A Survey on Mixed-Integer Programming Techniques in Bilevel Optimization

Kleinert, Thomas; Labbé, Martine; Ljubić, Ivana; Schmidt, Martin

doi:10.1016/j.ejco.2021.100007

Cited by 105 publications

(48 citation statements)

References 202 publications

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“…Besides, we link the MIP formulations that we derive from the bilevel framework to the ones of [Fer+16]. We obtain them by standard and generic reformulations, as suggested by Kleinert et al in their review of bilevel programming [Kle+21]. By comparison with all these works, the main novelty is the introduction of the quadratic regularized model and the evidences that it has the same good features as the logit model, in terms of economic realism and robustness, while being computationally more tractable.…”

Section: Contributionmentioning

confidence: 99%

“…The special structure of these problems can be generally cast into the bilevel framework, that have been extensively studied in the last decades [Bar13;Dem+15]. As detailed in Kleinert et al [Kle+21], two classical approaches consist in reformulating the problem as a single-level one, either using strong-duality or the KKT conditions, to express the optimality of the lower decision and constrain the upper problem. Formulations based on on KKT conditions lead to Mathematical Programs with Complementarity Constraints (MPCC), a class of optimization problems whose interest has been growing in recent years [Dus+20] and particularly in the energy sector [Afş+16; Ale+19; Aus+20; ARR21].…”

Section: Introduction 1contextmentioning

confidence: 99%

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Quadratic Regularization of Unit-Demand Envy-Free Pricing Problems and Application to Electricity Markets

Jacquet¹,

Ackooij²,

Alasseur³

et al. 2021

Preprint

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We consider a profit-maximizing model for pricing contracts as an extension of the unitdemand envy-free pricing problem: customers aim to choose a contract maximizing their utility based on a reservation price and multiple price coefficients (attributes). Classical approaches suppose that the customers have deterministic utilities; then, the response of each customer is highly sensitive to price since it concentrates on the best offer. To circumvent the intrinsic instability of deterministic models, we introduce a quadratically regularized model of customer's response, which leads to a quadratic program under complementarity constraints (QPCC). This provides an alternative to the classical logit approach, still allowing to robustify the model, while keeping a strong geometrical structure. In particular, we show that the customer's response is governed by a polyhedral complex, in which every polyhedral cell determines a set of contracts which is effectively chosen. Moreover, the deterministic model is recovered as a limit case of the regularized one. We exploit these geometrical properties to develop a pivoting heuristic, which we compare with implicit or non-linear methods from bilevel programming, showing the effectiveness of the approach. Throughout the paper, the electricity provider problem is our guideline, and we present a numerical study on this application case.

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Section: Contributionmentioning

confidence: 99%

Section: Introduction 1contextmentioning

confidence: 99%

Quadratic Regularization of Unit-Demand Envy-Free Pricing Problems and Application to Electricity Markets

Jacquet¹,

Ackooij²,

Alasseur³

et al. 2021

Preprint

View full text Add to dashboard Cite

show abstract

“…When it comes to the situation of a lowerlevel problem with continuous nonlinearities there is not too much literature-in particular in comparison to the case in which the lower-level problem is convex; see, e.g., Kleniati and Adjiman (2011, 2014a,b, 2015, Mitsos (2010), Mitsos et al (2008), , Paulavičius, Gao, et al (2020), and Paulavičius et al (2016). Due to the brevity of the article, we do not go into the details of the literature but refer to the seminal book by Dempe (2002) as well as the recent survey by Kleinert, Labbé, Ljubić, et al (2021) for further discussions of the relevant literature.…”

Section: Introductionmentioning

confidence: 99%

On a Computationally Ill-Behaved Bilevel Problem with a Continuous and Nonconvex Lower Level

Beck¹,

Schmidt²,

Thürauf³

et al. 2022

Preprint

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It is well known that bilevel optimization problems are hard to solve both in theory and practice. In this short note, we highlight a further computational difficulty when it comes to solving bilevel problems with continuous but nonconvex lower levels. Even if the lower-level problem is solved to ε-feasibility regarding its nonlinear constraints for an arbitrarily small but positive ε, the obtained bilevel solution can be arbitrarily far away from the unique bilevel solution that one obtains for exactly feasible points in the lower level. Since the consideration of ε-feasibility cannot be avoided for nonconvex problems, our result shows that computational bilevel optimization with continuous and nonconvex lower levels needs to be done with great care.

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“…Games can broaden the modeling capabilities of MIP , and extend classical combinatorial and decision-making problems to multi-agent settings that can account for interactions among multiple decision-makers. For instance, bilevel programming [3,8,29,36,39,41]) and Integer Programming Games (IPGs) [9,11,19,24,33,40]. This recent research direction suggests that the joint endeavor between game theory and MIP can widen their theoretical understanding and practical impact.…”

Section: Introductionmentioning

confidence: 99%

The Cut and Play Algorithm: Computing Nash Equilibria via Outer Approximations

Carvalho¹,

Dragotto²,

Lodi³

et al. 2021

Preprint

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The concept of Nash equilibrium enlightens the structure of rational behavior in multi-agent settings. However, the concept is as helpful as one may compute it efficiently. We introduce the Cut-and-Play, an algorithm to compute Nash equilibria for a class of non-cooperative simultaneous games where each player's objective is linear in their variables and bilinear in the other players' variables. Using the rich theory of integer programming, we alternate between constructing (i.) increasingly tighter outer approximations of the convex hull of each player's feasible set -by using branching and cutting plane methods -and (ii.) increasingly better inner approximations of these hulls -by finding extreme points and rays of the convex hulls. In particular, when these convex hulls are polyhedra, we prove the correctness of our algorithm and leverage the mixed integer programming technology to compute equilibria for a large class of games. Further, we integrate existing cutting plane families inside the algorithm, significantly speeding up equilibria computation. We showcase a set of extensive computational results for Integer Programming Games and simultaneous games among bilevel leaders. In both cases, our framework outperforms the state-of-the-art in computing time and solution quality.

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A Survey on Mixed-Integer Programming Techniques in Bilevel Optimization

Cited by 105 publications

References 202 publications

Quadratic Regularization of Unit-Demand Envy-Free Pricing Problems and Application to Electricity Markets

Quadratic Regularization of Unit-Demand Envy-Free Pricing Problems and Application to Electricity Markets

On a Computationally Ill-Behaved Bilevel Problem with a Continuous and Nonconvex Lower Level

The Cut and Play Algorithm: Computing Nash Equilibria via Outer Approximations

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