Branch-and-bound for biobjective mixed-integer linear programming

We present a generic branch-and-bound method for finding all the Pareto solutions of a biobjective mixed integer program. Our main contribution is new algorithms for obtaining dual bounds at a node, for checking node fathoming, presolve and duality gap measurement. Our various procedures are implemented and empirically validated on instances from literature and a new set of hard instances. We also perform comparisons against the triangle splitting method of Boland et al. [\emph{INFORMS Journal on Computing}, \textbf{27} (4), 2015], which is a objective space search algorithm as opposed to our variable space search algorithm. On each of the literature instances, our branch-and-bound is able to compute the entire Pareto set in significantly lesser time. Most of the instances of the harder problem set were not solved by either algorithm in a reasonable time limit, but our algorithm performs better on average on the instances that were solved.Comment: 35 pages, 12 figures. Original preprint at Optimization Online, October 201

Efficient Storage of Pareto Points in Biobjective Mixed Integer Programming

In biobjective mixed integer linear programs (BOMILPs), two linear objectives are minimized over a polyhedron while restricting some of the variables to be integer. Since many of the techniques for finding or approximating the Pareto set of a BOMILP use and update a subset of nondominated solutions, it is highly desirable to efficiently store this subset. We present a new data structure, a variant of a binary tree that takes as input points and line segments in $\R^2$ and stores the nondominated subset of this input. When used within an exact solution procedure, such as branch-and-bound (BB), at termination this structure contains the set of Pareto optimal solutions. We compare the efficiency of our structure in storing solutions to that of a dynamic list which updates via pairwise comparison. Then we use our data structure in two biobjective BB techniques available in the literature and solve three classes of instances of BOMILP, one of which is generated by us. The first experiment shows that our data structure handles up to $10^7$ points or segments much more efficiently than a dynamic list. The second experiment shows that our data structure handles points and segments much more efficiently than a list when used in a BB

Adaptive Improvements of Multi-Objective Branch and Bound

Branch and bound methods which are based on the principle "divide and conquer" are a well established solution approach in single-objective integer programming. In multi-objective optimization branch and bound algorithms are increasingly attracting interest. However, the larger number of objectives raises additional difficulties for implicit enumeration approaches like branch and bound. Since bounding and pruning is considerably weaker in multiple objectives, many branches have to be (partially) searched and may not be pruned directly. The adaptive use of objective space information can guide the search in promising directions to determine a good approximation of the Pareto front already in early stages of the algorithm. In particular we focus in this article on improving the branching and queuing of subproblems and the handling of lower bound sets. In our numerical test we evaluate the impact of the proposed methods in comparison to a standard implementation of multiobjective branch and bound on knapsack problems, generalized assignment problems and (un)capacitated facility location problems

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