An Algorithm for Biobjective Mixed Integer Quadratic Programs

Multiobjective quadratic programs (MOQPs) are appealing since convex quadratic programs have elegant mathematical properties and model important applications. Adding mixed-integer variables extends their applicability while the resulting programs become global optimization problems. Thus, in this work, we develop a branch and bound (BB) algorithm for solving biobjective mixed-integer quadratic programs (BOMIQPs). An algorithm of this type does not exist in the literature. The algorithm relies on five fundamental components of the BB scheme: calculating an initial set of efficient solutions with associated Pareto points, solving node problems, fathoming, branching, and set dominance. Considering the properties of the Pareto set of BOMIQPs, two new fathoming rules are proposed. An extended branching module is suggested to cooperate with the node problem solver. A procedure to make the dominance decision between two Pareto sets with limited information is proposed. This set dominance procedure can eliminate the dominated points and eventually produce the Pareto set of the BOMIQP. Numerical examples are provided. Solving multiobjective quadratic programs (MOQPs) is fundamental to our research. Therefore, we examine the algorithms for this class of problems with different perspectives. The scalarization techniques for (strictly) convex MOPs are reviewed and the available algorithms for computing efficient solutions for MOQPs are discussed. These algorithms are compared with respect to four properties of MOQPs. In addition, methods for solving parametric multiobjective quadratic programs are studied. Computational studies are provided with synthetic instances, and examples in statistics and portfolio optimization. The real-life context reveals the interplay between the scalarizations and provides an additional insight into the obtained parametric solution sets