New necessary optimality conditions in optimistic bilevel programming
TL;DR: This work reduces a basic optimistic model in bilevel programming to a one-level framework of nondifferentiable programs formulated via (nonsmooth) optimal value function of the parametric lower-level problem in the original model and derives new necessary optimality conditions for bileVEL programs reflecting significant phenomena that have never been observed earlier.
Abstract: The article is devoted to the study of the so-called optimistic version of bilevel programming in finite-dimensional spaces. Problems of this type are intrinsically nonsmooth (even for smooth initial data) and can be treated by using appropriate tools of modern variational analysis and generalized differentiation. Considering a basic optimistic model in bilevel programming, we reduce it to a one-level framework of nondifferentiable programs formulated via (nonsmooth) optimal value function of the parametric lower-level problem in the original model. Using advanced formulas for computing basic subgradients of value/marginal functions in variational analysis, we derive new necessary optimality conditions for bilevel programs reflecting significant phenomena that have never been observed earlier. In particular, our optimality conditions for bilevel programs do not depend on the partial derivatives with respect to parameters of the smooth objective function in the parametric lower-level problem. We present ef...
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TL;DR: A comprehensive review on bilevel optimization from the basic principles to solution strategies is provided in this paper, where a number of potential application problems are also discussed and an automated text-analysis of an extended list of papers has been performed.
Abstract: Bilevel optimization is defined as a mathematical program, where an optimization problem contains another optimization problem as a constraint. These problems have received significant attention from the mathematical programming community. Only limited work exists on bilevel problems using evolutionary computation techniques; however, recently there has been an increasing interest due to the proliferation of practical applications and the potential of evolutionary algorithms in tackling these problems. This paper provides a comprehensive review on bilevel optimization from the basic principles to solution strategies; both classical and evolutionary. A number of potential application problems are also discussed. To offer the readers insights on the prominent developments in the field of bilevel optimization, we have performed an automated text-analysis of an extended list of papers published on bilevel optimization to date. This paper should motivate evolutionary computation researchers to pay more attention to this practical yet challenging area.
588 citations
Posted Content•
TL;DR: An automated text-analysis of an extended list of papers published on bilevel optimization from the basic principles to solution strategies; both classical and evolutionary is performed.
Abstract: Bilevel optimization is defined as a mathematical program, where an optimization problem contains another optimization problem as a constraint. These problems have received significant attention from the mathematical programming community. Only limited work exists on bilevel problems using evolutionary computation techniques; however, recently there has been an increasing interest due to the proliferation of practical applications and the potential of evolutionary algorithms in tackling these problems. This paper provides a comprehensive review on bilevel optimization from the basic principles to solution strategies; both classical and evolutionary. A number of potential application problems are also discussed. To offer the readers insights on the prominent developments in the field of bilevel optimization, we have performed an automated text-analysis of an extended list of papers published on bilevel optimization to date. This paper should motivate evolutionary computation researchers to pay more attention to this practical yet challenging area.
268 citations
Cites background from "New necessary optimality conditions..."
...See [58], [60], [80], [105], [106], [126] for further discussion on existence of optimistic bilevel optimum and additional results on optimality conditions....
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TL;DR: This work considers the bilevel programming problem and its optimal value and KKT one level reformulations and shows how KKT type optimality conditions can be obtained under the partial calmness, using the differential calculus of Mordukhovich.
Abstract: We consider the bilevel programming problem and its optimal value and KKT one level reformulations. The two reformulations are studied in a unified manner and compared in terms of optimal solutions, constraint qualifications and optimality conditions. We also show that any bilevel programming problem where the lower level problem is linear with respect to the lower level variable, is partially calm without any restrictive assumption. Finally, we consider the bilevel demand adjustment problem in transportation, and show how KKT type optimality conditions can be obtained under the partial calmness, using the differential calculus of Mordukhovich.
126 citations
TL;DR: This paper combines the classical and the value function approaches to derive new necessary optimality conditions under rather weak conditions of the bilevel program.
Abstract: The bilevel program is a sequence of two optimization problems where the constraint region of the upper level problem is determined implicitly by the solution set to the lower level problem. The classical approach to solving such a problem is to replace the lower level problem by its Karush-Kuhn-Tucker (KKT) condition and solve the resulting mathematical programming problem with equilibrium constraints (MPEC). In general the classical approach is not valid for nonconvex bilevel programming problems. The value function approach uses the value function of the lower level problem to define an equivalent single level problem. But the resulting problem requires a strong assumption, such as the partial calmness condition, for the KKT condition to hold. In this paper we combine the classical and the value function approaches to derive new necessary optimality conditions under rather weak conditions. The required conditions are even weaker in the case where the classical approach or the value function approach alone is applicable.
121 citations
Cites methods from "New necessary optimality conditions..."
...Under the partial calmness condition this simpler KKT condition was proved to hold under the assumption of inner semicontinuity of the solution mapping of the lower level program [11]....
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TL;DR: An exact finite algorithm is proposed based on an optimal-value-function reformulation for bilevel mixed-integer programs whose constraints and objective functions depend on both upper- and lower-level variables and can be tailored to accommodate either optimistic or pessimistic assumptions on the follower behavior.
Abstract: We examine bilevel mixed-integer programs whose constraints and objective functions depend on both upper- and lower-level variables. The class of problems we consider allows for nonlinear terms to appear in both the constraints and the objective functions, requires all upper-level variables to be integer, and allows a subset of the lower-level variables to be integer. This class of bilevel problems is difficult to solve because the upper-level feasible region is defined in part by optimality conditions governing the lower-level variables, which are difficult to characterize because of the nonconvexity of the follower problem. We propose an exact finite algorithm for these problems based on an optimal-value-function reformulation. We demonstrate how this algorithm can be tailored to accommodate either optimistic or pessimistic assumptions on the follower behavior. Computational experiments demonstrate that our approach outperforms a state-of-the-art algorithm for solving bilevel mixed-integer linear programs.
89 citations
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Book•
01 Nov 1996TL;DR: Results in the book are expected to have significant impacts in such disciplines as engineering design, economics and game equilibria, and transportation planning, within all of which MPEC has a central role to play in the modelling of many practical problems.
Abstract: This book provides a solid foundation and an extensive study for an important class of constrained optimization problems known as Mathematical Programs with Equilibrium Constraints (MPEC), which are extensions of bilevel optimization problems. The book begins with the description of many source problems arising from engineering and economics that are amenable to treatment by the MPEC methodology. Error bounds and parametric analysis are the main tools to establish a theory of exact penalisation, a set of MPEC constraint qualifications and the first-order and second-order optimality conditions. The book also describes several iterative algorithms such as a penalty-based interior point algorithm, an implicit programming algorithm and a piecewise sequential quadratic programming algorithm for MPECs. Results in the book are expected to have significant impacts in such disciplines as engineering design, economics and game equilibria, and transportation planning, within all of which MPEC has a central role to play in the modelling of many practical problems.
1,830 citations
01 Jan 2006
1,528 citations
Book•
31 May 2002
TL;DR: This paper presents a meta-modelling framework that automates the very labor-intensive and therefore time-heavy and expensive process of solving linear and Discrete Bilevel Problems.
Abstract: 1. Introduction. 2. Applications. 3. Linear Bilevel Problems. 4. Parametric optimization. 5. Optimality Conditions. 6. Solution Algorithms. 7. Non-unique Lower Level Solution. 8. Discrete Bilevel Problems. References. Notations. Index.
1,216 citations
"New necessary optimality conditions..." refers background in this paper
...Our notation is basically standard; see the books 7 , 12 , 19 ....
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Book•
05 Mar 2013
TL;DR: This chapter discusses the theory and applications of Elliptic Boundary Value problems, and some of the algorithms used to solve these problems have been described.
Abstract: Preface. List of Notations. List of Acronyms. Part I: Theory. 1.Introduction. 2. Auxiliary Results. 3. Algorithms of Nonsmooth Optimization. 4. Generalized Equations. 5. Stability of Solutions to Perturbed Generalized Equations. 6. Derivatives of Solutions to Perturbed Generalized Equations. 7. Optimality Conditions and a Solution Method. Part II: Applications. 8. Introduction. 9. Membrane with Obstacle. 10. Elasticity Problems with Internal Obstacles. 11. Contact Problem with Coulomb Friction. 12. Economic Applications. Appendices: A. Cookbook. B. Basic Facts on Elliptic Boundary Value problems. C. Complementarity Problems. References. Index.
661 citations
"New necessary optimality conditions..." refers background in this paper
...The resulting upper-level problem belongs to the class of the so-called mathematical programs with equilibrium constraints ( MPECs ): see, eg, 14 , 26 and the references therein....
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