P_y)\), depending on the parameter \(y\). Accordingly, \((P_y)\) and \((P)\) are called lower level and upper level problems, respectively. If \((P_y)\) is uniquely solvable for all \(y\in \mathbb{R}^m\), then \((P)\) is a mathematical program whose objective function is implicitly defined. Otherwise, given \(y\in\mathbb{R}^m\), the set \[ \bigl\{f(x,y)\mid x\in\Psi(y),\;g_i(x,y)\leq 0,\;i\in I\bigr\} \subset \mathbb{R} \] is not necessarily a singleton and the task ``min'' can be interpreted in different ways, for instance, as \(\min_{y\in\mathbb{R}^m} \min_{x\in F(y)} f(x,y)\) \((\min_{y\in\mathbb{R}^m} \max_{x\in F(y)} f(x, y))\), corresponding to the optimistic perspective (pessimistic perspective, respectively). As the applications collected in Chapter 2 show, BP problems naturally arise in economy, engineering, chemistry, data analysis and many other fields. The book is mainly focussed on BP theory (specially optimality conditions), and its main novelties (in comparison with previous monographs) are the elimination of the strong uniqueness assumption on \((P_y)\) and the discussion of discrete BP problems. This is a valuable expository text for graduate and advanced undergraduate students which provides detailed information on the wide BP literature, and can be read with different purposes: 1. A short introduction to BP (Chapters 1--3, preferably complemented with Section 5.1). 2. An introduction to parametric optimization (Chapter 4), which is the key tool in BP theory. 3. An overview of BP (Chapters 1--5 and 5--7, skipping the proofs, which have been conveniently concentrated at the end of each chapter). 4. A …
Mathematics Subject Classification ID: 90-02
Mathematics Subject Classification ID: 90C31
Mathematics Subject Classification ID: 90C46
Mathematics Subject Classification ID: 90C27
Mathematics Subject Classification ID: 49J52
zbMATH DE Number: 1788462
zbMATH Keywords: Bilevel programming
zbMATH Keywords: parametric programming
zbMATH Keywords: optimality conditions
zbMATH Keywords: numerical methods
zbMATH Keywords: applications
author: Stephan Dempe
reviewed by: Miguel Angel Goberna
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1788462