Block linear majorants in quadratic 0-1 optimization

A usual technique to generate upper bounds on the optimum of a quadratic 0-1 maximization problem is to consider a linear majorant (LM) of the quadratic objective function f and then solve the corresponding linear relaxation. Several papers have considered LM’s obtained by termwise bounding, but the possibility of bounding groups of terms simultaneously does not appear to have been explored so far. In the present paper we develop this idea by suggesting the following approach: First, a suitable collection of “elementary” quadratic functions of few variables (typically, 3 or 4) is generated. All the coefficients of any such function (block) are either 1 or –1, and agree in sign with the corresponding coefficients of the given quadratic function. Next, for each block, a tightest LM (i.e., one having the same value as the block in as many points as possible), or a closest LM (i.e., one minimizing the sum of slacks) is computed. This can be accomplished through the solution of a small mixed-integer program, or a small linear program, respectively. Finally, the objective function is written as a weighted sum of blocks, with non-negative weights. Replacing in this expression each block by the corresponding LM, an LM of f can be obtained. We shall choose the weights in this process so that the maximum value of the resulting linear function is as small as possible. This amounts to a large-scale linear program, which can be solved by column generation. The encouraging results of a preliminary set of numerical tests are presented.