Efficiently solving linear bilevel programming problems using off-the-shelf optimization software

Pineda, Salvador; Bylling, Henrik C.; Morales, Juan M.

doi:10.1007/s11081-017-9369-y

Cited by 52 publications

(34 citation statements)

References 56 publications

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“…We show, using a counterexample, that the trial-and-error procedure that is presently employed to tune the big Ms in many works published in PES journals may actually fail and provide highly suboptimal solutions. We advocate, instead, for the use of more sophisticated methods like the one proposed in [18] to properly tune the values of the big-Ms when solving LBP.…”

Section: Discussionmentioning

confidence: 99%

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

Pineda

Morales

2019

IEEE Trans. Power Syst.

119

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The most common procedure to solve a linear bilevel problem in the PES community is, by far, to transform it into an equivalent single-level problem by replacing the lower level with its KKT optimality conditions. Then, the complementarity conditions are reformulated using additional binary variables and large enough constants (big-Ms) to cast the single-level problem as a mixed-integer linear program that can be solved using optimization software. In most cases, such large constants are tuned by trial and error. We show, through a counterexample, that this widely used trial-and-error approach may lead to highly suboptimal solutions. Then, further research is required to properly select big-M values to solve linear bilevel problems.Index Terms-Bilevel programming, optimality conditions, mathematical program with equilibrium constraints (MPEC).

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Section: Discussionmentioning

confidence: 99%

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

Pineda

Morales

2019

IEEE Trans. Power Syst.

119

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“…Another alternative to big-M constraints is to use special order set constraints of type 1 (SOS1) as discussed in [34,39]. An SOS1 constraint is a set of variables in which at most one member can be strictly positive.…”

Section: Mplcc Formulation Of H(a)mentioning

confidence: 99%

New characterizations of Hoffman constants for systems of linear constraints

2020

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We give a characterization of the Hoffman constant of a system of linear constraints in R n relative to a reference polyhedron R ⊆ R n . The reference polyhedron R represents constraints that are easy to satisfy such as box constraints. In the special case R = R n , we obtain a novel characterization of the classical Hoffman constant.More precisely, suppose R ⊆ R n is a reference polyhedron, A ∈ R m×n , and A(R) :Our characterization is stated in terms of the largest of a canonical collection of easily computable Hoffman constants. Our characterization in turn suggests new algorithmic procedures to compute Hoffman constants.Furthermore, observe that if J ∈ S(A) andṽ ∈ ext{v ∈ R J + , A T J v * ≤ 1} then A J ′ must have full row rank for J ′ := {i :ṽ i > 0} ⊆ J. Therefore Proposition 2 also implies thatwhich is precisely the characterization (4) of H(A). In addition, Proposition 2 implies the following bound on H(A) in terms of the χ(A) condition measure [41,44,46] established in [50] for the special case when A ∈ R m×n is full column rank and both R m and R n are endowed with Euclidean norms:|J |=n A J non-singular max{ v : v ∈ R J , A T J v ≤ 1} = max J ⊆[m],|J |=n A J non-singular A −1 J = χ(A). The last step follows from [50, Prop. 3.7].

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“…In this note, we discuss how (1.1) can be solved in practice. Even in the case of g being linear (in this case (1.1) becomes a bilevel linear programme (BLP)), problem (1.1) is not convex in general and solving it poses a hard task [1,4,13,23]. A common approach for solving general bilevel problems is to replace the inner optimisation problem-in case it is convex and a constraint qualification is satisfied-with its KKT conditions (see [11,14]).…”

Section: Introductionmentioning

confidence: 99%

“…Handling the resulting non-linear complementary slackness condition is done by using a branch-andbound approach [2,3] or by splitting the complementary condition into two linear constraints involving binary variables [15]. The latter approach involves 'big-M' constraints, and it is a challenging task to find appropriate constants a priori (compare [23]). Alternatively, nonlinear solvers could be used to directly solve the KKT-reformulated problem.…”

Section: Introductionmentioning

confidence: 99%

“…However, local optima of the regularised KKT-reformulation acquired by this method are not local optima of the original bilevel problem in general [10]. In [23], a local optimum of the KKT-reformulation obtained by the regularisation approach is used for finding appropriate parameters (big-M constant and initial values for binary variables) of the mixed integer reformulation from [15].…”

Section: Introductionmentioning

confidence: 99%

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Solving bilevel problems with polyhedral constraint set

2019

J. Appl. Numer. Optim.

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In this paper, we study the relationship between bilevel programmes and polyhedral projection problems. Extending a well-known result by Fülöp, we show that solving a bilevel problem with polyhedral constraints is equivalent to optimise the upper level objective over certain facets of an associated polyhedral projection problem. Utilising this result, we show how solutions to such bilevel problems can be computed.

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Efficiently solving linear bilevel programming problems using off-the-shelf optimization software

Cited by 52 publications

References 56 publications

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

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

New characterizations of Hoffman constants for systems of linear constraints

Solving bilevel problems with polyhedral constraint set

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