Philipp M. Christophel

arXiv2017

Shaping and Trimming Branch-and-bound Trees

We present a new branch-and-bound type search method for mixed integer linear optimization problems based on the concept of offshoots (introduced in this paper). While similar to a classic branch-and-bound method, it allows for changing the order of the variables in a dive (shaping) and removing unnecessary branching variables from a dive (trimming). The regular branch-and-bound algorithm can be seen as a special case of our new method. We also discuss extensions to our new method such as choosing to branch from the top or the bottom of an offshoot. We present several numerical experiments to give a first impression of the potential of our new method.

math.OC↗

arXiv2014

New symmetries in mixed-integer linear optimization: Symmetry heuristics and complement-based symmetries

We present two novel applications of symmetries for mixed-integer linear programming. First we propose two variants of a new heuristic to improve the objective value of a feasible solution using symmetries. These heuristics can use either the actual permutations or the orbits of the variables to find better feasible solutions. Then we introduce a new class of symmetries for binary MILP problems. Besides the usual permutation of variables, these symmetries can also take the complement of the binary variables. This is useful in situations when two opposite decisions are actually symmetric to each other. We discuss the theory of these symmetries and present a computational method to compute them. Examples are presented to illustrate the usefulness of these techniques.

math.OC↗