A surrogate relaxation based algorithm for a general quadratic multi-dimensional knapsack problem

Exact solution method to solve large scale integer quadratic multidimensional knapsack problems

“…We will compare the computational performance of the exact solution method we proposed with these three other branch-and-bound algorithms in Section 4. [4]. These authors develop a branch-and-bound algorithm based on the computation of an upper bound of better quality than the one provided by solving the LP-relaxation of (QMKP ).…”

“…This upper bound method involves two improvements of the algorithm proposed by Djerdjour et al (1988) [4] summarized in Section 2. We also develop a heuristic which provides a good feasible solution we use as a starting point of the process.…”

“…The bounds u j of variables x j are integers such that u j ≤ ( c j 2d j ) because of the concavity of each f j , x * j ≤ ( c j 2d j ), where x * j is the optimal solution of the program max x j ≥0 f j (x j ). Problem (QMKP ) has wide applications, including financial models [4] (Djerdjour et al, 1988), [5] (Faaland, 1974) production and inventory management [2] (Bretthauer and Shetty, 2002). However, most of the knapsack problems literature has addressed attention to specialized versions.…”

“…In a previous study [17] (Quadri et al, 2007), we present an effective upper bound method based on a linearization and surrogate relaxation. We have compared analytically and computationally our upper bound to the bound suggested by Djerdjour et al (1988) [4], the basic LP-relaxation of (QMKP ), and the LP-relaxation of the equivalent linearized formulation (MKP ). Simulation results, over a set of large instances (up to 2000 variables and constraints), showed that our upper bound is of good quality and closer to the optimum than the three other.…”

“…There have been a limited number of papers studying (QMKP ). Djerdjour et al (1988) [4] proposed a branch-and-bound algorithm specifically designed to solve (QMKP ), unfortunately the computational results were limited to instances up to 20 variables and constraints. However, the real-world applications require effective methods which are able to solve large scale instances of the problems.…”