A note on linearized reformulations for a class of bilevel linear integer problems

Zare, M. Hosein; Borrero, Juan S.; Zeng, Bo; Prokopyev, Oleg A.

doi:10.1007/s10479-017-2694-x

Cited by 38 publications

(13 citation statements)

References 43 publications

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“…These MIP models typically exploit either subtour‐elimination or flow‐based ideas. On the other hand, single‐level reformulations of bilevel problems mostly focus on using the optimality conditions (e.g., strong duality of LPs) to replace the lower‐level problem with additional linear or nonlinear constraints . The models proposed in this section represent a combination of these ideas in the context of the BST problem.…”

Section: The Bst(xs) Problemmentioning

confidence: 99%

On bilevel minimum and bottleneck spanning tree problems

2019

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We study a class of bilevel spanning tree (BST) problems that involve two independent decision‐makers (DMs), the leader and the follower with different objectives, who jointly construct a spanning tree in a graph. The leader, who acts first, selects an initial subset of edges that do not contain a cycle, from the set under her control. The follower then selects the remaining edges to complete the construction of a spanning tree, but optimizes his own objective function. If there exist multiple optimal solutions for the follower that result in different objective function values for the leader, then the follower may choose either the one that is the most (optimistic version) or least (pessimistic version) favorable to the leader. We study BST problems with the sum‐ and bottleneck‐type objective functions for the DMs under both the optimistic and pessimistic settings. The polynomial‐time algorithms are then proposed in both optimistic and pessimistic settings for BST problems in which at least one of the DMs has the bottleneck‐type objective function. For BST problem with the sum‐type objective functions for both the leader and the follower, we provide an equivalent single‐level linear mixed‐integer programming formulation. A computational study is then presented to explore the efficacy of our reformulation.

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Section: The Bst(xs) Problemmentioning

confidence: 99%

On bilevel minimum and bottleneck spanning tree problems

2019

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“…These terms can only be linearized if all linking variables are integer. Recently, in [55], a numerical study is provided that compares the KKT approach with the strong-duality approach for linear bilevel problems with integer linking variables and continuous lower-level problems. The authors conclude that the strong-duality reformulation works significantly better than the KKT reformulation for problems with large lower-level problems.…”

Section: A Convex Single-level Reformulationmentioning

confidence: 99%

“…The products of binary and continuous variables can then be linearized by several techniques. According to the numerical study in [55], the following approach works best in a bilevel context. We express the integer variables x j with the help ofr j = log 2 (x + j ) + 1 many auxiliary binary variables s jr :…”

Section: Convexification Of the Strong-duality Constraintmentioning

confidence: 99%

Outer approximation for global optimization of mixed-integer quadratic bilevel problems

2021

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Bilevel optimization problems have received a lot of attention in the last years and decades. Besides numerous theoretical developments there also evolved novel solution algorithms for mixed-integer linear bilevel problems and the most recent algorithms use branch-and-cut techniques from mixed-integer programming that are especially tailored for the bilevel context. In this paper, we consider MIQP-QP bilevel problems, i.e., models with a mixed-integer convex-quadratic upper level and a continuous convex-quadratic lower level. This setting allows for a strong-duality-based transformation of the lower level which yields, in general, an equivalent nonconvex single-level reformulation of the original bilevel problem. Under reasonable assumptions, we can derive both a multi- and a single-tree outer-approximation-based cutting-plane algorithm. We show finite termination and correctness of both methods and present extensive numerical results that illustrate the applicability of the approaches. It turns out that the proposed methods are capable of solving bilevel instances with several thousand variables and constraints and significantly outperform classical solution approaches.

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“…The main drawback of this approach is the bilinear term λ C x of primal upperlevel and dual lower-level variables. When considering only integer linking variables, as, e.g., in [25], linearizations can be applied yielding mixed-integer linear reformulations. Here, however, we study purely continuous bilevel problems.…”

Section: A New Valid Primal-dual Inequalitymentioning

confidence: 99%

Closing the gap in linear bilevel optimization: a new valid primal-dual inequality

2020

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Linear bilevel optimization problems are often tackled by replacing the linear lower-level problem with its Karush–Kuhn–Tucker conditions. The resulting single-level problem can be solved in a branch-and-bound fashion by branching on the complementarity constraints of the lower-level problem’s optimality conditions. While in mixed-integer single-level optimization branch-and-cut has proven to be a powerful extension of branch-and-bound, in linear bilevel optimization not too many bilevel-tailored valid inequalities exist. In this paper, we briefly review existing cuts for linear bilevel problems and introduce a new valid inequality that exploits the strong duality condition of the lower level. We further discuss strengthened variants of the inequality that can be derived from McCormick envelopes. In a computational study, we show that the new valid inequalities can help to close the optimality gap very effectively on a large test set of linear bilevel instances.

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A note on linearized reformulations for a class of bilevel linear integer problems

Cited by 38 publications

References 43 publications

On bilevel minimum and bottleneck spanning tree problems

On bilevel minimum and bottleneck spanning tree problems

Outer approximation for global optimization of mixed-integer quadratic bilevel problems

Closing the gap in linear bilevel optimization: a new valid primal-dual inequality

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