Inductive Linearization for Binary Quadratic Programs with Linear Constraints: A Computational Study

The computational performance of inductive linearizations for binary quadratic programs in combination with a mixed-integer programming solver is investigated for several combinatorial optimization problems and established benchmark instances. Apparently, a few of these are solved to optimality for the ﬁrst time.


Introduction
Given a binary quadratic programs (BQP) comprising linear (and possibly quadratic) constraints, the inductive linearization technique [18] may serve as a computationally attractive compromise between the wellknown "standard" linearization, and a complete application of the Reformulation Linearization Technique (RLT) [2]. In various relevant cases, inductive linearizations are more constraint-side compact than the "standard" linearization and provide a continuous relaxation that is at least as tight. Structural proofs of the latter property have been derived for some of these cases, e.g. when all the factors of products x i x j are involved in at least one linear less-or-equal inequality or equation whose non-zero left hand coefficients and right hand side are equal to one [18]. Prominent combinatorial optimization problems where this applies are e.g. the Quadratic Assignment and the Quadratic Matching Problem. Further conditions for provably strong relaxations exist, covering for instance the Quadratic Traveling Salesman Problem.
In this light, the central contribution of this paper is a systematic computational study in order to address a number of research questions: Does the mentioned compactness without any tightness trade-off concerning the continuous relaxation also translate into a faster solution time of the corresponding mixed-integer programs? For which (further) problem structures is the inductive linearization technique (not) well-suited and why? Can an inductive linearization be obtained quickly in practice, and if so, how? To this end, we investigate the performance of the inductive as well as the "standard" linearization in combination with (and in comparison to) a professional mixed-integer programming solver on various BQPs with linear constraints. More precisely, we look at the Quadratic Assignment Problem, the Quadratic Knapsack Problem, the Quadratic Matching Problem, the Quadratic Shortest Path Problem, and on further instances of the well-established QPLIB 1 and MINLPLib 2 . In all conscience, a few instances from the latter libraries are solved to optimality for the first time using the proposed inductive linearization. Besides that, an algorithmic frame is presented to derive inductive linearizations in practice.
The outline of this paper is as follows: In the beginning of Sect. 2, we briefly review the inductive linearization technique, in order to strongly emphasize on its practical application in the subsequent subsections. In Sect. 3, we present the aforementioned applications along with the respective problem formulations as well as the benchmark instances used for the computational study, and a discussion of the respective results. Finally, a conclusion is formulated in Sect. 4.

Inductive Linearization
Let n, m L ∈ N \ {0} and let m Q ∈ N. Suppose further we are given matrices Q k ∈ R n×n for k ∈ {0, . . . , m Q } and A ∈ R m L ×n , vectors h k ∈ R n for k ∈ {1, . . . , m Q } and c ∈ R n , and finally scalars β k ∈ R for k ∈ {1, . . . , m Q }. Then the inductive linearization technique addresses the following general form of optimization problems: x ∈ {0, 1} n Throughout this paper, we will denote by N := {1, . . . , n} the index set of the binary variables x ∈ {0, 1} n . The set of products P truly present in the problem is determined by the matrices Q k ∈ R n×n , k ∈ {0, . . . , m Q }, in the objective function and in the m Q ∈ N 0 quadratic restrictions as follows: Naturally, we will assume that P is non-empty and, since we have for binary solutions that x i x j = x j x i for i, j ∈ N, i = j, and x i = x 2 i for all i ∈ N, we may further assume the matrices Q k , k ∈ {0, . . . , m Q }, to be strictly upper triangular, and thus i < j for (i, j) ∈ P.
While quadratic constraints may or may not exist, some linear constraints on the binary variables are actually necessary to apply the inductive linearization technique. We assume w.l.o.g. that these are given as equations and less-or-equal inequalities, hence denoted Ax b. More precisely, the requirement is that each binary variable x i , i ∈ N, being a factor of a product in P (to be linearized with the technique proposed), appears in at least one of these (with a non-zero coefficient). Clearly, this can be assumed (or established) for a binary problem without loss of generality. Indeed, if there is a factor x i , i ∈ N, that is entirely free (unconstrained) in the original problem then in principle any equation or less-or-equal inequality may be employed that is valid for its feasible set (even, though rather not so desirable, x i ≤ 1). Moreover, if this requirement is not fulfilled for some factor, this does not affect a successful inductive linearization of all the products whose factors do. For simplicity, we thus assume from now on that all products in P are linearly constrained.
The inductive linearization technique is a generalization of a principle proposed by Liberti in [14] for the special case of equations with right hand side and left hand side coefficients equal to one, and of its later revision [16]. In his original article, Liberti coined the name "compact linearization" because it typically adds fewer constraints to the mentioned problems than the "standard" linearization whose general form reads ≥ 0 x ∈ {0, 1} n for our original problem. With m := |P|, we here use y ∈ R m to denote the linearized products, d ∈ R m to denote the quadratic objective coefficients, and g k ∈ R m to express the quadratic constraint coefficients for all k = 1, . . . , m Q . Thereby, we use the subscript notation y i j for i < j and analogously define d i j = (Q O ) i j + (Q 0 ) ji and g ki j = (Q k ) i j + (Q k ) ji . Of course, if d i j is less than zero, (4) can be omitted, and if d i j is larger than zero, (2) and (3) can be omitted. As we will see, in many cases, the generalized approach achieves constraint-side compactness as well. However, this cannot be guaranteed for any kind of BQP with linear constraints. Moreover, depending on how the method is applied, more than |P| linearization variables may be induced (although this can, in principle, always be circumvented as described in Sect. 2.6). Therefore, and to have a clear distinction from other linearizations being called "compact", as well as to emphasize that the proposed method aims at "inducing" the products associated to the set P by multiplying original constraints with original variables, the technique is referred to as "inductive linearization" since its generalization first presented in [18].

Mathematical Derivation of Inductive Linearizations
Given a problem as introduced at the beginning of this section, suppose that we identify a working (sub-)set of the linear constraints Ax b to actually induce the linearization with. Let us denote the index set of the selected equations and inequalities with K E and K I , respectively. That is, we consider the constraints where I k := {i ∈ N | a i k = 0} denotes the respective support index set for each k ∈ K E or k ∈ K I . As already mentioned, we require w.l.o.g. a choice of K := K E ∪ K I such that there exist indices k, ∈ K with i ∈ I k and j ∈ I for all (i, j) ∈ P. To refer to the respective constraints, we will use the notation K(i) := {k ∈ K : i ∈ I k }, as well as K E (i) and K I (i) analogously defined if more preciseness is in order. Moreover, although we do not require this for the original problem, let us temporarily assume in addition that b k > 0 for all k ∈ K and a i k > 0 for all i ∈ I k , k ∈ K. We will elaborate in Sect. 2.4 on how to handle constraints not fulfilling these prerequisites.
The first step of the inductive linearization approach now associates to each equation k ∈ K E another index set M E k ⊆ N that is supposed to specify original variables used as multipliers. To each inequality k ∈ K I , two such index sets M + k , M − k ⊆ N are associated. The corresponding interpretation is as follows: This leads to the following subset of the first level RLT constraints: Then Q := {(i, j) | i ≤ j and ∃k ∈ K : i ∈ I k and j ∈ M k , or j ∈ I k and i ∈ M k } is the index set of the products induced by (7)- (9).
For ease of reference, we also define M := k∈K M k which is to be regarded as a multiset of multiplier indices.
Remark 1. The induced set Q may contain tuples that correspond to squares. Eliminating them is a simple and worthwhile optimization (see also Theorem 4 in Sect. 2.2). Recognition is already possible when squares are (or rather would be) induced: If x j is used as a multiplier for a constraint with j ∈ I k , the result may be instantly strengthened to 3 : One has to ensure, however, that the coefficient of x j is not turned zero by this operation. In this case the generated constraint would still be valid, but it would not link the products on its left hand side to their factor x j as required (from the conditions described below).
If we now rewrite (7)-(9) by substituting for each (i, j) ∈ Q the product x i x j by a continuous linearization variable y i j that has explicit lower and upper bounds, i.e., 0 ≤ y i j ≤ 1, we obtain the linearization constraints: Now, as is expressed by the following theorem, for all the induced (i, j) ∈ Q and binary x i , x j , one has y i j = x i x j if the following three consistency conditions are met: There is a k ∈ K(i) such that j ∈ M E k or j ∈ M + k , respectively. Condition 2. There is a k ∈ K( j) such that i ∈ M E k or i ∈ M + k , respectively.

Condition 3.
There is a k ∈ K(i) such that j ∈ M E k or j ∈ M − k , respectively, or a k ∈ K( j) such that i ∈ M E k or i ∈ M − k , respectively. Theorem 1. ( [18]) For any integer solution x ∈ {0, 1} n , the linearization constraints (10)- (12) imply y i j = x i x j for all (i, j) ∈ Q if and only if Conditions 1-3 are satisfied.
So altogether, if we choose M consistently in terms of the conditions and such that Q contains P, we obtain a linearization for our original problem. In fact, at the potential expense of losing some continuous relaxation strength, it is always possible to have Q = P as described in Sect. 2.6.

Linear Relaxation Strength of Inductive Linearizations
The linear programming relaxation obtained from an inductive linearization is provably at least as tight as the one obtained from the "standard" linearization if the constraints employed to satisfy the consistency conditions have only zero-one coefficients on the left, and a right hand side of one.
Theorem 2. ( [18]) Consider a (sub-)set K E of equations (5) with b k = 1 for all k ∈ K E , and a i k = 1 for each i ∈ I k , k ∈ K E . Let Q ⊆ Q be the set of tuples induced by the multipliers M E k , k ∈ K E , and suppose that these multipliers satisfy the Conditions 1 and 2 for all (i, j) ∈ Q . Then, for any 0 ≤ x ≤ 1, we have y i j ≤ x i , y i j ≤ x j and y i j ≥ Theorem 3. ( [18]) Consider a (sub-)set K I of inequalities (6) with b k = 1 for all k ∈ K I , and a i k = 1 for each i ∈ I k , k ∈ K I . Let Q ⊆ Q be the set of tuples induced by the multipliers M + k and M − k , k ∈ K I , and suppose that these multipliers satisfy the Conditions 1-3 for all (i, j) ∈ Q . Then, for any 0 ≤ x ≤ 1, we have y i j ≤ x i , y i j ≤ x j and y i j ≥ x i + x j − 1 for all (i, j) ∈ Q .
A provably strong inductive linearization is also achieved for equations with a right hand side of two and left hand side coefficients one if these equations are multiplied by all variables on their left hand sides (i.e., M k = I k ) and squares are ruled out as described in Remark 1.
Theorem 4. ( [18]) Consider a (sub-)set K E of equations (5) with b k = 2 for all k ∈ K E , a i k = 1 for each i ∈ I k , k ∈ K E , and suppose that M E k = I k for all k ∈ K E . Let Q ⊆ Q be the set of tuples induced by these multipliers after eliminating squares. Then, for any 0 ≤ x ≤ 1, we have y i j ≤ x i , y i j ≤ x j and y i j ≥ x i + x j − 1 for all (i, j) ∈ Q .
Remark 2. The referenced proofs of these theorems make apparent that the tightness of the relaxations of inductively linearized BQPs relates (besides other criteria) to the ratio between the right hand side and the left hand side coefficients. The referenced article also provides an example for a case where an inductive linearization provably has a strictly stronger continuous relaxation than the "standard" linearization.

Practical Derivation of Inductive Linearizations
Given a problem formulation on sheet, a multiset M that induces a set Q ⊇ P and establishes consistency can often be derived by inspection once the necessary implications of Conditions 1-3 are understood. This is likely in particular for combinatorial optimization problems. Nevertheless, especially if P is sparse, or if the formulation is complicated, it may be non-trivial to find a combination of constraints and multipliers that best suits compactness or other objectives. Moreover, an automated derivation is desirable especially for larger problem instances and allows for a linearization framework to be coupled with a mixed-integer programming solver.
Concerning the computation of an inductive linearization, it has been shown in [18] on the negative side, that the associated optimization problem (allowing e.g. to derive a linearization that is as compact as possible in terms of additional variables and constraints) is NP-hard in its general form. On the positive side, the problem can be solved well in practice, frequently even exactly, and there exist polynomial-time algorithms for specifically structured BQPs. Moreover, in the exact as well as in the heuristic case, computation times can be reduced by carefully preselecting the set K of original constraints considered for inductions. For many applications, the number of candidate constraints to induce a certain product, respectively to satisfy one of the three conditions, is anyway rather small.
We first review the mixed-integer program from [18] to model and solve the exact case: Algorithm 1 is an extension of this algorithm to the general case and to incorporate "weights" in a similar way as the mixed-integer program does. It may serve as a heuristic to quickly derive inductive linearizations in practice that worked quite well in our experiments. In this context, it is also worth to mention that the presolve routines of a MIP solver may well eliminate some of the variables and constraints imposed by an inductive linearization that is not "most compact". Moreover, a few additional constraints may sometimes improve the relaxation strength, so compactness need not necessarily be an ultimate goal.

Normalization, Inductive Linearizations with General Linear Constraint Sets
We shall now discuss how to deal with the case that some (or all) of the original constraints k ∈ K to be employed, i.e., (5) and (6), do not satisfy b k > 0 and a i k > 0 for all i ∈ I k . To this end, suppose that is an equation (the following also holds for a ≤-inequality) in K, and let I − k ⊆ I k be the set of variable indices such that a i k < 0 for each i ∈ I − k . For ease of notation, define also I + k = I k \ I − k . The explicit approach to deal with such constraints is to define a new complement variablex i for each i ∈ I − k , k ∈ K, along with the corresponding equation: Apparently, the equations (23) have only positive coefficients on the left hand side and a positive right hand side. Moreover, we may now replace any of the original equations with where the term − ∑ i∈I − k a i k on the left and on the right hand side is non-negative as well. We will also refer to the latter as the normalized right hand side.
Carrying out this procedure for every equation or inequality with negative coefficients on the left hand side clearly gives a system with only non-negative coefficients on the left. Now if any of the normalized right hand sides is negative, the system is obviously infeasible. Furthermore, if any of them is zero then all the variables on the respective left hand side can be fixed (original ones to zero, complemented ones to one) and thus be removed from the formulation.
As a consequence, it is possible to satisfy the prerequisites b k > 0 for all k ∈ K and a i k ≥ 0 for all i ∈ I k , k ∈ K, without loss of generality. A clear drawback of the explicit approach is however that it may add up to n variables and equations while the new variables become potential multipliers and the new equations are rather undesirable candidates for multiplications. Further, complementing variables "beforehand" may also incur avoidable (or only tediously removable) overhead if it is not a priori clear that the resulting normalized constraints will be at all employed for multiplications.
A more economical strategy is to keep the original constraints as they are, and to consider them for multiplication with each x j (and (1 − x j ) for case (9)) and eachx j (and (1 −x j ) for case (9)) without ever really introducing the complement variables. Instead, the idea of this implicit approach is to choose, for each i ∈ I k , the "right" of the four possible combinations x i x j ,x i x j , x ix j , andx ix j to be induced, such that the respective linearization constraint imposes the necessary implications on their value.
Algorithm 1 A simple heuristic to construct an inductive linearization.
More precisely, as can be verified from the proof of Theorem 4 in [18], the inequalities (8) (respectively their linearized counterparts (11)) have the effect of enforcing all the products (respectively, linearization variables) on the left hand side to be zero if the multiplier x j is zero. Similarly, the constraints (9) (respectively, (12)) enforce any x i x j (y i j ) on the left hand side to coincide with x i if x j is one. The equations (7) respectively (10) even directly impose both relationships at once. These implications are exactly what is established by Conditions 1-3 if the coefficients and right hand sides of the original constraints are non-negative respectively positive.
To achieve the same in the general case, one has to replace (8) by Here, one may again verify that, if x j respectivelyx j is equal to one, then the inequalities enforce the linearization variables to be substituted for the products to equal As a result, we achieved that complement variables and the associated equations need not be introduced. The handling of negative coefficients becomes even almost oblivious as it certainly does not matter whether to introduce a linearization variable for x i x j or for any ofx i x j , x ix j , andx ix j instead. Moreover, if (i, j) ∈ P but x i x j is not induced, quadratic constraints and terms in the objective function referring to this product can still be expressed using the following equations for (implicit) substitutions 4 : Nonetheless, even with the implicit normalization approach, negative coefficients are still not entirely free of charge, as it may be (like also in the explicit approach) that not just one but several of the four possible products need to be induced for i, j ∈ N, i < j. Moreover, negative coefficients increase the complexity of the mixed-integer program from Sect. 2.3 (besides that, for the implicit approach equations (13) need to be relaxed to the enforcement of at least one of the four combinations) as well as the complexity of implementing Algorithm 1.

Optional Preparations: Equation Splitting and Replication
Depending on the problem structure, one might consider to split equations into two inequalities before starting to derive an inductive linearization (see also the next subsection). Moreover, if an equation has positive and negative coefficients on the left hand side, one may consider both the original and the equation after multiplication with −1 as candidates to create linearization constraints (and induce linearization variables).

Postprocessing: Variable Elimination by Possible Constraint Weakening
It is possible to eliminate (all) variables in Q \ P as a postprocessing step if (all) of these have been generated from inequalities. As is clear from inequalities (11) and (12) as well as from the discussion in Sect. 2.4, removing summands on their left hand sides will neither harm their validity nor their necessary implications on the respective remaining linearization variables. In general, however, the feasible region of the continuous relaxation may of course be enlarged by this procedure.

A Preliminary Computational Study
Some evidence for a computational utility of inductive linearizations for specific applications is already available in the literature (e.g. [7,14,15,16,17]). Here, our aim is a systematic study and a structured overview of the computational performance on various well-known and well-suited as well as not so well-suited binary quadratic programs with linear constraints. To this end, we identified a number of prominent combinatorial optimization problems and benchmark instances commonly used by the community in order to evaluate inductive linearizations in comparison with "standard" linearizations. Concerning the former, we distinguish a usual inductive linearization (IL) and, if inequalities are present, a weakened inductive linearization (ILW) which employs the postprocessing from Sect. 2.6. We further distinguish the complete "standard" linearization (SLC) causing |P| additional variables, 3|P| additional inequalities, and 7|P| additional non-zeros, and the reduced "standard" linearization (SLR) which only contains those of the inequalities (2)-(4) that are required due to the objective coefficients.
While specific details are given in the respective subsections, the following paragraphs subsume those parts of the setting that apply to all the experiments.
As a preprocessing, all linear greater-or-equal inequalities were turned into less-or-equal ones, and if a left hand side has only integer coefficients, fractional right hand sides were rounded down.
To derive the inductive linearizations, we employed Algorithm 1 with implicit normalization as described in Sect. 2.4. In each iteration of the function Append, the weights of the constraint-multiplier combinations w E k, j and w + k, j (w − k, j ) are recomputed as the negated number of products in Q add for which Conditions 1, 2 or 3 would be satisfied if constraint k was multiplied with x j (1 − x j ). Absolute coefficients or right hand sides as well as the number of non-zero coefficients are however not taken into account. Due to the order of looking up constraints being candidates for multiplications, a slight implicit preference of equations over inequalities is inherent to the implementation. The table columns titled "Get [s]" in the following subsections will display the running time of this algorithm (i.e., the wall clock time to derive the respective inductive linearization) in seconds. As the results eventually show, they are adequate with only very few exceptions. Nevertheless, it is just a simple prototype implementation that could be tailored and optimized in several ways.
In order to finally solve the resulting MIPs, we employed Gurobi 5 in version 9.03 with its seed parameter set to one. All computations were carried out using a single thread on a Debian Linux system equipped with an Intel Xeon E5-2690 CPU (3 GHz) and 128 GB RAM. Each run had a time limit of 48 hours. If it was exceeded, this is indicated by "-" in the respective table column. Apart from that, we would like to emphasize that the displayed (wall clock) running times should only be considered as an indicator for which kind of problems the respective methods appear particularly suited or rather not suited, especially as the absolute running times for each single method may vary significantly depending on various influences (as e.g. different seeds, parameters, branching choices, solver versions) whose possible combinations would justify a computational study on their own. Further, the results in this preprint are also preliminary in the sense that access to the mentioned system was not always exclusive to a single experiment.

The Quadratic Assignment Problem
Given T, D ∈ R n×n and c ∈ R n , a quadratic assignment problem (QAP) in the form by Koopmans and Beckmann [13] can be written as follows.
Frieze and Yadegar [8] derived a linearization of the QAP that is actually an inductive one, but that is not yet most compact. To characterize a most compact inductive linearization for the case where all (meaningfully) possible products are of interest, observe first that each of the variables X := {x ip | i, p ∈ {1, . . . , n}} occurs exactly once in the equation set (24) and exactly once in the equation set (25). Thus, in order to induce all products and to satisfy Conditions 1 and 2 for them, it would suffice to either multiply all of the constraints (24) with X, or to multiply all of the constraints (25) with X. Moreover, since objective coefficients for x 2 ip may be moved to the linear part, and since the variables y ipiq for all p, q ∈ {1, . . . , n} as well as all variables y ip j p for all i, j ∈ {1, . . . , n} can be eliminated, it suffices to formulate the resulting constraints only for i = j, or p = q, respectively. If one further identifies y jqip with y ip jq whenever i < j, a most compact inductive linearization of all meaningfully possible products is: In this formulation, (26) could also be replaced by the constraints which resembles once more the original freedom to choose one of (24) and (25) as the basis for inductions. The total number of additional equations thus amounts to only n 3 − n 2 instead of 3 · 1 2 (n 2 − n)(n 2 − n) = 3 2 (n 4 − 2n 3 + n 2 ) inequalities when using the complete "standard" linearization and creating y ip jq only for i < j and p = q as well. However, these most compact formulations have a weaker linear programming relaxation than the ones by Frieze and Yadegar, and Adams and Johnson, that comprise more constraints. If P does not contain all meaningfully possible products, it is clear that once more only a subset of the possible linearization constraints is required.
For our experiments, we employed the established QAPLIB ( [6]) instances. In Table 1, we list those of them that could be solved within 48 hours using either IL or one of the standard linearizations SLC and SLR. For each instance, the corresponding value of n is part of the instance name. The SLR has of course |P| additional variables, and -as all the listed instances have positive objective coefficients only -|P| additional inequalities, and 3|P| additional non-zeros (column "NZ+"). For IL, we display the number of linearization variables, equations, and the associated number of additional non-zeros in the constraint matrix. The task to find a good combination satisfying Conditions 1 and 2 for all induced products could be solved very quickly with Algorithm 1. In some further experiments, the mixed-integer program from Sect. 2.3 could also be solved by Gurobi within the root of its branch-and-bound tree.
The results obtained using IL are clearly better than with the two "standard" linearizations while of course still not competitive to state-of-the-art methods for the QAP. Nevertheless, it is apparent that most QAPLIB instances with up to about 20000 products can be handled within the time limit. The tendential superiority compared to the "standard linearization" is also in line with the results that have been obtained earlier for other optimization problems with "(semi-)assignment" (or rather "one-out-of-many selection") constraints. Some examples are graph partitioning ( [16]), multiprocessor scheduling ( [15]), graph layering ( [17]), and quadratic semi-assignment problems ( [4]). Nevertheless, additional constraint sets and the structure of the objective coefficients may always impact the outcome significantly as will also become apparent in Sections 3.5 and 3.6.

The Quadratic 0-1 Knapsack Problem
The possibly simplest inequality-only application for the inductive linearization technique is the quadratic 0-1 knapsack problem (QKP). It is particularly interesting for the computational study because here the left hand side coefficients (item sizes) and the right hand sides (knapsack capacity) vary considerably and can also be large.
Starting from the canonical formulation with a capacity b ∈ R, and a variable x j for each item j of a ground set J with size a j ∈ R, it has been observed in the literature that inequalities of type (8) could be used in combination with the "standard" linearization (see e.g. [3]), and also inequalities of type (9) have been applied in the context of semidefinite relaxations to improve the obtained dual bounds ( [11]). A corresponding square-reduced (cf. Remark 1) inductive linearization is the following mixed-integer program: Once more, the formulation is more compact than a "standard" linearization, especially if all possible products are of interest: Assuming |J| = n, the n 2 products are then linearized using only 2n − 1 (n − 1 inequalities of type (9) suffice to satisfy Condition 3 for all of them while n of (8) are needed for Conditions 1 and 2) instead of 3 n 2 inequalities. However, in the general case of arbitrary a j , j ∈ J, and b, an implication of the "standard" linearization inequalities (2)-(4) cannot be expected.
Moreover and irrespective of the cardinality of P, since there is only one original inequality, the satisfaction of Conditions 1-3 must be established using (27) for each product induced, and it is hence entirely predetermined which products will be induced. It is thus not surprising that Algorithm 1 as well as the MIP solver could derive the unique solution quickly in our experiments.
For these, we employed the randomly generated instances by [5]. The results are displayed in Tables 2  and 3. The respective number of items is given by the first number in the instance name. The presentation is restricted to ILW that clearly outperformed IL. The SLR has here |P| additional variables, 2|P| additional inequalities, and 4|P| additional non-zeros as all listed instances have positive objective coefficients only, and the objective is to be maximized. In case of jeu 200 100 5 we needed to change Gurobi's MIPGap parameter to 10 −6 (while the default used instead is 10 −4 ) in order to retrieve an optimum solution with the SLC.
As could be expected from the large constraint coefficients and right hand sides, the performance of the MIP solver using the inductive linearizations and the bounds of their root relaxations are here significantly weaker than with the "standard" linearization. Moreover, while the numbers of linearization constraints are much smaller for the former, the converse is true for the induced numbers of non-zeros. We observe that there is a considerable variation in the solution times even within each size and product density group. For the larger instances, only those with a comparably small capacity remain solvable using ILW. As the left hand side coefficient ranges do (almost) not vary, this once more underlines the inductive linearization's sensitivity to the "ratio" between the right hand side and the left hand side coefficients.

The Quadratic Matching Problem
Given an undirected graph G = (V, E), a canonical BQP that models the quadratic matching problem (QMP) on G (see e.g. [12]) can be expressed as It is easily observed that each edge (factor) occurs in exactly two constraints, namely those inequalities (28) associated with its endpoints. In this sense, the situation is similar as in case of the QAP. However, the constraints need not be as regular as they depend on the structure of G, and the mixed-integer programs to compute the inductive linearizations turned out to be more difficult for the MIP solver employed. On the contrary, the heuristic delivered good linearizations quickly.
For our experiments, we employed the instances used by [12]. The presentation in the Tables 4, 5, and 6 is restricted to those instances called "BM" in this reference, as the other ones with less variables or products could be solved very quickly using all linearization approaches, and the other instances of about the same size produced quite similar results as has been the case also in the reference. In particular, nearly all instances could be solved within 48 hours by all methods, only in three exceptions this was not the case with SLR. The exact number of additional inequalities and non-zeros for SLR here vary with the (objective coefficients in the) different instances which is why they are displayed in the tables.
From a coarse perspective, Gurobi could solve the problems better using the inductive linearizations than when using the standard linearizations. Taking a closer look, the ILW led to the best running time in most of the cases.

The Quadratic Shortest Path Problem
Given a directed graph G = (V, A), a source node s ∈ V and a target node t ∈ V , a canonical BQP that models the quadratic shortest s-t-path problem (QSPP) on G (see e.g. [19]) can be expressed as One can see immediately that constraints (29) have −1 and 1 coefficients on their left hand sides, requiring normalization. As it has been the case with the QMP, each potential factor (arc) occurs on the left hand sides of two constraints (those associated with its endpoints), but now with opposite signs. This impacts the MIP solution times to derive an inductive linearization negatively while this is not the case for Algorithm 1. Not surprisingly, for the the largest instances with |P| > 10 6 , a noticeable increase in the derivation time takes place nevertheless.
For the experiments, we generated instances as described by [19], three for each type and size. However, even though ruling out some products that are implied to be zero by the problem structure, the number of products remained relatively large and only a few instances could be solved within an 48 hour time limit when passing the inductive linearization to the MIP solver. As one can see from Table 7, the standard linearization clearly works better with Gurobi for these instances. The SLR has |P| additional variables, |P| additional inequalities, and 3|P| additional non-zeros as the instances have positive objective coefficients only. Using IL, there are less linearization constraints but the number of induced products and non-zeros is significantly larger. We found that we can further reduce the number of linearization constraints and non-zeros for IL by a factor of about three when investing more time to derive the inductive linearization using the MIP. For the grid1 10 instances, this led to a faster solution of the resulting MIP by a factor of about six. On the large grid2 16 16 instances, the size reduction was the same but the solution times were less than halved only. Hence, there is evidently room for improvement but it currently appears to be too challenging to compete with the "standard" linearization and especially specialized methods for the QSPP as in [19] that work even much better.

MINLPLib
As another benchmark and to further broaden the experimental setting, we incorporated library collections such as the MINLPLib providing also instance formats that we could pass on directly to Gurobi (we used the .lp format) in order to have an additional comparison (table columns "GRB"). More precisely, we identified all linearly constrained BQPs except for six QSPP instances and qap.lp (as similar instances have been covered above). We remark that all except ten of these instances have equations only. The first two exceptions are crossdock 15x7 which has 30 equations and 14 inequalities, and crossdock 15x8 which has 30 equations and 16 inequalities. Also for these two instances, however, only equations are employed to derive the inductive linearizations. Finally, there are the pbXXYYZZ instances which have XX equations and YY inequalities, and only equations are employed for the inductive linearizations as well. Especially, partly when investing some more computational resources and time, the following of these instances in Table 8 were solved using IL -to the best of our knowledge for the first time.  As opposed to that, the instances in Table 9 have been solved before (respectively celar6-sub0 is solved as well by Gurobi as a standalone solver), but still confirmations of the claimed optima found are demanded whence we list the values that we retrieved.  Tables 10 and 11 display the entire results. A first observation is that even among those instances with only equations and left hand side coefficients as well as right hand sides equal to one, the inductive as well as the "standard" linearization(s) can be clearly superior or inferior. For instance, for celar6-sub0, the inductive linearization is orders of magnitude faster even though there are many products and induced non-zeros, and e.g. for color lab3 3x0 the converse is true. Similar extremes are also identified when comparing with Gurobi as a standalone solver. Although there is a more comprehensive study in [16], the graph partitioning problems were not sorted out from the experiments for two reasons. Firstly, these instances differ from those in the reference in that they lack an additional normalization constraint, and there are a few outliers in the results where IL turns out to be not as robust despite the typically good results for optimization problems with similar constraints as mentioned already in Sect. 3.1. Secondly, it becomes apparent that the graphpart-clique-instances become especially challenging with increasing size. In these instances, each node must be assigned to one of three partitions and highly symmetric costs are associated to placing any pair of nodes in the same partition. On the maximum constraint satisfiability problems, the SLR performs clearly best with Gurobi.          First of all, the overall results support once more the impression from the last subsection that even among the instances with left hand side coefficients as well as right hand sides equal to one, the inductive as well as the standard linearization can potentially outperform the other, or solve the respective instance at all within the time limit. To exemplify an extreme, when using IL the running times are orders of magnitude faster than with SLR or SLC and even Gurobi as a stand-alone solver is clearly outperformed for QPLIB 3413. On the other hand, IL cannot solve QPLIB 0633, which has just one equation asking to select 15 out of 75 variables such that the objective is minimum, within the time limit while with SLR less than twenty minutes are needed.
For the instances with equations in Tables 12 and 15 the overall performance of IL compares rather favorably. Here, it is often even faster than Gurobi standalone, and e.g. the instance QPLIB 3709 is solved to optimality (according to the QPLIB website, this was not achieved before). The optimal value is 5710645 which is also found by Gurobi when applying it standalone and with many threads on an according system, but even then the search tree becomes too large so that an optimality proof seems yet out of reach. On the other hand, on the inequality-only instances in Tables 13 and 14, IL is outperformed first by ILW and often also by the standard linearizations and Gurobi as a standalone solver. Thus, only ILW is displayed in these tables. As a remark, the "standard" linearization and WIL results for the depicted instances QPLIB 59XX could be significantly improved using a preprocessing that rules out products guaranteed to be zero, and the results for QPLIB 0067 are in line with those from Sect. 3.2 as this is a small QKP. Moreover, it becomes once more visible that especially large coefficients and right hand sides (as in the QPLIB 100XX instances), as well as negative left hand side coefficients, yet lead to less effective inductive linearizations.

Conclusion
A framework to derive inductive linearizations in practice has been outlined and it has been demonstrated that it can be effectively applied to a variety of binary quadratic programs with linear constraints. The experiments covering the Quadratic Assignment, Knapsack, Matching and Shortest Path Problems, as well as instances from the MINLBLib and QPLIB, show that the performance that can be expected when combining an inductive linearization with a professional MIP solver depends on the target application. As is also predicted from theory, as a rule of thumb, small ratios between right hand sides and (non-negative) left hand side coefficients (ideally, ratios of one) of the employed constraints are favorable, and especially equations appear to be effective to obtain a compact and strong linearization at the same time. But also inequalities can be effectively used to build well-suited inductive linearizations as especially the results for the Quadratic Matching Problem showed. Conversely, large ratios and negative coefficients are not yet handled as routinely. This is a clear field for further research, as well as additional computational studies providing even more mosaics and tailored approaches for particular applications. There is also room for developments and improvements regarding the exact and heuristic derivation of inductive linearizations that has just been prototyped so far. Yet, depending also on further parameters such as the rate of truly present product terms and the distribution of their factors over the set of (employed) constraints, the well-known "standard" linearization as well as commercial solvers may either be clearly outperformed or superior. Notwithstanding that, in some fortunate cases, the methodology allows to solve established benchmark library instances to optimality for the first time as is the case with the instances pb302035, pb302095, and pb351535 from the MINLPLib, and apparently also with QPLIB 3709 from the QPLIB.