Solving Linear Bilevel Problems Using Big-Ms: Not All That Glitters Is Gold

Pineda, Salvador; Morales, Juan M.

doi:10.1109/tpwrs.2019.2892607

Cited by 111 publications

(42 citation statements)

References 20 publications

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“…The need for such bounds is also a major drawback in the KKT-based reformulation (2). In [46], it is shown that wrong big-Ms can lead to suboptimal solutions or points that are actually bilevel infeasible. Unfortunately, even verifying that the bounds are correctly chosen is, in general, at least as hard as solving the original bilevel problem; see [37].…”

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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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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“…However, most of the papers that use it do not explicitly mention a method to determine the Big-M values. In fact, as mentioned in [102] if these values are small, suboptimal solutions can appear, and conversely, too large Big-Ms can lead to numerical issues (when different variable magnitudes are reflected in dual variables), such as unstable solutions or large execution times. In [99], a method is proposed to define Big M values by mixing the regularisation and KKT-MIP previously commented methods.…”

Section: 31mentioning

confidence: 99%

“…In the case of MPEC problems, there is still an active field of research for finding efficient methods to solve these optimisation problems. As mentioned before in [102], Big M is the most common technique to solve the one-level reformulation of bi-level programs. However, small values of Big Ms can lead to suboptimal solutions and large constants can lead to numerical issues (if different orders of magnitude are present in the dual variables of the lower level).…”

Section: Solution Technique Gapsmentioning

confidence: 99%

Review on generation and transmission expansion co‐planning models under a market environment

Gonzalez-Romero

Wogrin

Gómez

2020

IET Generation, Transmission & Distribution

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This study presents a review of the state-of-the-art on the coordination of generation and transmission expansion planning. First, the authors present the different investment and operation modelling approaches, with an emphasis on the centralised co-optimisation problem. Second, a comprehensive review of co-planning hierarchical equilibrium models, under a market environment, is carried out. The authors categorise the distinctive market approaches that usually represent the lower level of co-planning problems. They offer an updated and detailed classification of multilevel equilibrium models, based on their hierarchical and regulatory structure versus their equivalent reduced structure. Finally, the authors identify research gaps in the literature of each one of the mentioned model categories.

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“…The problem can then be solved by standard mixed-integer solvers. However, big-Ms that are chosen too small can yield suboptimal or infeasible solutions [21] and verifying the correctness of a big-M constant is as hard as solving the original bilevel problem; see [16]. From today's point of view, this method should only be used if correct big-Ms can be obtained via problem-specific knowledge.…”

Section: Assumptionmentioning

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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Solving Linear Bilevel Problems Using Big-Ms: Not All That Glitters Is Gold

Cited by 111 publications

References 20 publications

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

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

Review on generation and transmission expansion co‐planning models under a market environment

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

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