Global Optimization of QCPs Using MIP Relaxations with a Base-2 Logarithmic Partitioning Scheme

Several problemsof interest to Process Systems Engineering canbe formulated as nonconvex quadratically constrained programs (QCPs).These are challenging to solve to global optimality due to the existenceof many local optimal solutions and the difficulty of computing atight dual bound. Although recent research has shown that mixed-integerprogramming (MIP) relaxations of the bilinear terms in the constraintscan play an important role addressing this challenge, because theytake considerably more time to solve than their linear programmingcounterparts, state-of-the art commercial solvers have found limiteduse for them. We now propose a global optimization algorithm relyingextensively on MIP relaxations, for reducing the variable domain throughoptimality-based bound tightening and improving the dual bound. MIPrelaxations are derived from the multiparametric disaggregation technique,using a logarithmic partitioning scheme to ensure tractability andbase-2 to minimize the jump in complexity when increasing the numberof intervals in the partition for a subset of the bilinear variables.Major improvements are achieved doing this sequentially, with refinementsoccurring within a spatial branch and bound procedure. Using a setof 54 benchmark instances from the literature, we show that the newalgorithm significantly outperforms GUROBI 9.5.2 and BARON 22.7.23,in terms of optimality gap at termination (reduction up to four ordersof magnitude), and number of problems solved.