We consider the separable quadratic multi-knapsack problem (QMKP) which consists in maximizing a concave separable quadratic integer function subject to m linear capacity constraints. The aim of this paper is to develop an effective method to compute an upper bound for (QMKP) from a surrogate relaxation originally proposed in Djerdjour et al. (1988). We evaluate the quality of three other upper bounds for (QMKP) and compare them theoretically and experimentally with the bound we suggest. We also present an effective heuristic method to obtain a good feasible solution for (QMKP). Finally, we report computational experiments that assess the efficiency of our upper bound for instances up to 2000 variables and constraints.