A framework for generalized Benders’ decomposition and its application to multilevel optimization

Bolusani, Suresh; Ralphs, Ted K.

doi:10.1007/s10107-021-01763-7

Cited by 12 publications

(9 citation statements)

References 53 publications

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“…[50] propose a projection-based single-level reformulation that implicitly enumerates the follower's integer variables. Finally, [10] utilizes MILP duality theory to develop a Benders-like decomposition algorithm that approximates the follower's value function. MIBLP has successfully applied to several optimization problems [13,26].…”

Section: Bilevel Programmingmentioning

confidence: 99%

“…The most widely studied BP problem is bilevel linear programming (BLP), in which both leader's and follower's problems are linear programs (LPs) [22]. Although BLPs are non-convex and strongly NP-hard, the optimality of the follower's decisions can be enforced through the Karush-Kuhn-Tucker (KKT) or strong duality conditions of the follower's problem [10]. The resulting single-level nonlinear problem can be solved by a combination of branch-and-bound and penalty methods [26].…”

Section: Bilevel Programmingmentioning

confidence: 99%

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Managing Product Transitions: A Bilevel Programming Approach

Rahman

Özaltın

Kempf

et al. 2022

INFORMS Journal on Computing

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We model the hierarchical and decentralized nature of product transitions using a mixed-integer bilevel program with two followers, a manufacturing unit and an engineering unit. The leader, corporate management, seeks to maximize revenue over a finite planning horizon. The manufacturing unit uses factory capacity to satisfy the demand for current products. The demand for new products, however, cannot be fulfilled until the engineering unit completes their development, which, in turn, requires factory capacity for prototype fabrication. We model this interdependency between the engineering and manufacturing units as a generalized Nash equilibrium game at the lower level of the proposed bilevel model. We present a reformulation where the interdependency between the followers is resolved through the leader’s coordination, and we derive a solution method based on constraint and column generation. Our computational experiments show that the proposed approach can solve realistic instances to optimality in a reasonable time. We provide managerial insights into how the allocation of decision authority between corporate leadership and functional units affects the objective function performance. This paper presents the first exact solution algorithm to mixed-integer bilevel programs with interdependent followers, providing a flexible framework to study decentralized, hierarchical decision-making problems.

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Section: Bilevel Programmingmentioning

confidence: 99%

Section: Bilevel Programmingmentioning

confidence: 99%

Managing Product Transitions: A Bilevel Programming Approach

Rahman

Özaltın

Kempf

et al. 2022

INFORMS Journal on Computing

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“…While this case can be handled with techniques similar to those we describe in the paper from an algorithmic perspective, it does require a more complicated notation which, for the sake of clarity, we prefer not to adopt. Detailed descriptions of algorithms for this more general case in the bilevel setting are provided in Tahernejad et al [2016], Bolusani and Ralphs [2020].…”

Section: Technical Assumptionsmentioning

confidence: 99%

“…More details regarding the Benders-type algorithm are contained next in Section 7. Further details on the structure of and methods for approximating ρ and Ξ can be found in Bolusani and Ralphs [2020], which describes a Benders-type algorithm for solving (2SMILP).…”

Section: Reaction and Risk Functionsmentioning

confidence: 99%

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A Unified Framework for Multistage Mixed Integer Linear Optimization

Bolusani

Coniglio

Ralphs

et al. 2020

Bilevel Optimization

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We introduce a unified framework for the study of multilevel mixed integer linear optimization problems and multistage stochastic mixed integer linear optimization problems with recourse. The framework highlights the common mathematical structure of the two problems and allows for the development of a common algorithmic framework. Focusing on the two-stage case, we investigate, in particular, the nature of the value function of the second-stage problem, highlighting its connection to dual functions and the theory of duality for mixed integer linear optimization problems, and summarize different reformulations. We then present two main solution techniques, one based on a Benders-like decomposition to approximate either the risk function or the value function, and the other one based on cutting plane generation.

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