Representations of quadratic combinatorial optimization problems: A case study using quadratic set covering and quadratic knapsack problems

Abstract

The objective function of a quadratic combinatorial optimization problem (QCOP) can be represented using two data items, a quadratic cost matrix Q and a linear cost vector c. Different, but equivalent, representations of the pair (Q, c) for the same QCOP are well known in literature. These representations however have inherently different properties. Popular general purpose 0–1 integer programming solvers such as GUROBI and CPLEX do not suggest a preferred representation of Q and c. Our experimental analysis discloses that GUROBI prefers the upper triangular representation of the matrix Q whereas CPLEX prefers the symmetric representation, in a statistically significant manner, for the quadratic set covering problem (QSCP) and the quadratic unconstrained binary optimization problem. However, the same behavior was not observed in the case of the quadratic knapsack problem. This shows that the structure of feasible solutions are also important in choosing a preferred equivalent representation, if any. Equivalent representations, although preserve optimality, they could alter the corresponding lower bound values obtained by various lower bounding schemes. For the Gilmore–Lawler type lower bound of a QSCP, the symmetric representation of Q produced tighter bounds, in general. Effects of equivalent representations when CPLEX and GUROBI are run in a heuristic mode, are also explored. Further, we review various equivalent representations of a QCOP from the literature that have theoretical basis to be viewed as ‘strong’. We also provide new theoretical insights on generating ‘strong’ equivalent representations by making use of the constant value property of a linear objective function and diagonalization (linearization) of a quadratic cost matrix.