Bilevel Knapsack with Interdiction Constraints

Caprara, Alberto; Carvalho, Margarida; Lodi, Andrea; Woeginger, Gerhard J.

doi:10.1287/ijoc.2015.0676

Cited by 72 publications

(47 citation statements)

References 19 publications

(28 reference statements)

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“…For many real-world problems that require a bilevel or even multilevel modeling, application-specific solution techniques have been developed. This includes but is not limited to fields such as energy markets [8,13,15], pricing problems [18,19], or network interdiction problems [4,10]. In a more general setting in which no problem-specific structure can be exploited, most solution techniques resort to an equivalent single-level reformulation.…”

Section: Assumptionmentioning

confidence: 99%

“…Thus, every primal-dual feasible point satisfying Inequality (4) fulfills the strong-duality equation and is primal-dual optimal for the lower level. An alternative formulation of the single-level reformulation (3) can hence be obtained by replacing the KKT complementarity condition (3d) with the strong duality condition (4). The main drawback of this approach is the bilinear term λ C x of primal upperlevel and dual lower-level variables.…”

Section: A New Valid Primal-dual Inequalitymentioning

confidence: 99%

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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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Section: Assumptionmentioning

confidence: 99%

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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“…This is often the case in the security domain (An et al 2011;Kiekintveld et al 2009), where a defender, aiming to protect a set of valuable targets from the attackers, plays first, while the attackers, acting as followers, make their move only after observing the leader's defensive strategy. Other noteworthy cases are interdiction problems (Caprara et al 2016;Matuschke et al 2017), toll setting problems (Labbé and Violin 2016), network routing problems (Amaldi et al 2013) and (singleton) congestion games (Castiglioni et al 2018;Marchesi et al 2018).…”

Section: Introductionmentioning

confidence: 99%

Bilevel programming methods for computing single-leader-multi-follower equilibria in normal-form and polymatrix games

Basilico

Coniglio

Gatti

et al. 2020

EURO Journal on Computational Optimization

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The concept of leader-follower (or Stackelberg) equilibrium plays a central role in a number of real-world applications bordering on mathematical optimization and game theory. While the single-follower case has been investigated since the inception of bilevel programming with the seminal work of von Stackelberg, results for the case with multiple followers are only sporadic and not many computationally affordable methods are available. In this work, we consider Stackelberg games with two or more followers who play a (pure or mixed) Nash equilibrium once the leader has committed to a (pure or mixed) strategy, focusing on normal-form and polymatrix games. As customary in bilevel programming, we address the two extreme cases where, if the leader's commitment originates more Nash equilibria in the followers' game, one which either maximizes (optimistic case) or minimizes (pessimistic case) the leader's utility is selected. First, we show that, in both cases and when assuming mixed strategies, the optimization problem associated with the search problem of finding a Stackelberg equilibrium is N P-hard and not in Poly-APX unless P = N P. We then consider different situations based on whether the leader or the followers can play mixed strategies or are restricted to pure strategies only, proposing exact nonconvex mathematical programming formulations for the optimistic case for normal-form and polymatrix games. For the pessimistic problem, which cannot be tackled with a (single-level) mathematical programming formulation, we propose a heuristic black-box algorithm. All the methods and formulations that we propose are thoroughly evaluated computationally.

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“…A prominent example is that one of security games, where a defender, acting as leader, is tasked to allocate scarce resources to protect valuable targets from an attacker, who acts as follower [3,17,28]. Besides the security domain, applications can be found in, among others, interdiction games [10,23], toll-setting problems [19], and network routing [2].…”

Section: Introductionmentioning

confidence: 99%

Computing a Pessimistic Stackelberg Equilibrium with Multiple Followers: The Mixed-Pure Case

2019

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The search problem of computing a Stackelberg (or leader-follower) equilibrium (also referred to as an optimal strategy to commit to) has been widely investigated in the scientific literature in, almost exclusively, the single-follower setting. Although the optimistic and pessimistic versions of the problem, i.e., those where the single follower breaks any ties among multiple equilibria either in favour or against the leader, are solved with different methodologies, both cases allow for efficient, polynomialtime algorithms based on linear programming. The situation is different with multiple followers, where results are only sporadic and depend strictly on the nature of the followers' game. In this paper, we investigate the setting of a normal-form game with a single leader and multiple followers who, after observing the leader's commitment, play a Nash equilibrium. When both leader and followers are allowed to play mixed strategies, the corresponding search problem, both in the optimistic and pessimistic versions, is known to be inapproximable in polynomial time to within any multiplicative polynomial factor unless P = NP. Exact algorithms are known only for the optimistic case. We focus on the case where the followers play pure strategies-a restriction that applies to a number of real-world scenarios and which, in principle, makes the problem easier-under the assumption of pessimism (the optimistic version of the problem can be straightforwardly solved in polynomial time). After casting this search problem (with followers playing pure strategies) as a pessimistic bilevel programming problem, we show that, with two followers, the problem is NP-hard and, with three or more followers, it cannot be approximated in polynomial time to within any multiplicative factor which is polynomial in the size of the normal-form game, nor, assuming utilities in [0, 1], to within any constant additive loss stricly smaller than 1 unless P = NP. This shows that, differently from what happens in the optimistic version, hardness and inapproximability in the pessimistic problem are not due to the adoption of mixed strategies. We then show that the problem admits, in the general case, a supremum but not a maximum, and we propose a single-level mathematical programming reformulation which asks for the maximization of a nonconcave All the authors contributed equally to this manuscript. Extended author information available on the last page of the article quadratic constraints. Since, due to admitting a supremum but not a maximum, only a restricted version of this formulation can be solved to optimality with state-of-theart methods, we propose an exact ad hoc algorithm (which we also embed within a branch-and-bound scheme) capable of computing the supremum of the problem and, for cases where there is no leader's strategy where such value is attained, also an α-approximate strategy where α > 0 is an arbitrary additive loss (at most as large as the supremum). We conclude the paper by evaluating the scalability of our algorithms via computat...

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Bilevel Knapsack with Interdiction Constraints

Cited by 72 publications

References 19 publications

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

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

Bilevel programming methods for computing single-leader-multi-follower equilibria in normal-form and polymatrix games

Computing a Pessimistic Stackelberg Equilibrium with Multiple Followers: The Mixed-Pure Case

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