arXiv subjects

Nathan Adelgren

Publications and source records attributed to Nathan Adelgren.

3 recordsLinked to original sources

arXiv2022

A Unified Tool for Solving Uni-Parametric Linear Programs, Convex Quadratic Programs, and Linear Complementarity Problems

We introduce a new technique for solving uni-parametric versions of linear programs, convex quadratic programs, and linear complementarity problems in which a single parameter is permitted to be present in any of the input data. We demonstrate the use of our method on a small, motivating example and present the results of a small number of computational tests demonstrating its utility for larger scale problems.

math.OC↗

arXiv2017

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

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, for checking node fathoming, presolve and duality gap measurement. Our branch-and-bound is predominantly a decision space search method since 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 also on a new set of instances that we generate using the MIPLIB benchmark library for single objective problems. We also perform comparisons against the triangle splitting method from literature, which is an objective space search algorithm.

math.OC↗

arXiv2014

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.

cs.DS↗