We present a generic branch-and-bound algorithm for finding all the Pareto solutions of a biobjective mixed-integer linear program. The main contributions are new algorithms for obtaining dual bounds at a node, checking node fathoming, presolve, and duality gap measurement. Our branch-and-bound is predominantly a decision space search method because the branching is performed on the decision variables, akin to single objective problems, although we also sometimes split gaps and branch in the objective space. The various algorithms are implemented using a data structure for storing Pareto sets. Computational experiments are carried out on literature instances and on a new set of instances that we generate using a benchmark library (MIPLIB2017) for single objective problems. We also perform comparisons against the triangle splitting method from literature, which is an objective space search algorithm. Summary of Contribution: Biobjective mixed-integer optimization problems have two linear objectives and a mixed-integer feasible region. Such problems have many applications in operations research, because many real-world optimization problems naturally comprise two conflicting objectives to optimize or can be approximated in such a manner and are even harder than single objective mixed-integer programs. Solving them exactly requires the computation of all the nondominated solutions in the objective space, whereas some applications may also require finding at least one solution in the decision space corresponding to each nondominated solution. This paper provides an exact algorithm for solving these problems using the branch-and-bound method, which works predominantly in the decision space. Of the many ingredients of this algorithm, some parts are direct extensions of the single-objective version, but the main parts are newly designed algorithms to handle the distinct challenges of optimizing over two objectives. The goal of this study is to improve solution quality and speed and show that decision-space algorithms perform comparably to, and sometimes better than, algorithms that work mainly in the objective-space.

Abstract. Biobjective mixed integer linear programs (BOMILP) are optimization problems where two linear objectives are optimized over a polyhedron while restricting some of the variables to be integer. Since many of the techniques for solving BOMILP (or approximating its solution set) are iterative processes which utilize data discovered during early iterations to aid in the discovery of improved data during later iterations, it is highly desirable to efficiently store the nondominated subset of a given set of data. This problem has not received considerable attention in the context of BOMILP; only naive methods have been implemented. We seek to bridge this gap by presenting a new data structure in the form of a modified binary tree that stores, updates, searches and returns nondominated solutions. This structure takes points and line segments in R2 as input and stores the nondominated subset of this input. We note that when used alongside an exact solution procedure, such as branch-and-bound (BB), at termination the data stored by this structure is precisely the set of Pareto optimal solutions. We perform two experiments. The first is designed to compare the utility of our structure for storing nondominated data to that of a dynamic list which updates via pairwise comparison. In the second we use our data structure alongside the biobjective BB techniques available in the literature and solve specific instances of BOMILP. The results of our first experiment suggest that the data structure performs reasonably well in handling input of up to 107 points or segments and does so much more efficiently than a dynamic list. The results of the second experiment show that when our structure is utilized alongside BB fathoming is enhanced and running times improve slightly. 1.