Efficient Storage of Pareto Points in Biobjective Mixed Integer Programming

Abstract

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.

Branch-and-Bound for Biobjective Mixed-Integer Linear Programming

“…The Pareto set of a mixed-integer problem is a finite union of graphs of piecewise linear functions, whereas that for a pure integer problem is a finite set of points, and hence Pareto computation and certification of Pareto optimality of a given subset is far more complicated in the former case. In fact, mixed-integer problems can benefit immensely from sophisticated data structures for storing Pareto sets, as shown recently by Adelgren et al [ABG18]. Most of the BB algorithms in literature are designed specifically for problems where all the integer variables are binary; see the literature reviews in [PG17;GNE19].…”

“…The two methods differ in the way fathoming rules are implemented. Firstly, we utilize the data structure of Adelgren et al [ABG18] to store and dynamically update the set N s throughout the BB process. In [BSW12; BSW16], fathoming rules are checked at a node s of the BB tree by: (i) using N s to generate U s by adding a set of local nadir points to N s , (ii) selecting the subset R := U s ∩ ((Y s ) ideal + R 2 ≥0 ), and (iii) solving auxiliary LPs to determine whether R and L s can be separated by a hyperplane.…”

“…To aid in understanding this idea, observe Unfortunately the nonconvex nature of U s makes the Hausdorff metric difficult to use since it cannot be computed using a linear program. In our implementation U s * is stored as the individual line segments and singletons comprising N s * using the data structure of [ABG18]. DB s * is computed by generating the points and line segments comprising its nondominated subset, which are also stored using the same data structure.…”

“…This greatly simplifies checking bound dominance for BOMILPs since given two line or piecewise linear curves in general, in R 2 , one can easily identify the dominated portion through pairwise comparisons. The data structure [ABG18] stores nondomimated line segments and efficiently checks if a new line segment is dominated by what is currently stored. This enabled the node processing step in this paper to perform fathoming efficiently.…”

Branch-and-Bound for Bi-objective Integer Programming

“…Improved fathoming rules are introduced in order to discard more nodes during tree search. Adelgren et al (2014) propose an efficient data structure to store and update upper bound sets in the context of mixed integer programming and illustrate its efficiency using the algorithms of Belotti et al (2013) and of Adelgren and Gupte (2016).…”