Rewriting integer variables into zero-one variables: Some guidelines for the integer quadratic multi-knapsack problem

This paper is concerned with the integer quadratic multidimensional knapsack problem (QMKP) where the objective function is separable. Our objective is to determine which expansion technique of the integer variables is the most appropriate to solve (QMKP) to optimality using the upper bound method proposed by Quadri et al. (2007). To the best of our knowledge the upper bound method previously mentioned is the most effective method in the literature concerning (QMKP). This bound is computed by transforming the initial quadratic problem into a 0–1 equivalent piecewise linear formulation and then by establishing the surrogate problem associated. The linearization method consists in using a direct expansion initially suggested by Glover (1975) of the integer variables and in applying a piecewise interpolation to the separable objective function. As the direct expansion results in an increase of the size of the problem, other expansions techniques may be utilized to reduce the number of 0–1 variables so as to make easier the solution to the linearized problem. We will compare theoretically the use in the upper bound process of the direct expansion (I) employed in Quadri et al. (2007) with two other basic expansions, namely: (II) a direct expansion with additional constraints and (III) a binary expansion. We show that expansion (II) provides a bound which value is equal to the one computed by Quadri et al (2007). Conversely, we provide the proof of the non applicability of expansion (III) in the upper bound method. More specifically, we will show that if (III) is used to rewrite the integer variables into 0–1 variables then a linear interpolation can not be applied to transform (QMKP) into an equivalent 0–1 piecewise linear problem.


Introduction
This paper deals with the integer quadratic multi-knapsack problem (QMKP) where the objective function is separable. Problems of this structure arise in numerous industrial and economic situations, for instance in production planning [12], reliability allocation [10] and finance [5]. These include the main application of (QMKP) which is in the portfolio management area where the investments are independent, see [4] and [5]. Nevertheless, solving (QMKP) efficiently will constitute a starting point to solve the more general and realistic portfolio management problem where the investments are dependent, i.e. the objective function is non separable.
The integer quadratic multi-knapsack problem (QMKP) where the objective function is separable consists in maximizing a concave separable quadratic integer function subject to m linear capacity constraints. It may be stated mathematically as follows: The problem (QMKP) which is a NP-hard problem [3] is a generalization of both the integer quadratic knapsack problem [2] and the 0-1 quadratic knapsack problem where the objective function is subject to only one constraint [1].
Since, (QMKP) is NP-hard, one should not expect to find a polynomial time algorithm for solving it exactly. Hence, we are usually interested in developing branch-andbound algorithms. A key step in designing an effective exact solution method for such a maximization problem is to establish a tight upper bound on the optimal value. Basically, the available upper bound procedures for (QMKP) may be classified as attempting either to solve efficiently the LP-relaxation of (QMKP) (see [2] and [8]) or to find a good upper bound, of better quality than the LP-relaxation of (QMKP), transforming (QMKP) into a 0-1 linear problem easier to solve (see [4] and [9]). To the best of our knowledge, the upper bound method we have proposed in a previous work [11] is better than the existing methods (Djerdjour, Mathur and Salkin algorithm [4], a 0-1 linearization method, a classical LP-relaxation of (QMKP)) from both a qualitative and a computational standpoint. We have first used a direct expansion of the integer variables, originally suggested by Glover [7], and apply a piecewise interpolation to the objective function: an equivalent 0-1 linear problem is thus obtained. The second step of the algorithm consists in establishing and solving the surrogate relaxation problem associated to the equivalent linearized formulation.
Nevertheless the transformed linear formulation encounters numerous 0-1 variables because of the direct expansion used (denoted by expansion (I) in the following).
Consequently, other expansions techniques may be utilized to reduce the number of 0-1 variables so as to make easier the solution to the linearized problem. Let us consider the three basic expansions for rewriting the integer variables of (QMKP) into 0-1 variables: • Expansion (I): direct expansion The purpose of this note is to evaluate the impact of the use of the above expansion techniques, on the upper bound computation developed in [11]. More specifically, we will determinate which expansion is the most appropriate to be used in the upper bound method for (QMKP) [11]. We will compare theoretically the use of the direct expansion (I) with the direct expansion with additional constraints (II) and with the binary expansion (III). We will show that the use of (II) provides a bound which value is equal to the one computed in [11]. Conversely, we provide the proof of the non applicability of both (III) and a linear interpolation to transform (QMKP) into an equivalent 0-1 piecewise linear problem.
The paper is organized as follows. The next section summarizes the upper bound method developed in [11] detailing the direct expansion (I) of integer variables and the piecewise interpolation. In Section 3, the direct expansion with additional constraints (II) is applied to (QMKP) so as to compute the upper bound suggested by Quadri et al. [11]. Section 4 is dedicated to the binary expansion (III). We finally conclude in Section 5.
In the remainder of this paper, we adopt the following notations: letting (P) be an integer or a 0-1 program, we will denote by ( P ) the continuous relaxation problem of (P). We let Z[P] be the optimal value of the problem (P) and Z[ P ] the optimal value of ( P ). Finally ⎡ ⎤ x (resp. ⎣ ⎦ x ) will denote the smallest (resp. highest) integer greater (resp. lower) than or equal to x.

Section 2. Direct expansion of the integer variables
In this section we summarize the upper bound method for (QMKP) proposed by Quadri et al. [11]. First, an equivalent formulation is obtained by using a direct expansion (I) of the integer variables x j as originally proposed by Glover [7] and by applying a piecewise interpolation to the initial objective function as discussed in [4].  In the second step of the algorithm, a surrogate relaxation is applied to the LPrelaxation of (MKP). The resultant formulation (KP, w) is the surrogate relaxation problem of (MKP) and can be written as: As proved by Glover [7], (KP, w) is a relaxation of (MKP). For any value of the optimal value of ( ) proposed in [11].

Section 3. Direct expansion of integer variables with additional constraints
In this section we apply to the integer variables of (QMKP) the direct expansion with additional constraints (II). That is each variable x j is replaced by the following , k =1,…,u j and j=1,…,n. Since the integer variables are now replaced by 0-1 variables, we transform the resultant 0-1 quadratic program into a 0-1 linear problem as follows: we replace each objective function term The problem (QMKP) is thus equivalent to the following problem (MKP 2 ): Following the upper bound method developed in [11] we then establish the surrogate problem (KP 2 ,w) associated to (MKP 2 ). The problem (KP 2 ,w) can be written as:  . We denote: x stands for the highest integer lower than or equal to the real x. j p is the integer part of j x . Note that We define the solution y as follows:   . We first note that: In the following we name j Λ the quantity ( ) . We thus have to prove that: ), we find with some easy algebra: is a degree 2 polynomial in k. Its discriminant is equal to 2 j d , so admits two roots: j p and j p +1. Since the coefficient of 2 is also nonnegative for k outside the roots, namely for j p k ≤ and 1 + ≥ j p k . Since we only consider integer values of k, we get: , • We now show that: The proof is analogous to the one of the previous point. Let y be an optimal solution for ( ) w KP, . We derive from y a solution ' Since ( ) is an integer lower than or equal to j u . We then finish to define ' y :

Section 4. Binary expansion of integer variables
This section is dedicated to the use of a binary expansion (referred as expansion (III)) of this integer variables in the upper bound procedure developed in [11]. Such expansion consists in rewriting each integer variable x j (j=1,…,n) as: (1) Using (1)  As previously mentioned, the aim of this study is to compare other expansion techniques in the upper bound process developed in [11]. Since, the variables are now binary the next step of the algorithm of Quadri et al. [11] concerns with a piecewise If (H) is true then it should exist coefficients g jk such that: The assumption (H) must be satisfied for all problem data. Let us set u j = 3, c j = 10 and d j = 4.
The equations system (I), (II) and (III) has clearly no solution. Consequently there is a contradiction with (H). Proposition 4.1 shows the non applicability of a binary expansion of the integer variables for (QMKP) so as to transform the initial problem into a 0-1 linear program.
Consequently, the upper bound method proposed in [11] can not be applied together with expansion (III).

Section 5. Concluding remarks
In this paper we have theoretically compared the use of three techniques to rewrite integer variables into zero-one variables in an upper bound procedure for (QMKP) developed by Quadri et al. [11], which provides, to the best of our knowledge a bound closer to the optimum than the existing methods. More specifically, we have compared a direct expansion of the integer variables, originally employed in [11] with a direct expansion with additional constraints (II) and with a binary expansion (III).
We have proved that (II) provides the same upper bound as the one computed in [11] whereas it involves n additional constraints. We therefore do not expect an technique is of worst quality than the one computed in [11]. Moreover, this process seems to be more time consuming than [11].