Convex hull representations for bounded products of variables

It is well known that the convex hull of \(\{{(x,y,xy)}\}\), where (x, y) constrained to lie in a box, given by reformulation-linearization technique (RLT) constraints. Belotti et al. (Electron Notes Discrete Math 36:805–812, 2010) and Miller (SIAG/OPT Views News 22(1):1–8, 2011) showed if there are additional upper and/or lower bounds on product \(z=xy\), then can be represented adding an infinite family inequalities, requiring separation algorithm implement. Nguyen (Math Progr 169(2):377–415, 2018) derived hulls for \(\{(x,y,z)\}\) with \(z=xy^b\), \(b\ge 1\). We focus case \(b=1\) show either bound or RLT constraints, z single second-order cone (SOC) constraint. With both product, using no more than three SOC each applicable subset values. In addition characterizations, volumes calculated compared relaxation imposes only As application these volume results, we how spatial branching applied variable so as minimize sum two resulting subproblems.