Fractional row bound for quadratic knapsack ($q_i=\max \{{\sum }_{j\ne i}{p}_{ij}{y}_j:{\sum }_{j\ne i}{w}_j{y}_j\le B-w_i,0\le y_j\le 1\}$)

Conditioning on selecting item $i$ leaves capacity $B-w_i$. Its fractional knapsack problem bounds that item's total interaction with the other selected items. Thus the ordered-pair quadratic objective is at most ${\sum }_i(v_i+q_i)x_i$. Remove overweight items before defining their residual-capacity problems.