Abstract
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 mixedinteger linear bilevel problems and the most recent algorithms use branchandcut techniques from mixedinteger programming that are especially tailored for the bilevel context. In this paper, we consider MIQPQP bilevel problems, i.e., models with a mixedinteger convexquadratic upper level and a continuous convexquadratic lower level. This setting allows for a strongdualitybased transformation of the lower level which yields, in general, an equivalent nonconvex singlelevel reformulation of the original bilevel problem. Under reasonable assumptions, we can derive both a multi and a singletree outerapproximationbased cuttingplane 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.
Introduction
Bilevel optimization problems are used in various applications, e.g., in energy markets [9, 12, 20, 29, 31, 34, 39], in critical infrastructure defense [11, 16], or in pricing problems [15, 40] to model hierarchical decision processes. As such, they embed one optimization problem, the socalled lowerlevel problem, into the constraints of a socalled upperlevel problem. This leads to inherent nonconvexities, which render already linear bilevel problems with a linear upper and lowerlevel problem NPhard; see, e.g., [14, 32, 35].
In this work, we study mixedinteger quadratic bilevel problems of the form
where \(H_u\in \mathbb {R}^{{n_x}\times {n_x}}\), \(G_u\in \mathbb {R}^{{n_y}\times {n_y}}\), and \(G_l\in \mathbb {R}^{{n_y}\times {n_y}}\) are symmetric and positive semidefinite matrices. Furthermore, we have vectors \(c_u\in \mathbb {R}^{n_x}\), \(d_u, d_l\in \mathbb {R}^{n_y}\), matrices \(A \in \mathbb {R}^{{m_u}\times {n_x}}\), \(B \in \mathbb {R}^{{m_u}\times {n_y}}\), \(C \in \mathbb {R}^{{m_l}\times I}\), \(D\in \mathbb {R}^{{m_l}\times {n_y}}\), as well as righthand side vectors \(a\in \mathbb {R}^{m_u}\) and \(b\in \mathbb {R}^{m_l}\). The variables \(x=(x_I, x_R)\) denote the integer (\(x_I\)) and continuous (\(x_R\)) upperlevel variables and y denotes the (continuous) lowerlevel variables. Note that we w.l.o.g. ordered the integer and continuous upperlevel variables for the ease of presentation. In this setup, the upperlevel problem is a convex mixedinteger quadratic problem (MIQP) and for fixed integer upperlevel variables \(x_I\), the lower level is a convex quadratic problem (QP), i.e., it is a parametric convex QP. In total, we are facing an MIQPQP bilevel problem with the following key properties:

(i)
All upperlevel integer variables \(x_I\) are bounded.

(ii)
All linking variables, i.e., upperlevel variables that appear in the lowerlevel constraints, are integer.
We note that in Problem (1), we implicitly assume that all integer upperlevel variables are linking variables. However, this formulation also contains the more general case of integer upperlevel variables that do not appear in the lowerlevel problem by setting some columns in the matrix C to zero.
The main motivation of this work is to exploit the two properties above to develop a multi and a singletree solution approach for Problem (1) based on outer approximation for convex mixedinteger nonlinear problems (MINLP) [18, 25]. The abovementioned two properties are required by many other stateoftheart algorithms for linear bilevel problems with purely integer linear (ILPILP) or mixedinteger linear upper and lowerlevel problems (MILPMILP); see, e.g. [21, 23, 43, 50, 54]. These methods successfully use bileveltailored branchandbound or branchandcut methods to solve quite large instances of hundreds or thousands of variables and constraints. However, they cannot directly deal with continuous lowerlevel problems and/or have not yet been extended to the convexquadratic case—if this is possible at all. An extension of [54] to the general class of mixedinteger nonlinear bilevel problems with integer linking variables is proposed in [41] but the computational study therein only covers mixedinteger linear bilevel problems. In [3], the authors use multiparametric programming techniques to solve bilevel problems with convex MIQPs on both levels. This algorithm explicitly allows for continuous linking variables. The backbone of the approach is the computation of the optimal solutions of the lowerlevel problem as a function of the upperlevel decisions. This step is very costly and, according to the authors, it is evident that this algorithm is not intended for the solution of larger problems. The computational study included in the paper deals with problems up to 65 variables. In addition, the algorithm is only applicable for problems that possess unique lowerlevel solutions.
Aside from mixedinteger bilevel problems, algorithms for various classes of continuous convexquadratic bilevel problems have been proposed. However, in contrast to algorithms for mixedinteger bilevel problems, reported numerical results for continuous convexquadratic bilevel problems seem to cover only rather small instances. A branchandbound algorithm for bilevel problems with a convex upperlevel problem and a strictly convex lowerlevel problem is proposed in [5]. The author demonstrates the effectiveness of his method for problems with up to 15 variables and 20 constraints on each level. In [6], a convexquadratic lowerlevel problem is replaced by its Karush–Kuhn–Tucker (KKT) conditions and then a branching on the complementarity constraints is applied. The authors report results for problems with up to 60 upperlevel and 40 lowerlevel variables. This approach is generalized from linear upperlevel to convex upperlevel problems in [19]. Two different descent algorithms for bilevel problems with a strictly convex lower level and a concave or convex upper level are proposed in [53]. However, the authors do not provide computational results. Recently, also neural networks are used to tackle continuous convexquadratic bilevel problems; see [33, 42].
To the best of our knowledge, tailored algorithms for mixedinteger quadratic bilevel problems of the form (1) are neither reported nor has their efficiency been demonstrated in a comprehensive computational study. In fact, there exist no code packages that can be used as a benchmark for our proposed solution techniques. Thus, we use wellknown singlelevel reformulations based on KKT conditions and strong duality as a benchmark in our computational study in Sect. 4.
Our contribution is the following. We consider bilevel problems with a mixedinteger convexquadratic upper level and a convexquadratic lower level. For this nonconvex problem class, we provide an equivalent reformulation to a convex MINLP that uses strong duality of the lower level; see Sect. 2. Further, in Sect. 3 we propose a multi and a singletree solution approach that are both inspired by outerapproximation techniques for convex MINLPs. We prove the correctness of the methods and discuss further extensions. We are not aware of any other work that applies outerapproximation techniques from the area of convex MINLP to mixedinteger bilevel programming. In Sect. 4, we evaluate the effectiveness of the proposed approaches in an extensive numerical study, in which we solve instances with up to several thousand variables and constraints. We conclude in Sect. 5.
A convex singlelevel reformulation
Most solution techniques for bilevel problems rely on a reformulation of the bilevel problem to a singlelevel problem [14]. For problems with a convex lower level, the lowerlevel problem can be replaced by its nonconvex KKT conditions. Especially for problems with linear lowerlevel constraints, this approach is very popular, because it allows for a mixedinteger linear reformulation of the KKT complementarity conditions using additional binary variables and bigM values; see, e.g., [26]. With this approach the bilevel problem (1) can be equivalently transformed to the following mixedinteger singlelevel problem
where Constraints (2b) and (2c) model primal feasibility of the upper and lower level, respectively. Constraint (2d) models dual feasibility of the lowerlevel problem with dual lowerlevel variables \(\lambda \in \mathbb {R}^{m_l}_{\ge 0}\). Constraints (2e) ensure KKT complementarity via additional binary variables \(v_j\) and sufficiently large numbers \(M_1\) and \(M_2\). Similarly, a strongdualitybased reformulation can be derived by replacing KKT complementarity (2e) by the strongduality equation of the lower level. This approach is significantly less used in practice because even for linear bilevel problems one obtains nonconvex bilinear terms due to products of primal upperlevel and dual lowerlevel variables. These terms can only be linearized if all linking variables are integer. Recently, in [55], a numerical study is provided that compares the KKT approach with the strongduality approach for linear bilevel problems with integer linking variables and continuous lowerlevel problems. The authors conclude that the strongduality reformulation works significantly better than the KKT reformulation for problems with large lowerlevel problems. Other contributions that successfully apply the strongduality reformulation to linear bilevel problems include, e.g., [2, 27, 28, 30, 40, 44]. Except from Bard, who briefly sketches the idea in [5], we are not aware of any works that use strong duality for bilevel problems with quadratic lower levels. One reason might be that the resulting strongduality equation is quadratic. Opposed to the linearized KKT reformulation, the strongdualitybased reformulation thus yields a quadratically constrained program. Despite this drawback, we derive such a strongdualitybased singlelevel reformulation after introducing some notation.
General notation
The bilevel constraint region is denoted by
Throughout this paper, we assume that P is bounded. This set corresponds to the set obtained by relaxing the optimality of the lowerlevel problem. Its projection onto the decision space of the upper level is given by
For fixed \(\bar{x}=(\bar{x}_R,\bar{x}_I) \in P_u\), the lowerlevel feasible region is given by
and the rational reaction set of the lower level reads
Since \(G_l\) is semidefinite, the lower level may not have a unique solution, i.e., \(M(\bar{x}_I)\) may not be a singleton. In such a case, we assume the optimistic bilevel solution, i.e., \({\bar{y}}\in M(\bar{x}_I)\) is chosen in favor of the upper level; see, e.g., Chapter 1 in [14]. In Problem (1), this is indicated by “\(\min _{x,{\bar{y}}}\)” in the upperlevel objective, i.e., the upperlevel minimizes over x and \({\bar{y}}\). We emphasize that singlelevel reformulations, like, e.g., Problem (2), implicitly assume the optimistic bilevel solution, such that there is no need to distinguish between y and \({\bar{y}}\). Further, note that our solution approach explicitly allows for \(M(\bar{x}_I)=\emptyset \) for some \(\bar{x} \in P_u\). Finally, the bilevel feasible set is given by
If \(\mathcal {F}= \emptyset \), then Problem (1) is infeasible.
Strongdualitybased nonconvex singlelevel reformulation
We now use strong duality of the lowerlevel problem to transform Problem (1) into an equivalent nonconvex singlelevel problem. The parametric Lagrangian dual problem of the parametric lower level
with parameter \(x_I\) is given by
with \(g(x_I;\lambda )= \inf _y \mathcal {L}(x_I;y,\lambda )\); see [10]. In our setup, the Lagrangian \(\mathcal {L}\) reads
Since \(\mathcal {L}(x_I;y,\lambda )\) is convex and differentiable in y, the infimum is given by
In order to denote the Lagrangian dual problem in its general form (4), we could use Expression (6) to obtain \(y = G_l^{1}(D^\top \lambda  d_l)\). This can then be used to substitute the primal variable y in the Lagrangian (5) to obtain
However, this only works if \(G_l\) is regular, e.g., if \(G_l\) is strictly definite. In the more general case of semidefinite matrices, we can explicitly keep the primal variable y and substitute \(D^\top \lambda = G_ly + d_l\) in the Lagrangian (5). This yields the dual problem
Note that \(\bar{g}(x_I;\cdot )\) is a concavequadratic function in y and \(\lambda \) because \(G_l\) is positive semidefinite. Thus, the dual problem (7) is a parametric concavequadratic maximization problem over affinelinear constraints. Since Problem (7) does not involve the inverse \(G_l^{1}\), we use formulation (7) also in the case when \(G_l\) is strictly positive definite.
In the following, we only consider \(x \in P_u\), i.e., upperlevel variables for which the parametric lowerlevel problem is feasible. The parametric lowerlevel problem (3) is a convexquadratic minimization problem over affinelinear constraints. Consequently, duality conditions apply without requiring additional constraint qualifications; see Chapter 5.2.3 in [10] and Theorem 24.1 in [52]. For every primaldual feasible point \((y,\lambda )\), i.e., \(y \in P_l(x_I)\) and \(\lambda \ge 0\) that fulfill \(G_ly + d_l= D^\top \lambda \), weak duality
holds. Furthermore, for optimal primaldual points \(({y}^{*}, {\lambda }^{*})\), strong duality holds, i.e., \(q_l({y}^{*}) = \bar{g}(x_I;{y}^{*},{\lambda }^{*})\). Together with weak duality (8), strong duality can be enforced by the constraint \(q_l(y)  \bar{g}(x_I;y,\lambda ) \le 0\), i.e.,
This strongduality inequality is convex in y but the bilinear term \(\lambda ^\top C x_I\) is nonconvex. Using (9), the bilevel problem (1) can be recast as the equivalent singlelevel nonconvex mixedinteger quadratically constrained quadratic program (MIQCQP):
Convexification of the strongduality constraint
The only nonconvexity in the strongduality inequality (9) is the bilinear product \(\lambda ^\top C x_I\). Since the variables \(x_I\) are integer, this product can be reformulated using a binary expansion. For the ease of presentation, we w.l.o.g. assume in the following that \(x_i^ = 0\) for all \(i \in I\). The products of binary and continuous variables can then be linearized by several techniques. According to the numerical study in [55], the following approach works best in a bilevel context. We express the integer variables \(x_j\) with the help of \(\bar{r}_j = \lfloor \log _2(x_j^+) \rfloor + 1\) many auxiliary binary variables \(s_{jr}\):
With this we obtain
Now, we replace the binarycontinuous products \(s_{jr} \sum _{i=1}^{{m_l}} c_{ij} \lambda _i\) by introducing auxiliary continuous variables \(w_{jr}\) and enforce
by the additional constraints
In this formulation, we need bounds \(\overline{\lambda }_j \ge \sum _{i=1}^{{m_l}} c_{ij} \lambda _i\), and \(\underline{\lambda }_j \le \sum _{i=1}^{{m_l}} c_{ij} \lambda _i\) which are, in practice, often derived by some suitable bigM. The need for such bounds is also a major drawback in the KKTbased reformulation (2). In [46], it is shown that wrong bigMs 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, bigMs should be derived using problemspecific knowledge. However, we point out that our solution approaches introduced in Sect. 3 compute bilevelfeasible points independent of the bigMs. Thus, in case of too small bigMs, our algorithms may terminate with a suboptimal solution but never compute bilevelinfeasible points. We discuss this also later in Sect. 3.3.
Using (11) and (13), we rewrite Constraint (9) as
which is convex in y and linear in \(\lambda \) and \(w\). Note that \(2^{r1} \le x_j^+\) holds, i.e., for reasonable bounds \(x_j^+\), the exponential coefficients \(2^{r1}\) can be considered as numerically stable.
Finally, the singlelevel problem (10) can be stated equivalently as
This is a convex MIQCQP that has more binary variables and constraints compared to the nonconvex MIQCQP (10). We denote the feasible set of Problem (15) by \(\Omega \), feasible points by \(z=(x,y,\lambda ,w,s) \in \Omega \) and optimal points by \({z}^{*}\). By construction of Problem (15), we have the following equivalence result.
Lemma 1
The feasible set of Problem (15) projected on the (x, y)space equals the bilevel feasible set of the bilevel Problem (1), i.e., \(\mathcal {F}= {{\,\mathrm{Proj}\,}}_{(x,y)}(\Omega )\) holds. In addition, for every global optimal solution \({z}^{*}=({x}^{*}, {y}^{*}, {\lambda }^{*}, {w}^{*}, {s}^{*})\) of Problem (15), \(({x}^{*},{y}^{*})\) is a global optimal solution for Problem (1) and every global optimal solution \(({x}^{*}, {y}^{*})\) of Problem (1) is part of an optimal solution \({z}^{*}\) of Problem (15).
Two outerapproximation solution approaches
Problem (15) is an MIQCQP in which the single quadratic constraint is convex. Such problem classes can be solved, e.g., directly by modern solvers like Gurobi or CPLEX. On the other hand, Problem (15) belongs to the broader class of convex MINLPs. For such problems, a variety of approaches exist, e.g., nonlinear branchandbound or multi and singletree methods based on outer approximation, generalized Benders decomposition, and extended cuttingplanes; see [7] for a detailed survey of these methods. In this section, we introduce outerapproximation techniques that are tailored for mixedinteger quadratic bilevel problems. The general idea is to relax the convexquadratic strongduality inequality (14) of the lower level in Problem (15) to obtain an MIQP. Strong duality is then resolved by iteratively adding linear outerapproximation cuts. In its simplest form, this is a direct application of Kelley’s cuttingplanes approach [36], which would add linear outerapproximation cuts until the strongduality inequality (14) is satisfied up to a certain tolerance. Our preliminary numerical results indicate that this requires an enormous amount of iterations. We thus discuss a more sophisticated multitree approach and its singletree variant in Sects. 3.1 and 3.2.
A multitree outerapproximation approach
The wellknown multitree outer approximation for convex MINLPs was first proposed in [18] and has been enhanced in [8, 25]. It alternatingly solves a mixedinteger linear master problem and a convex nonlinear problem (NLP) as a subproblem. The master problem is a mixedinteger linear relaxation of the original convex MINLP and is tightened subsequently by adding linear outerapproximation cuts for the convex nonlinearities. The convex nonlinear subproblem results from fixing all integer variables to the solution of the master problem in the original convex MINLP. Under suitable assumptions, every feasible integer solution of the master problem is visited at most once, and the algorithm terminates after a finite number of iterations with the correct solution. The algorithm proposed in this subsection is very much inspired by this scheme. The master problem that we solve in every iteration \(p\ge 0\) is given by
where \(\bar{c}(\bar{y}^\ell ;y, \lambda , w) \le 0\) is the linear outerapproximation cut that is added to the master problem after every iteration. Thus, in iteration \(p\), the master problem (\(\text {M}^{p}\)) contains \(p\) outerapproximation cuts. We shed some more light on \(\bar{c}\) later, but first emphasize that Problem (\(\text {M}^{p}\)) is a convex MIQP. This is in contrast to the standard outerapproximation literature, where the objective function is relaxed and iteratively approximated as well, resulting in a mixedinteger linear master problem. The rationale is that, in our implementation, the main working horse is a stateoftheart solver like Gurobi or CPLEX. In recent years, these solvers made significant progress in solving convex MIQPs effectively. Thus, we want to exploit these highly evolved solvers as much as possible.
Now, we give some more details on the linear function \(\bar{c}\), which is derived using the firstorder Taylor approximation of the convex strongduality inequality (14). For a general convex function h(v), the firstorder Taylor approximation at \(\bar{v}\) reads
Applied to (14), this gives
Since \(\hat{{c}}(y,\lambda ,w)\) is linear in \(\lambda \) and \(w\), the firstorder Taylor approximation (16) is parameterized solely by \(\bar{y}\). The effectiveness of the proposed outerapproximation approach will depend on the actual selection of \(\bar{y}\), but \(\bar{c}(\bar{y};y,\lambda , w) \le 0\) is a valid inequality no matter how \(\bar{y}\) is obtained. This is shown in the following lemma, in which \(\mathcal {M}^p\) denotes the feasible set of (\(\text {M}^{p}\)).
Lemma 2
For every iteration \(p\ge 1\), \(\Omega \subseteq \mathcal {M}^p\subseteq \mathcal {M}^{p1}\) holds.
The consequence of Lemma 2 is that every master problem (\(\text {M}^{p}\)) is a relaxation of the singlelevel reformulation (15).
Proof
Let \(z=(x,y,\lambda ,w,s) \in \Omega \) be a feasible point of Problem (15). In particular, z fulfills strong duality, i.e., \(\hat{{c}}(y,\lambda ,w) = 0\). Obviously, \(z \in \mathcal {M}^0\) holds because \(\mathcal {M}^0\) corresponds exactly to \(\Omega \) without the strongduality inequality (14). In addition, since \(\hat{{c}}\) is convex, its firstorder Taylor approximation is a global underestimator at any point \(\bar{y}\), i.e.,
This holds in particular for the choice \(\bar{y}=\bar{y}^\ell \) for any \(\ell =1,\ldots ,p\) and \(p\ge 1\). Hence, \(z \in \mathcal {M}^p\). Further, \(\mathcal {M}^p\subseteq \mathcal {M}^{p1}\) follows by construction. \(\square \)
In the following, we give details on how to select the linearization points \(\bar{y}\). We therefore assume that the master problem (\(\text {M}^{p}\)) is solvable and denote its solution by \(z^p=(x^p, y^p, \lambda ^p, w^p, s^p)\). According to Kelley’s cutting plane approach [36], one would add an outerapproximation cut (16) at the solution of the master problem, i.e., one would add the inequality \(\bar{c}(y^p;y,\lambda , w) \le 0\). Kelley has shown that this yields a convergent approach. However, in many cases this will turn out to be inefficient, because these cuts are rather weak. The outerapproximation methods in the spirit of [8, 18, 25] additionally solve a convex nonlinear subproblem that results from fixing the integer variables in the original convex MINLP (or an auxiliary feasibility problem if the subproblem is infeasible) to obtain suitable linearization points \(\bar{y}\). In our context, the subproblem is given by fixing \(x_I=x_I^p\) and \(s=s^p\) in the convex MINLP (15), which yields the convex quadratically constrained quadratic problem (QCQP):
We denote the feasible set of (\(\text {S}^p\)) by \(\mathcal {S}^p\) and first assume \(\mathcal {S}^p\ne \emptyset \). The infeasible case is discussed afterward. Let \((\bar{x}_R^p, \bar{y}^p, \bar{\lambda }^p, \bar{w}^p)\) be the solution of the subproblem (\(\text {S}^p\)). For the correctness of our proposed algorithms, we need a technical assumption that is also used in [8, 18, 25].
Assumption 1
For every feasible subproblem (\(\text {S}^p\)), the Abadie constraint qualification holds at the solution \((\bar{x}_R^p, \bar{y}^p, \bar{\lambda }^p, \bar{w}^p)\), i.e., the tangent cone and the linearized tangent cone coincide at \((\bar{x}_R^p, \bar{y}^p, \bar{\lambda }^p, \bar{w}^p)\).
A formal description of standard cones in nonlinear optimization along with the corresponding theory that is also required in the proof of the following lemma can be found, e.g., in [45]. In theory, Assumption 1 is crucial for the termination of the methods described in this section. An indication that this assumption is not fulfilled in practice is cycling, i.e., a certain integer solution is computed more than once. In our preliminary numerical tests, cycling hardly ever occurred for any of the proposed approaches, such that we disabled an expensive cycling detection (e.g., storing all integer solutions and adding a nogoodcut if an integer solution is computed for the second time) in our computations.
We now show that it is indeed a good idea to linearize the strongduality inequality (14) at the solution of the subproblem \(\bar{y}^p\) instead of at the solutions of the master problem \(y^p\).
Lemma 3
Let \(z^{p}=(x_I^p,x_R^p, y^p, \lambda ^p, w^p, s^p)\) be an optimal solution of the master problem (\(\text {M}^{p}\)) and assume that the subproblem (\(\text {S}^p\)) is feasible and has the optimal solution \((\bar{x}_R^p, \bar{y}^p, \bar{\lambda }^p, \bar{w}^p)\). Suppose further that Assumption 1 holds and consider the new master problem that is obtained by adding the outerapproximation cut \(\bar{c}(\bar{y}^{p};y,\lambda , w) \le 0\) to (\(\text {M}^{p}\)). Then, for any feasible point of the form \(z=(x_I^p, x_R, y,\lambda ,w,s^p)\) of this problem the following holds:
Proof
We consider \(x_I^p\) and \(s^p\) fixed and assume that (\(\text {S}^p\)) is feasible with optimal solution \((\bar{x}_R^p, \bar{y}^{p},\bar{\lambda }^{p}, \bar{w}^p)\). Thus, \((\bar{y}^{p},\bar{\lambda }^{p}, \bar{w}^p)\) fulfills the convex strongduality inequality (14). Since weak duality holds anyway, we obtain \(\hat{{c}}(\bar{y}^p, \bar{\lambda }^p, \bar{w}^p) = 0\). Now, let \(z=(x_I^p, x_R, y,\lambda ,w,s^p)\) be feasible for (\(\text {M}^{p}\)) and let \(\bar{c}(\bar{y}^p; y, \lambda , w) \le 0\), i.e., z is feasible for a suitably chosen master problem in iteration \(p+ 1\). In the following, we abbreviate the vector \(v=(y,\lambda , w)\) for the ease of presentation. Then, we have
i.e., \( \nabla _{v} \hat{{c}}(\bar{v}^{p})^\top (v  \bar{v}^{p}) \le 0, \) which means that \((v  \bar{v}^p)\) is in the linearized tangent cone \({T}^{\mathrm {lin}}_{\mathcal {S}^p}(\bar{v}^p)\). Due to Assumption 1, \({T}^{\mathrm {lin}}_{\mathcal {S}^p}(\bar{v}^p)\) equals the tangent cone \({T}_{S^p}(\bar{v}^p)\), which gives \((v  \bar{v}^p) \in {T}_{S^p}(\bar{v}^p)\). For all directions \(d \in {T}_{S^p}(\bar{v}^p)\), we know that the property
holds. Thus, we have
The first inequality follows because \(q_u\) is convex, i.e., its firstorder Taylor approximation is a global underestimator. The second inequality follows from Inequality (17). \(\square \)
In contrast to the slightly different setting in [25], using the solution of the subproblem (\(\text {S}^p\)) as the linearization point of the outerapproximation inequality (14) does not explicitly cut off the related integer solution \(x_I^p\). The reason is our modified master problem, that does not linearize and approximate the convex objective function. Nevertheless, Lemma 3 lets us conclude that every integer assignment \(x_I\) that yields a feasible subproblem (\(\text {S}^p\)) needs to be visited only once, because the objective cannot be improved by visiting such a solution for a second time. This will be one of the key properties to prove finite termination of our algorithm.
We now consider the case of an infeasible subproblem (\(\text {S}^p\)). In [18], it is argued that in order to eliminate \(x_I^p\) from further consideration, an integer nogoodcut must be introduced. In our application, this is a straightforward task. The subproblem (\(\text {S}^p\)) is fully parameterized by fixed upperlevel variables \(x_I^p\). For these variables we have a binary expansion available anyway (the variables \(s\)) so that a simple binary nogoodcut on \(s\) can be used. However, such nogoodcuts are known to cause numerical instabilities. As a remedy, in [25], it is proposed to derive cutting planes from an auxiliary feasibility problem that indeed cut off the integer solution \(x_I^p\). The feasibility problem minimizes the constraint violations of the infeasible subproblem in some suitable sense, e.g., via the \(\ell ^1\) or the \(\ell ^\infty \)norm. Recap that \(z^p\) is a solution of the master problem (\(\text {M}^{p}\)). In particular, \((y^p,\lambda ^p)\) is primaldual feasible for the lowerlevel problem (3) with fixed \(x_I^p\). Thus, the latter problem also has an optimal solution that fulfills strong duality. Since the subproblem (\(\text {S}^p\)) is infeasible, this optimal solution must be infeasible for the upperlevel constraints. On the other hand, \(z^p\) must be feasible for the subproblem (\(\text {S}^p\)) without the strongduality inequality, because \(z^p\) is feasible for Problem (\(\text {M}^{p}\)). Thus, a simple feasibility problem in the sense of [25] is given by
whose objective value is strictly greater than zero, since otherwise the subproblem (\(\text {S}^p\)) would be feasible. For a solution of (\(\text {F}^p\)), we obtain the following lemma by adapting Lemma 1 of [25].
Lemma 4
Let \(z^p\) be a solution of the master problem (\(\text {M}^{p}\)), let the subproblem (\(\text {S}^p\)) be infeasible, and let \((\bar{x}_R^p, \bar{y}^p, \bar{\lambda }^{p}, \bar{w}^{p})\) be a solution of the feasibility problem (\(\text {F}^p\)). Then, \(\hat{{c}}(\bar{y}^p, \bar{\lambda }^p, \bar{w}^p) > 0\) and every \(z=(x_I^p, x_R, y, \lambda , w, s^p) \in \mathcal {M}^p\) is infeasible for the constraint
Proof
Consider a fixed \(x_I^p\) and assume (\(\text {S}^p\)) to be infeasible, which means that (\(\text {F}^p\)) has an optimal solution \((\bar{x}_R^p, \bar{y}^p, \bar{\lambda }^p, \bar{w}^p)\) with \(\hat{{c}}(\bar{y}^p, \bar{\lambda }^p, \bar{w}^p) > 0\). For the ease of presentation, we again use the abbreviation \(v=(y,\lambda , w)\) and we rewrite the linear constraint set of (\(\text {F}^p\)) to obtain
Problem (\(\text {F}^p\)), and hence Problem (19), minimizes a convex function over affinelinear constraints. Thus, \((\bar{x}_R^p,\bar{v}^p)\) fulfills the KKT conditions of Problem (19), i.e., primal feasibility, stationarity, nonnegativity of multipliers \(\alpha \) of inequality constraints, and complementarity. With \(\delta \) denoting the multipliers of the equality constraints, the KKT conditions read
We recap that \(\bar{c}\) is derived from the firstorder Taylor approximation, i.e., it holds
This can be expanded to
by using KKT stationarity (20c) (for (21a)) and reordering the terms (for (21b)). Further, we replaced \(\alpha ^\top \tilde{B} \bar{v}^p\) by \(\alpha ^\top \tilde{b}\) according to (20e) and (20b) and \(\tilde{D} \bar{v}^p\) by \(\tilde{d}\) according to (20a) to obtain (21c). Now, let \(z=(x_I^p, x_R, v, s^p) \in \mathcal {M}^p\). This implies that v must be feasible for (\(\text {F}^p\)), respectively (19). In particular, we know that \(\tilde{B}v  \tilde{b} \le  \tilde{A}x_R\) and \(\tilde{D} v  \tilde{d} = 0\). Applying this to (21) and using (20b) yields
Thus, v violates (18). \(\square \)
We recap that, with the last lemma, we can derive cutting planes that cut off \(x_I^p\) if the subproblem (\(\text {S}^p\)) is infeasible. This is another key property for the finite termination of our approach. With this result, we are ready to present the multitree outer approximation in Algorithm 1.
In every iteration \(p\), Algorithm 1 first solves the master problem (\(\text {M}^{p}\)) to obtain a solution \(z^p\). According to Lemma 2, \(\mathcal {M}^p\subseteq \mathcal {M}^{p 1}\) and we can update the lower bound \(\phi \) by \(q_u(x_I^p, x_R^p, y^p)\). Next, either the subproblem (\(\text {S}^p\)) or, in case of infeasibility, the feasibility problem (\(\text {F}^p\)) is solved. If the subproblem is feasible with solution \((\bar{x}_R^p, \bar{y}^p, \bar{\lambda }^p, \bar{w}^p)\), then the point \((x_I^p, \bar{x}_R^p, \bar{y}^p, \bar{\lambda }^p, \bar{w}^p, s^p)\) is feasible for the convex singlelevel reformulation (15) and the upper bound is updated if \(q_u(x_I^p, \bar{x}_R^p, \bar{y}^p) < \Phi \). In this case, also \({z}^{*}\) is updated. We terminate when \(\phi \ge \Phi \) is achieved and return the best solution \({z}^{*}\). We now show the correctness of this approach.
Theorem 1
Algorithm 1 terminates after a finite number of iterations at an optimal solution of Problem (1) or with an indication that the problem is infeasible.
The following proof is adapted from [25].
Proof
First of all, note that all master problems are bounded since P is assumed to be bounded. We next show finiteness of Algorithm 1. If Problem (1) is feasible, then we can follow from Lemma 3 that at most one integer solution is visited twice. Whenever an integer solution is visited for a second time, \(\phi \ge \Phi \) holds and the algorithm terminates. On the other hand, if Problem (1) is infeasible, then every subproblem (\(\text {S}^p\)) is infeasible. According to Lemma 4, the integer solution \(x_I^p\) is infeasible for the master problem in iteration \(p+ 1\), which results in an infeasible master problem after a finite number of iterations. Thus, finiteness follows from the finite number of integer solutions for Problem (1).
Second, we show that Algorithm 1 always terminates at a solution of Problem (1), if it is feasible. We denote a (possibly nonunique) optimal solution of Problem (1) by \(({x}^{*}_I, {x}^{*}_R, {y}^{*})\) with objective function value \(q_u^*=q_u({x}^{*}_I, {x}^{*}_R, {y}^{*})\). Now assume that Algorithm 1 terminates with a solution \((x_I^\prime , x_R^\prime , y^\prime , \lambda ^\prime , w^\prime , s^\prime )\) and objective function value \(\Phi = q_u(x_I^\prime , x_R^\prime , y^\prime )\). It is obvious that \((x_I^\prime , x_R^\prime , y^\prime , \lambda ^\prime )\) is feasible for the nonconvex singlelevel reformulation (10). According to Lemma 1, \((x_I^\prime , x_R^\prime , y^\prime )\) is then feasible for the original bilevel problem, which gives \(q_u^*\le \Phi = q_u(x_I^\prime , x_R^\prime , y^\prime )\). On the other hand, Lemma 2 together with Lemma 1 state that every master problem (\(\text {M}^{p}\)) is a relaxation of the original bilevel problem (1), i.e., \(\phi \le q_u^*\). Since Algorithm 1 only terminates when \(\phi \ge \Phi \), we obtain \(q_u({x}^{*}_I, {x}^{*}_R, {y}^{*}) = q_u(x_I^\prime , x_R^\prime , y^\prime )\). \(\square \)
In the remainder of this subsection we discuss some enhancements of Algorithm 1.
 Additional outerapproximation cuts:

Since the outerapproximation cuts of the form of (16) are globally valid, we can add cuts also for points other than the solution of the subproblem. One point that comes for free is the solution \(z^p\) of the master problem, i.e., we can add a cut (16) for the point \(y^p\). This is an outerapproximation cut in the sense of Kelley [36]. Further, we can add outerapproximation cuts for all feasible solutions that are encountered in the process of solving the master problem.
 Early termination of the master problem:

It is sufficient for the correctness of the entire algorithm that the master problem provides a new integerfeasible point. Since the outerapproximation cuts are constructed in a way that already visited integer solutions of the previous iterations have an objective value worse or equal to the incumbent \(\Phi \), we can stop the master problem with the first improving integerfeasible solution, i.e., a solution that has an objective value better than the incumbent \(\Phi \); see also [25]. This strategy mimics the singletree approach that is stated in Sect. 3.2.
 Warmstarting the master problem:

Warmstarting mixedinteger problems can be a very effective strategy because it may produce a tight initial upper bound and help to keep the branchandbound trees small. Since the incumbent solution \(z^*\) is feasible for every master problem, it is reasonable to warmstart the master problems with this solution.
A singletree outerapproximation approach
The multitree outerapproximation approach from Sect. 3.1 can also be cast into a single branchandbound tree. In the context of general convex MINLPs, this approach is known as LP/NLPbased branchandbound (LP/NLPBB) and was first introduced in [47]. LP/NLPBB avoids the timeconsuming solution of subsequently updated mixedinteger master problems by branching on the integer variables of an initial master problem and solving continuous relaxations of the subsequently updated master problem at every branchandbound node. Whenever such a relaxation results in a new integer solution with a better objective value than the incumbent, the solution process is interrupted. In this event the convex nonlinear subproblem with fixed integer variables is solved and the master problem is updated by an outerapproximation cut derived from the solution of the subproblem. In this view, LP/NLPBB can be interpreted as a branchandcut algorithm that requires the solution of NLPs to separate cuts. For additional implementation details we refer to [1, 8].
If we apply such a singletree approach to our setup, the initial master problem is given by Problem (\(\text {M}^{p}\)) for \(p=0\), i.e., this corresponds exactly to the initial multitree master problem. In this view, the problem that is solved at every branchandbound node is the continuous relaxation of Problem (\(\text {M}^{p}\)) with bounds \(\ell ,u \in \mathbb {R}^{I}\) on the integer variables, i.e., the QP:
The index \(p\) in (\(\text {N}^p(l, u)\)) corresponds to the number of added strongduality cuts. The specific values for l and u follow from branching. As opposed to the textbook LP/NLPBB that solves an LP at every branchandbound node, we solve a QP. Thus, we rather perform a QP/NLPBB.
We now state a tailored singletree approach for bilevel problems of the form (1) in Algorithm 2.
The rationale is the following. The algorithm subsequently solves QPs of the form (\(\text {N}^p(l, u)\)), starting with the rootnode problem for \(p=0\), \(l=x^\), and \(u=x^+\). Whenever such a QP is infeasible or its objective function value can not improve the incumbent, then this problem can be removed from the set of open problems \(\mathcal {O}\) once and for all. In case the solution of Problem (\(\text {N}^p(l, u)\)) is integer feasible, then the corresponding subproblem (\(\text {S}^p\))—or the feasibility problem (\(\text {F}^p\)) if the subproblem is infeasible—is solved. One key difference of LP/NLPBB compared to a standard branchandbound is that this solved subproblem must not be pruned but needs to be updated with an appropriate outerapproximation cut. Moreover, all other open problems in \(\mathcal {O}\) need to be updated as well. Finally, if Problem (\(\text {N}^p(l, u)\)) is feasible but the solution is not integer feasible, one branches on a fractional integer variable to obtain two new open problems. Similar to the multitree approach, only one integer assignment—the optimal one—is computed twice during the solution process. Finiteness follows again from the finite number of possible integer assignments. At some point all problems in the set \(\mathcal {O}\) are infeasible, \(\mathcal {O}\) is emptied, and the algorithm terminates. If all subproblems turned out to be infeasible, then \({z}^{*}\) is never updated and the infeasibility of the bilevel problem is correctly detected. All together, we obtain the following correctness theorem.
Theorem 2
Algorithm 2 terminates after a finite number of iterations with an optimal solution of Problem (1) or with an indication that the problem is infeasible.
Since all arguments mainly follow from the proof of Theorem 1, we refrain from a formal proof of Theorem 2. We close this subsection by discussing some possible enhancements of Algorithm 2.
 Additional outerapproximation cuts:

Similar to the multitree approach, we can enhance Algorithm 2 by adding outerapproximation cuts for all integerfeasible solutions \(z^{l,u}\) with \(q_u(x^{l,u},y^{l,u}) \ge \Phi \), i.e., integerfeasible solutions that do not fulfill the ifcondition in Line 7 in Algorithm 2. This can be done, e.g., by storing these nonimproving integerfeasible solutions and adding outerapproximation cuts for these solutions together with the cut that is added in Line 13.
 Advanced initialization:

Initializing the singletree approach with a bilevelfeasible solution may be beneficial for various reasons. First, in the initial master problem (\(\text {M}^{p}\)) for \(p=0\), the constraints for the binary expansion (11) and for the linearization (13) are redundant, because they are not yet coupled by any outerapproximation cut. When the master problem is equipped with an initial outerapproximation cut, however, then all parts of the model are coupled and the solver can effectively presolve the entire model before solving the rootnode problem, i.e., (\(\text {N}^p(l, u)\)) for \(p=0\), \(l=x^\), and \(u=x^+\). In addition, this initial outerapproximation cut results in a tighter rootnode problem.
Second, an initial bilevelfeasible solution can be used to pass an incumbent solution \(z^*\) to Algorithm 2 and to compute an initial upper bound \(\Phi \). This may allow to prune parts of the search tree right in the beginning. An initial bilevelfeasible point can be obtained, e.g., by finding a feasible or optimal point for Problem (\(\text {M}^{p}\)) for \(p=0\) and solving the corresponding subproblem. This mimics the first iteration of the multitree approach.
Exploiting the bilevel structure
The two outerapproximation algorithms stated in the previous sections are an application of the approaches in [8, 25] and [47], respectively. The effectiveness of both algorithms will depend, among other aspects, on the following properties:

(i)
The ability to solve the master problem(s) effectively.

(ii)
The number of integerfeasible solutions of the master problem that need to be evaluated.

(iii)
The ability to solve the subproblems effectively.
Aspect (i) is addressed by the various enhancements stated in Sects. 3.1 and 3.2, respectively. For the latter two aspects, we can exploit the specific bilevel structure of Problem (1). We explain this on the example of the multitree method in Algorithm 1, but the same explanations hold for the singletree approach in Algorithm 2 as well.
We first discuss aspect (ii). In general, the number of integerfeasible solutions of the master problem coincides with the number of subproblems that need to be solved. In the worst case, the algorithm needs to consider every integerfeasible solution of the initial master problem. However, for bilevel problems of the form (1), there is hope that one needs to evaluate only a few subproblems. The hypothesis is the following. Both the upper and the lowerlevel objective functions are convexquadratic in y and are to be minimized. This means that explicit minmax problems for which the upper level minimizes a function that the follower maximizes, cannot arise for quadratic bilevel problems of the form (1), unless all matrices \(H_u\), \(G_u\), and \(G_l\) are 0. Thus, the solution of the early master problems (that mainly abstract from lowerlevel optimality) might already be a good estimate of the optimal lowerlevel solution, depending on how competitive the two objective functions are. As a consequence, it might be quite likely that the first few solutions of the master problem already contain a closetooptimal or even optimal integer upperlevel decision \(x_I\). Hence, the solution of the respective subproblem already provides a very tight upper bound \(\Phi \). This is, of course, instancespecific and we discuss this in more detail in Sect. 4.6.
We now turn to aspect (iii). Both Algorithm 1 and Algorithm 2 require the solution of the subproblem (\(\text {S}^p\)). It is easy to see that the variables \(w\) can be eliminated in this convex QCQP. The resulting problem is then equivalent to fixing \(x_I=x_I^p\) directly in the original nonconvex MINLP (10). Note that for fixed integer variables, (10) is a convex QCQP as well. While the full subproblem (\(\text {S}^p\)) is more in line with the standard literature on outer approximation for convex MINLPs and makes it easier to proof correctness, it is better to use Problem (10) with fixed integer variables in the actual implementation for the following reasons. First, Problem (10) is smaller than (\(\text {S}^p\)). Second, and more importantly, Problem (10) does not contain any bigM. As already discussed in Sect. 2, a wrong bigM can result in terminating with points that are actually bilevelinfeasible or suboptimal. When using Problem (10) as the subproblem, the former case can never appear. This is an huge advantage compared to solving the two singlelevel reformulations (2) and (15) directly, which may indeed terminate with points that are actually bilevelinfeasible.
Further, we can replace the subproblem (\(\text {S}^p\)) (respectively Problem (10) with fixed integers) by two easier problems. For the parametric lowerlevel problem (3), the optimal value function is given by
It is well known that the bilevel problem (1) can be reformulated as an equivalent singlelevel problem using the optimal value function (22); see, e.g., [13, Chapter 5.6]:
It thus follows directly from Lemma 1 that Problem (15) and Problem (23) are equivalent in the following sense.
Lemma 5
The feasible set of Problem (15) projected on the (x, y)space coincides with the feasible set of Problem (23). In addition, for every global optimal solution \({z}^{*}=({x}^{*}, {y}^{*}, {\lambda }^{*}, {w}^{*}, {s}^{*})\) of Problem (15), \(({x}^{*},{y}^{*})\) is a global optimal solution for Problem (23) and every global optimal solution \(({x}^{*}, {y}^{*})\) of Problem (23) can be extended to a global optimal solution \({z}^{*}\) of Problem (15).
This enables to solve the subproblem in a bilevelspecific way as follows.
Remark 1
We can replace Step 9 of Algorithm 1 (or Step 8 of Algorithm 2) by first solving the parametric lowerlevel problem (3) with fixed integer linking variables \(x_I=x_I^p\) to obtain a (possibly ambiguous) lowerlevel solution \(\tilde{y}^p\) and the corresponding objective function value \(q_l(\tilde{y}^p) = q_l^*(x_I^p)\). Then, we solve Problem (23) with fixed \(x_I=x_I^p\), which is a convex QCQP, to obtain an optimistic bilevel solution \((x_I^p, \bar{x}_R^p, \bar{y}^p)\). In other words, instead of solving subproblem (\(\text {S}^p\)), we can solve a convex QP and a convex QCQP that is considerably smaller than (\(\text {S}^p\)).
In case the lowerlevel problem has a unique solution, Remark 1 can be strengthened.
Remark 2
If the matrix \(G_l\) is positive definite, then the lowerlevel problem has a unique solution and \(M(\bar{x}_I)\) is a singleton. In this case we can replace Step 9 of Algorithm 1 (or Step 8 of Algorithm 2) by subsequently solving the parametric lowerlevel QP (3) with fixed integer upperlevel variables \(x_I^p\) to obtain the unique lowerlevel solution \(\bar{y}^p\), and solving the upperlevel problem
in which all integer upperlevel variables are fixed to \(x_I^p\) and all lowerlevel variables are fixed to the unique solution \(\bar{y}^p\). Problem (24) is a convex QP as well.
With this remark, the large QCQP (\(\text {S}^p\)) can be replaced by two considerably easier QPs. We discuss the effectiveness of Remarks 1 and 2 in Sect. 4.
Computational study
In this section, we provide detailed numerical results for the methods proposed in the previous sections. Besides mean and median running times and counts of solved subproblems, the evaluations and comparisons rely on performance profiles according to [17]. For every test instance i we compute ratios \(r_{i,s}= t_{i,s} / \min \{t_{i,s} : s \in S\}\), where S is the set of the solution approaches and \(t_{i,s}\) is the running time of a solver s for instance i, given in wallclock seconds. Each performance profile in this section shows the percentage of instances (yaxis) for which the performance ratio \(r_{i,s}\) of approach s is within a factor \(\tau \ge 1\) (logscaled xaxis) of the best possible ratio. Before we go into the details, we provide some information on the computational setup in Sect. 4.1. We then specify the test set that we use throughout the study in Sect. 4.2. We also compare the two benchmark approaches of solving the KKTbased reformulation (2) and the strongdualitybased reformulation (15) in this section. In Sects. 4.3 and 4.4, we evaluate the results for different variants of the multi and the singletree approach, respectively. In Sect. 4.5, we compare both methods and also test their performance against the benchmark. Finally, we evaluate the impact of different modifications of the test set on running times in Sect. 4.6.
Computational setup
We implemented all solution approaches using C++11 and used GCC 7.3.0 as the compiler. All optimization problems, i.e., all convex (MI)(QC)QP problems, are solved by Gurobi 9.0.1 using its C interface.^{Footnote 1} The singletree approach of Sect. 3.2 is realized using lazy constraint callbacks of Gurobi that are invoked whenever a new integerfeasible solution with an objective value better than the bilevel incumbent is found. Note however that using Gurobi’s lazy constraint callbacks requires to set the parameter LazyConstraints to 1, which avoids certain reductions and transformations during the presolve that are incompatible with lazy constraints. For all solution approaches we set the NumericFocus parameter to a value of 3, which results in increased numerical accuracy. Further, we tightened Gurobi’s integer feasibility tolerance from its default value \(10^{5}\) to \(10^{9}\) throughout all computations. The rationale is to prevent numerical inaccuracies caused by products of binary variables and bigM values, e.g., in the Constraints (13). All bigM values are fixed to \(10^5\). For each solution attempt, we set a time limit of \({3600}{\hbox {s}}\). The computational experiments have been executed on a compute cluster using compute nodes with Xeon E31240 v6 CPUs with 4 cores, \({3.7}{\hbox { GHz}}\), and \({32}{\hbox { GB}}\) RAM; see [49] for more details.
Selection of the test set and evaluation of the benchmarks
Our test set is based on a subset of the MIQPQP test set used in [38], but is extended by the additional instance classes DENEGRE, INTERASSIG, and INTERKP, which turned out to be too easy for the local optimality considerations in [38]. On the other hand, some of the instance classes used in [38] are too hard to be solved to global optimality, i.e., for most instances of the respective instance class, every tested solver exceeds the time limit. For this reason, we excluded the instance classes GENERALIZED, GK, KP, MIPLIB2010, MIPLIB2017, and OR from our test set. All instances used in this paper are based on MILPMILP instances from the literature; see the “Ref” column in Table 1 for a reference to the original MILPMILP test set. The MIQPQP instances were generated by relaxing all integrality conditions in the lowerlevel problem and by enforcing continuous linking variables to be integer. Further, we added quadratic terms to both objective functions. To this end, we randomly generated quadratic matrices Q, R, and S of suitable sizes and entries in \([\root 4 \of {\sigma }, \root 4 \of {\sigma }]\) with \(\sigma = \max \{\Vert c_u\Vert _{\infty }, \Vert d_u\Vert _{\infty }\}\) (or \(\sigma = \Vert d_l\Vert _{\infty }\)). We then set \(H_u=Q^\top Q\), \(G_u=R^\top R\), and \(G_l=S^\top S + D\), where D is a diagonal matrix with entries in \([1, \root 4 \of {\Vert d_l\Vert _{\infty }}]\). This approach renders \(H_u\) and \(G_u\) positive semidefinite and \(G_l\) positive definite.^{Footnote 2} This allows to evaluate both Remarks 1 and 2 on the full test set. Note that for the sake of completeness, we also provide results for semidefinite matrices \(G_l\) in Sect. 4.6. The full test set \(\mathcal {I}^\mathrm {full}\) contains 757 instances, which is—to the best of our knowledge—the largest test set of MIQPQP bilevel problems considered for approaches that compute global optimal solutions of these models. Before we thin out this test set to compare the various methods on a more balanced set, we first compare the two benchmark approaches introduced in Sect. 2 on the full test set \(\mathcal {I}^\mathrm {full}\).
We therefore briefly recap the two methods. The first approach (KKTMIQP) solves a reformulated and linearized singlelevel problem, in which the lower level is replaced by its KKT conditions. The KKT complementarity conditions are linearized with a bigM formulation. This yields the convex MIQP (2), which can be solved directly using solvers such as Gurobi or CPLEX. This approach is very popular and widely used in applied bilevel optimization. Similarly, using strong duality of the lower level and convexifying the strongduality inequality yields the convex MIQCQP (15). We call this approach SDMIQCQP in the following. Instead of applying outerapproximation algorithms to this problem as proposed in Sect. 3, one can also solve this problem directly. Solvers like Gurobi then either apply a linear outer approximation or solve a continuous QCP relaxation at every branchandbound node. The exact method can be set via the MIQCPMethod parameter that we left at its default value \(1\). This setting automatically chooses the best strategy. In rare cases (5 instances) this resulted in unsolved node relaxations and thus a suboptimal termination. We count these instances as unsolved by SDMIQCQP. We also emphasize again, that KKTMIQP and SDMIQCQP make use of a bigM value. This may result in bilevelinfeasible “solutions”, which is not the case for the proposed outerapproximation algorithms. Thus, we implemented an expost sanity check that computes the relative strongduality error of the lower level for a given solution \((x,y,\lambda )\):
Whenever the error \(\chi (x,y,\lambda )\) exceeds the tolerance of \(10^{4}\), we consider the instance as unsolved for the respective solver. This never occurred for KKTMIQP but happened in 19 cases for SDMIQCQP.
In Fig. 1 (left) we compare the running times of KKTMIQP and SDMIQCQP on those 561 instances in \(\mathcal {I}^\mathrm {full}\) that at least one of the two benchmark approaches solves. It is obvious that SDMIQCQP is the better and more reliable approach. It solves around \({95}\%\) of the instances of the subset and is the faster method for most of these instances. In contrast, KKTMIQP is only capable of solving around \({60}\%\) of the instances. Figure 1 (right) compares the running times of KKTMIQP and SDMIQCQP on the subset of 310 instances that both benchmark approaches solve. Also on this subset, SDMIQCQP is the dominating approach. This is interesting, since the KKT reformulation is the most used approach in applied bilevel optimization. Note that these results are in line with the study in [55], which reveals that a strongdualitybased reformulation outperforms a KKTbased reformulation for mixedinteger linear bilevel problems with integer linking variables and considerably large lowerlevel problems. Due to this clear dominance, we exclude KKTMIQP from our further considerations.
We now thin out the test set \(\mathcal {I}^\mathrm {full}\) to obtain a more balanced set for the outerapproximation methods that we evaluate in the following and for SDMIQCQP. We remove 7 instances that exceed the memory limit for all abovementioned approaches and 1 instance that is proven to be infeasible by all approaches. Further, we exclude 177 instances that are too easy, i.e., that all approaches solve within \({1}{\hbox {s}}\). In addition, we also exclude 149 instances that cannot be solved to global optimality by any of the approaches within the time limit of \({3600}{\hbox {s}}\). The resulting final test set \(\mathcal {I}\) contains \(\mathcal {I}=423\) instances with up to several thousand variables and constraints. Note that we checked that the objective function values and best bounds provided by different approaches are consistent for each instance. This is not guaranteed due to possibly wrong bigM values. All instances in \(\mathcal {I}\) passed this expost optimality check. More details on the instances in \(\mathcal {I}\) can be found in Table 1.
Besides the resulting size of each instance class (“Size”), we also specify the minimum and maximum number of upperlevel and lowerlevel variables (\({n_x}, {n_y}\)) and constraints (\({m_u}, {m_l}\)), as well as the minimum and maximum number of linking variables (\(I\)) and of the maximum upper bound of the linking variables (\(\max _{i \in I} x_i^+\)). The densities of the objective function matrices \(H_u\), \(G_u\), and \(G_l\) of the instances in \(\mathcal {I}\) are displayed in Fig. 2.
Finally, we mention that we later also analyze the performance of the various methods on the instance set \(\mathcal {I}^\text {hard}\) that contains those 149 instances that none of the tested approaches can solve within the time limit.
Evaluation of the multitree approach
We now evaluate the following different parameterizations of the multitree approach as described in Sect. 3.1:
 MT:

A basic variant without any enhancements, i.e., the plain Algorithm 1.
 MTK:

Like MT but additional Kelleytype cutting planes (“K”) are used.
 MTKF:

The master problem terminates as soon as a first (“F”) improving integerfeasible solution is found and after every iteration additional Kelleytype cutting planes are added for every nonimproving integerfeasible solution found by the master problem.
 MTKFW:

Like MTKF but every master problem is warmstarted (“W”) using the best available bilevelfeasible solution.
Figure 3 (left) shows the performance profile of the four variants on the 406 instances in \(\mathcal {I}\) that at least one of the four methods can solve.
It turns out that MTK clearly outperforms MT, which means that adding Kelleytype cutting planes improves the performance. In addition, MTKF dominates MTK in terms of reliability, i.e., it solves more instances. MTKF in turn is dominated by MTKFW, which obviously outperforms all other tested variants. In fact, MTKFW is the fastest method for almost \({50}\%\) of the instances. Further, it is the most reliable approach and solves almost every instance that any of the multitree approaches solves. Overall, according to Fig. 3 (left), MTKFW is the winner among the four tested variants. It is noteworthy, however, that the performance profiles suggest that the difference in MTK, MTKF, and MTKFW mostly lies in the reliability of the approaches, i.e., in the number of solved instances.
In Fig. 3 (right), we compare the “winner setting” MTKFW with a variant with the same settings than the former approach but that uses the “standard outerapproximation” subproblem (\(\text {S}^p\)) instead of the bileveltailored strategy proposed in Remark 2. We label the latter approach by MTSTD as an abbreviation for MTKFWSTD. Note that the underlying instance set covers those 404 instances in \(\mathcal {I}\) that can be solved by at least one of the two methods. The performance profile shows that MTSTD is clearly dominated by MTKFW, which is the faster method for around \({85}\%\) of the instances. This highlights the usefulness of applying Remark 2 to solve the subproblem in a bileveltailored way.
The conclusions drawn from the performance profiles are underlined by the mean and median running times displayed in Table 2. Note that, in order to have a fair comparison, we used for the computation of the numbers in Table 2 the 339 instances in \(\mathcal {I}\) that every multitree solver can solve. It can be seen that MTK, MTKF, and MTKFW have considerably shorter mean and median running times than MT. The differences across the former three approaches are however negligible. This supports the conclusion drawn from the performance profiles in Fig. 3 (left): The algorithmic enhancements mainly yield an increased number of solved instances but, on the instances that all approaches can solve, no significant differences can be observed. In fact, MTKFW neither has the shortest mean running time nor the shortest median running time. Table 2 also shows the mean and median number of solved subproblems (or feasibility problems), which corresponds to the number of evaluated integerfeasible solutions (or to the number of iterations minus 1, since per construction, the last iteration computes a specific integer solution for the second time). The table reveals that adding Kelleytype outerapproximation cuts reduces the number of solved subproblems by almost \({75}\%\) in the mean (MTK vs. MT). On the other hand, terminating the master problem early increases the number of solved subproblems again by almost \({300}\%\) in the mean (MTKF vs. MTK). This is expected and these additional iterations are obviously overcompensated by a reduction in the running time per iteration, i.e., the master problem terminates much faster. Note that the number of solved subproblems is more or less identical for MTKFW and MTSTD. This is expected, since apart from the solution routine for the subproblem, the algorithmic setting is the same for these two approaches. However, the time spent for solving the subproblems drastically increases, if the standard subproblem is used. The mean and median times spent in the subproblems clearly justify the bileveltailored solution of the subproblems as proposed in Remark 2.
Evaluation of the singletree approach
We now analyze the singletree approach described in Sect. 3.2 in the following variants:
 ST:

A basic variant without any enhancements as stated in Algorithm 2.
 STK:

Additional Kelleytype cutting planes (“K”) are added for every nonimproving integerfeasible solution found.
 STKC:

Like STK but an initial bilevelfeasible solution is computed (if available) to add an initial outerapproximation cut (“C”).
 STKCS:

Like STKC but the initial bilevelfeasible solution is used to set start values (“S”) for \(z^*\) and \(\Phi \).
We compare the running times of the four variants using the performance profiles in Fig. 4 (left) on those 409 instances in \(\mathcal {I}\) that can be solved by at least one of the four approaches.
The first observation is that the plots for ST and STK almost match. This means that additional Kelleytype cutting planes for all nonimproving integerfeasible solutions do not make a significant difference, which is in contrast to the results for the multitree approach. One explanation might be that the performance boost that is observed in the multitree method is mainly due to more effective presolving of the individual master problems when additional cuts are added. This is, of course, not possible in the singletree approach. On the contrary, adding an initial outerapproximation cut is very beneficial. The methods without this initial cut, ST and STK, are clearly dominated by STKC. The latter is in turn slightly dominated by STKCS, the variant that also sets starting values according to the initial bilevelfeasible solution. STKCS is the fastest method for around \({50}\%\) of the instances and it solves more instances than any other singletree variant. Thus, STKCS is the “winner setting” for the singletree approach.
In Fig. 4 (right), we compare this winner setting to a variant with the same settings but that uses the “standard outerapproximation” subproblem (\(\text {S}^p\)). We use the label STSTD as an abbreviation for STKCSSTD for this variant. The underlying instance set consists of the 401 instances in \(\mathcal {I}\) that can be solved by STKCS or STSTD. The performance profile shows that STKCS is the faster method for around \({80}\%\) of the instances and it also solves more instances than STSTD. This suggests, that it clearly makes sense to solve the subproblem in a bileveltailored way.
The conclusions drawn from the performance profiles in Fig. 4 are also visible in Table 3, which displays statistics on running times and on the number of solved subproblems. The instances underlying the analysis in Table 3 consist of the 359 instances in \(\mathcal {I}\) that all singletree variants (including STSTD) solve. Note that this renders a comparison of Tables 2 and 3 invalid, because the underlying instance sets are different. Looking at the running times in Table 3, we see that STKCS has the lowest mean running time, which is almost half of the mean running time of ST. Additionally, the median running times are dominated by STKCS, which supports the conclusion drawn from Fig. 4 (left): STKCS is the best parameterization of the singletree approach. Looking at the number of solved subproblems, it is interesting to see that the additional Kelleytype cutting planes decrease the mean number of solved subproblems significantly, although in terms of running times only a slightly positive effect can be observed (STK vs. ST). This may indicate that there is a handful of instances that require to solve many subproblems when no additional Kelleytype cutting planes are added. In contrast, the other two algorithmic enhancements show hardly any effect on the number of solved subproblems (STKC and STKCS vs. STK). Since STSTD is set up in the same way as STKCS, it is clear that the mean and median numbers of solved subproblems are similar. However, the time spent for the subproblems significantly increases when (\(\text {S}^p\)) is used (STSTD vs. STKCS), such that STSTD has significantly longer running times in the mean and median. Again, this justifies the bileveltailored solution of the subproblems as proposed in Remark 2.
Comparison of the multi and singletree approaches with the benchmark
We now compare the best parameterizations of the multi and singletree approach (MTKFW and STKCS) with the benchmark approach SDMIQCQP. Figure 5 (left) shows performance profiles of the running times for those 419 instances that at least one of the three methods solves. Obviously, both outerapproximation approaches dominate the benchmark SDMIQCQP. They are more reliable and solve around \({95}\%\) of the 419 instances compared to around \({85}\%\) solved by SDMIQCQP. The outerapproximation methods are also the faster methods. In particular, STKCS is the fastest approach for more than \({60}\%\) of the instances and is the dominating approach according to the performance profiles.
We analyze this in more detail by looking at mean and median running times displayed in Table 4. This table is based on those 338 instances in \(\mathcal {I}\) that all three approaches can solve. Restricted to these instances, SDMIQCQP is a factor of around 1.7 slower compared to MTKFW in the mean, but it is slightly the faster method in the median. However, as seen in Fig. 5 (left), MTKFW is the more reliable method. In contrast, SDMIQCQP is a factor of more than 2 slower in the mean and almost 2.5 in the median compared to STKCS—without taking the 55 instances into account that SDMIQCQP cannot solve within the time limit but that are solved by STKCS. Compared to MTKFW, the singletree method is almost 3 times faster in the median, although it needs to solve more subproblems in the mean and median and thus significantly spends more time in the subproblems. The reason for that may simply lie in the nature of the methods: The singletree approach needs to search only one branchandbound tree. Overall, the singletree approach STKCS is the winner approach on the test set \(\mathcal {I}\). For a more detailed evaluation of the three methods, we provide performance profiles and mean and median running times per instance class as well as tables with running times and gaps per instance in Appendix 1. The figures and tables therein underline the observations discussed in this subsection.
We also evaluate the performance of the three methods on the hard instances \(\mathcal {I}^\text {hard}\) that none approach can solve within the time limit. Therefore, we show plots of the empirical cumulative distribution functions (ECDF) of the optimality gaps of each method obtained after the time limit in Fig. 5 (right). The xaxis shows the gap in percent while the yaxis shows the percentage of instances. The figure reveals that after \({3600}{\hbox {s}}\), STKCS has the smallest optimality gap and is thus the preferable method also on the instances in \(\mathcal {I}^\text {hard}\). In addition, the two outerapproximation variants are more robust in the sense that they provide gaps within \({100}\%\) for more instances than SDMIQCQP does. However, the differences between the three methods are not very pronounced.
The effectiveness of our methods is also underlined in that extent that the charm of solving the MIQCQP (15) directly, i.e., applying SDMIQCQP, lies, among other things, in the exploitation of the numerical stability of modern solvers. This contains, e.g., an elaborate numerical polishing of the outerapproximation cuts and managing these cuts in cut pools—numerical details that we mostly abstracted from in our implementation. Incorporating such aspects in our implementation would certainly not harm our results, but it can be expected that a more elaborated implementation would lead to an even greater domination of our approaches compared to the benchmark approaches.
Sensitivity on specific test set properties
Finally, we analyze the performance of the outerapproximation algorithms MTKFW and STKCS in comparison to the benchmark SDMIQCQP under three different modifications of the test set \(\mathcal {I}\).
First, we adapt the matrix \(G_l\) to be positive semidefinite instead of positive definite for every instance in \(\mathcal {I}\). We label this adapted test set by \(\mathcal {I}^\text {psd}\). Note that with this modification, all instances in \(\mathcal {I}^\text {psd}\) may have ambiguous lowerlevel solutions such that Remark 2 is not applicable anymore. We thus equip the multi and singletree approaches with the subproblem routine according to Remark 1 and label these approaches by MTR1 and STR1 as abbreviations for MTKFWR1 and STKCSR1, respectively. Note that MTSTD and STSTD, i.e., MTKFW and STKCS equipped with the standard subproblem, as well as the benchmark approach SDMIQCQP, are also applicable for the instance set \(\mathcal {I}^\text {psd}\). We thus compare these five methods on the 417 instances in \(\mathcal {I}^\text {psd}\) that at least one of the five methods solves using the performance profiles shown in Fig. 6.
It can be seen that the singletree methods are still the dominating ones among the tested approaches, both in terms of running times and also in terms of reliability. Further, the multitree approaches still dominate the benchmark SDMIQCQP. Thus, the outerapproximation methods outperform the benchmark, although not as pronounced as for the standard test set \(\mathcal {I}\). The maximum factor \(\tau \) is approximately 14 in Fig. 6 compared to 100 in Fig. 5 (left). This can be expected, since the subproblem routines are more expensive compared to approaches that make use of Remark 2. Figure 6 also suggests that using Remark 1 is not beneficial over simply using the standard outerapproximation subproblem. Thus, for instances with ambiguous lowerlevel solutions, STSTD is the method of choice. This is underlined by mean and median running times as well as the numbers of solved subproblems that are shown in Table 5.
The table reveals also another interesting aspect. While the mean and median number of solved subproblems is very comparable for STR1 and STSTD (as well as for MTR1 vs. MTSTD), the mean and median times spent in the subproblems differ significantly. While the first aspect can be expected due to the same algorithmic setting, the latter aspect is interesting. STSTD spends more than twice the time of STR1 in the subproblems in the mean, but it only spends half of the time of STR1 in the subproblem in the median. Thus, there seem to be few instances for which the standard subproblem (\(\text {S}^p\)) is very challenging, but for the most instances it is much easier to solve than subsequently solving the two problems (3) and (23) as proposed by Remark 1.
Second, we choose the entries of the matrices Q, R, S, and D to be in \([\root 2 \of {\sigma }, \root 2 \of {\sigma }]\) respectively \([1, \sqrt{\sigma }]\) with \(\sigma = \max \{\Vert c_u\Vert _{\infty }, \Vert d_u\Vert _{\infty }\}\) (or \(\sigma = \Vert d_l\Vert _{\infty }\)) instead of \([\root 4 \of {\sigma }, \root 4 \of {\sigma }]\). In this setting, the coefficients of the resulting matrices \(H_u=Q^\top Q\), \(G_u=R^\top R\), and \(G_l=S^\top S + D\) have larger absolute values and an analysis of the resulting matrices revealed that also the size of the spectrum, i.e., the range between the smallest and largest eigenvalue, increases compared to the matrices described in Sect. 4.2. We label this modified test set as \(\mathcal {I}^\mathrm{{lc}}\) and compare the methods MTKFW, STKCS, and SDMIQCQO. Performance profiles of those 400 instances in \(\mathcal {I}^\mathrm{{lc}}\) that can be solved by at least one of the three methods is shown in Fig. 7.
In comparison to the standard test set \(\mathcal {I}\), see Fig. 5, the dominance of the outerapproximation approaches is even more pronounced. The reason for this behavior is that the outerapproximation methods need to solve significantly less subproblems on the test set \(\mathcal {I}^\mathrm{{lc}}\); see Table 6.
In fact, for at least half of the instances in \(\mathcal {I}^\mathrm{{lc}}\) the outerapproximation methods need to solve only 1 subproblem; see the median numbers of solved subproblems in Table 6. In addition, there is not much difference between the multi and the singletree method; see Fig. 7 and the mean and median running times in Table 6. One possible explanation follows the discussion in Sect. 3.3. For many instances, the linear parts in the upper and lowerlevel objective functions model a minmax structure. This structure cannot be “purely” present in our quadratic setting. Thus, choosing larger coefficients in the quadratic parts reduces the minmax structure and the two objective functions are more aligned, such that the first or second integer solution \(x_I\) of the master problem already yields the bilevel optimal solution. Overall, the properties of the involved matrices of the quadratic terms seem to have a significant impact on the effectiveness of the outer approximation algorithms.
Third, we highlight that for almost all instance classes in \(\mathcal {I}\), the linking variables are binary. The reason is that most of these instances originate from MILPMILP interdiction instances, in which the integer variables are binary by nature. In addition, for many instances in XUWANG and XULARGE that have lower and upper bounds of 0 and 10, the implied bounds are in fact 0 and 1, because the righthand side values of these randomly generated instances limit the feasible region of the linking variables. In [54] it is pointed out that enlarging the feasible set of the original MILPMILP instances by changing the righthand side vectors renders the instances very hard to solve. We observed the same pattern for the MIQPQP variants in a preliminary numerical test. In our notation, all entries in A, B, a, C, D, and b are nonpositive. Decreasing the righthand side vectors renders most of the instances too hard to be solved by any of the proposed outerapproximation or benchmark methods. On the one hand, this underlines the general hardness of MIQPQP bilevel problems with general integer (i.e., not binary) linking variables. On the other hand, the instances XUWANG and XULARGE are randomly generated and it is not clear whether these observations generalize to “realworld” bilevel instances.
Conclusion
In this paper, we considered bilevel problems with a convexquadratic mixedinteger upper level and a convexquadratic lower level. Further, all linking variables are assumed to be bounded integers. For such problems, we proposed an equivalent transformation to a singlelevel convex MINLP and developed a multi and a singletree outerapproximation algorithm that we derived from algorithms for general convex MINLPs. We further proposed enhancements of these algorithms that exploit the bilevelspecific structure of the problem. Finally, we proved the correctness of the methods and carried out an extensive numerical study.
The study revealed that the two proposed outerapproximation algorithms outperform known benchmark approaches. For bilevel problems with unique lowerlevel solutions, the proposed bileveltailored solution of the subproblem turned out to be very effective. Even for instances with ambiguous lowerlevel problems, the novel algorithms perform better than the benchmark approaches. In general, the singletree outer approximation implementation performs better than the multitree counterpart and is, in our opinion, the preferred method.
For both methods several questions remain open. Following the discussion in Sect. 4.6, the impact of the objective function matrices and their spectra as well as the performance on instances with nonbinary linking variables needs further assessment. Up to know, this is however not possible due to a lack of bilevel instances with such properties. An interesting direction for future research can be to drop the integrality condition on the linking variables. This would require, e.g., spatial branching on linking variables but may also introduce some pitfalls like unattainable bilevel solutions; see [43]. Another question is whether one can introduce integer variables to the lower level, i.e., considering MIQPMIQP bilevel problems. Certainly, the strongdualitybased reformulation for convex lowerlevel problems would not be applicable anymore, but one could use a more general singlelevel reformulation like the valuefunction reformulation, as it is done, e.g., for MILPMILP bilevel problems in [21].
Notes
 1.
Since Gurobi’s C interface offers more flexibility compared to the C++ interface, we wrote a C++ wrapper around Gurobi’s C interface.
 2.
The MATLAB function that we implemented to generate the matrices and a brief documentation thereof can be found in the GitHub repository under https://github.com/mschmidtmathopt/qpbilevelmatrixgenerator.
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Acknowledgements
This research has been performed as part of the Energie Campus Nürnberg and is supported by funding of the Bavarian State Government. We also thank the Deutsche Forschungsgemeinschaft for their support within project A05 and B08 in the “Sonderforschungsbereich/Transregio 154 Mathematical Modelling, Simulation and Optimization using the Example of Gas Networks”.
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Appendix A: Detailed results
Appendix A: Detailed results
In this section, we provide more detailed results for the benchmark approach SDMIQCQP as well as for MTKFW and STKCS. For each instance class specified in Table 1, we provide

(i)
performance profiles for all instances of the class that can be solved by at least one of the three methods;

(ii)
a table with mean and median running times and number of solved subproblems for all instances of the class that could be solved by all three approaches;

(iii)
a table with the exit status, gap, and running time of each solver for each instance of the class.
In the latter tables, a status “optimal” means that the instance has been solved to global optimality, “time limit” specifies that the time limit has been reached, “suboptimal” indicates a suboptimal termination due to unsolved node relaxations, and “numerics” denotes numerical issues detected in an expost feasibility check. Note that the last status is only relevant for SDMIQCQP. For each instance, we mark the best running time using bold font.
Results for CLIQUE
Figure 8, Tables 7 and 8 show the results for the instance class CLIQUE.
Results for DENEGRE
Figure 9, Tables 9 and 10 show the results for the instance class DENEGRE.
Results for IMKP
Figure 10, Tables 11 and 12 show the results for the instance class IMKP.
Results for INT0SUM
Figure 11, Tables 13 and 14 show the results for the instance class INT0SUM.
Results for INTERASSIG
Figure 12, Tables 15 and 16 show the results for the instance class INTERASSIG.
Results for INTERCLIQUE
Figure 13, Tables 17 and 18 show the results for the instance class INTERCLIQUE.
Results for INTERFIRE
Figure 14, Tables 19 and 20 show the results for the instance class INTERFIRE.
Results for INTERKP
Figure 15, Tables 21 and 22 show the results for the instance class INTERKP.
Results for MIPLIB
Figure 16, Tables 23 and 24 show the results for the instance class MIPLIB.
Results for XULARGE
Figure 17, Tables 25 and 26 show the results for the instance class XULARGE.
Results for XUWANG
Figure 18, Tables 27 and 28 show the results for the instance class XUWANG.
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Kleinert, T., Grimm, V. & Schmidt, M. Outer approximation for global optimization of mixedinteger quadratic bilevel problems. Math. Program. (2021). https://doi.org/10.1007/s10107020016012
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Keywords
 Bilevel optimization
 Outer approximation
 Quadratic programming
 Convex mixedinteger nonlinear optimization
Mathematics Subject Classification
 9008
 90C11
 90C30
 90C26
 90C46