Solving discrete linear bilevel optimization problems using the optimal value reformulation

Dempe, Stephan; Kue, Floriane Mefo

doi:10.1007/s10898-016-0478-5

Cited by 26 publications

(14 citation statements)

References 27 publications

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“…There are various solution approaches proposed for bilevel discrete optimization problems. Some of the recent works can be obtained in [6,7,10,19,23,29], where some of these methods often resulting in approximate solutions while others give exact solutions. However, due to the introduction of the middle-level follower in the problem, solution approaches developed for bi-level programming are not necessarily applicable to tri-level optimization problems.…”

Section: Branch-and-cut Methods For Tri-level Integer Linear Programming Problemsmentioning

confidence: 99%

A simplex-based branch-and-cut method for solving integer tri-level programming problems

Awraris¹,

Wordofa²,

Kassa³

2021

J. Math. Computer Sci.

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Optimization problems involving three decision makers at three hierarchical decision levels are referred to as tri-level decision making problems. In particular in the case where all objective functions and constraints are linear, and all involved variables are required to take integer values is known as integer tri-level programming problems (T-ILP). It has been known that the general T-ILP is an NP-hard problem. So far there are very few attempts in the literature that finds an exact global solution for integer programming hierarchical problems beyond two levels. This paper proposes a novel approach based on a branch-and-cut (B&C) framework for solving T-ILP. The convergence of the algorithm is proved. Numerical examples are used to demonstrate the performance of the proposed algorithm. Finally, the results of this study show that the proposed algorithm is promising and more efficient to solve T-ILP. Moreover, it has been indicated in the remark that the proposed algorithm can be modified and used to solve discrete multilevel programming problems of any number of levels.

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Section: Branch-and-cut Methods For Tri-level Integer Linear Programming Problemsmentioning

confidence: 99%

A simplex-based branch-and-cut method for solving integer tri-level programming problems

Awraris¹,

Wordofa²,

Kassa³

2021

J. Math. Computer Sci.

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“…This procedure is similar in spirit to the Gomory procedure for standard MILPs. It is used, for instance, in Dempe and Kue [2017]. We next describe the method with a brief example.…”

Section: Convexification-based Methodsmentioning

confidence: 99%

“…Following these early works, the focus shifted primarily to various special cases, especially those in which the lower-level problem has the integrality property. Dempe [2001] considers a special case characterized by continuous upper-level variables and integer lower-level variables and uses a cutting plane approach to approximate the lower-level feasible region (a somewhat similar approach is adopted in Dempe and Kue [2017] for solving a bilinear mixed integer bilevel problem with integer second-level variables). Wen and Yang [1990] consider the opposite case, where the lower-level problem is a linear optimization problem and the upper-level problem is an integer optimization problem, using linear optimization duality to derive exact and heuristic solutions.…”

Section: Algorithmsmentioning

confidence: 99%

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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“…Ref. [31]. However, as the here presented model aims to focus on short term time frames, it is reasonable to assume that a range of potential unit commitment schedules is already established.…”

Section: Th J |×Max(t )mentioning

confidence: 99%

Hydro-thermal power market equilibrium with price-making hydropower producers

Löschenbrand

Wei

Liu

2018

Energy

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This paper formulates an electricity market dominated by price-making hydrothermal generation. Generation companies optimize their unit commitment, scheduling and bidding decisions simultaneously as a Mixed Integer Programming problem and participate in a market under quantity competition, giving rise to a discontinuous Nash-Cournot game. Both hydropower and thermal units are considered as price-makers. The market equilibrium under uncertainty is computed via time stage decomposition and nesting of a Continuous Nash game into the original Discontinuous Nash game that can be solved via a search algorithm. To highlight applicability of the proposed framework, a case study on the Scandinavian power market is designed and suggests positive welfare effects of large scale storage, whereas the implications on scheduling of conventional units are subsequently discussed. Reformulation allows computationally efficient scaling of the problem and possible extensions to allow large scale applications are discussed.

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Solving discrete linear bilevel optimization problems using the optimal value reformulation

Cited by 26 publications

References 27 publications

A simplex-based branch-and-cut method for solving integer tri-level programming problems

A simplex-based branch-and-cut method for solving integer tri-level programming problems

A Unified Framework for Multistage Mixed Integer Linear Optimization

Hydro-thermal power market equilibrium with price-making hydropower producers

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