Technical Note—There’s No Free Lunch: On the Hardness of Choosing a Correct Big-M in Bilevel Optimization

Kleinert, Thomas; Labbé, Martine; Plein, Fränk; Schmidt, Martin

doi:10.1287/opre.2019.1944

Cited by 85 publications

(41 citation statements)

References 28 publications

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“…Where α is a binary variable and M is a large enough constant. However, choosing an appropriate value for M is important for numerical stability, but can be a challenging task in itself [22].…”

Section: A Linearization Methodsmentioning

confidence: 99%

A Stochastic MPEC Approach for Grid Tariff Design with Demand Side Flexibility

Askeland¹,

Burandt²,

Gabriel³

2019

Preprint

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<div>As the end-users increasingly can provide flexibility to the power system, it is important to consider how this flexibility can be activated as a resource for the grid. Electricity network tariffs are one option that can be used to activate this flexibility. Therefore, by designing efficient grid tariffs, it might be possible to reduce the total costs in the power system by incentivizing a change in consumption patterns.</div><div><br></div><div>This paper provides a methodology for optimal grid tariff design under decentralized decision-making and uncertainty in demand, power prices, and renewable generation. A bilevel model is formulated to adequately describe the interaction between the end-users and a distribution system operator. In addition, a centralized decision-making model is provided for benchmarking purposes. The bilevel model is reformulated as a mixed-integer linear problem solvable by branch-and-cut techniques.</div><div><br></div><div>Results for a deterministic example and a stochastic case study are presented and discussed.</div>

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Section: A Linearization Methodsmentioning

confidence: 99%

A Stochastic MPEC Approach for Grid Tariff Design with Demand Side Flexibility

Askeland¹,

Burandt²,

Gabriel³

2019

Preprint

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“…The resulting problem is an exact reformulation of the original problem, if the M parameters are appropriately chosen. However, as pointed out in [29], this is a non-trivial task. In our case, the nonlinearities in the objective function are not further linearized as in [2], because Gurobi's noncovnex solver handled the resulting MINLP efficiently and it guarantees global optimality.…”

Section: Bi-level Formulation Of the Optimal Bidding Problemmentioning

confidence: 97%

Capturing Electricity Market Dynamics in the Optimal Trading of Strategic Agents using Neural Network Constrained Optimization

Dolányi¹,

Bruninx²,

Toubeau³

et al. 2023

Preprint

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In competitive electricity markets the optimal trading problem of an electricity market agent is commonly formulated as a bi-level program, and solved as mathematical program with equilibrium constraints (MPEC). In this paper, an alternative paradigm, labeled as mathematical program with neural network constraint (MPNNC), is developed to incorporate complex market dynamics in the optimal bidding strategy. This method uses input-convex neural networks (ICNNs) to represent the mapping between the upper-level (agent) decisions and the lower-level (market) outcomes, i.e., to replace the lower-level problem by a neural network. In a comparative analysis, the optimal bidding problem of a load agent is formulated via the proposed MPNNC and via the classical bi-level programming method, and compared against each other.

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“…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]. Thus, if possible, big-Ms should be derived using problem-specific knowledge.…”

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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Technical Note—There’s No Free Lunch: On the Hardness of Choosing a Correct Big-M in Bilevel Optimization

Abstract: There's No Free Lunch in Bilevel Optimization

Cited by 85 publications

References 28 publications

A Stochastic MPEC Approach for Grid Tariff Design with Demand Side Flexibility

A Stochastic MPEC Approach for Grid Tariff Design with Demand Side Flexibility

Capturing Electricity Market Dynamics in the Optimal Trading of Strategic Agents using Neural Network Constrained Optimization

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

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