Section: New Results
Dual-feasible functions
Dual-feasible functions have been used in the past to
compute fast lower bounds and valid inequalities for different
combinatorial optimization and integer programming
problems. Until now, all the dual-feasible functions
proposed in the literature were 1-dimensional
functions, and were defined only for positive arguments. In [12] we extended the principles of dual-feasible functions to the m-dimensional case by introducing the
concept of vector packing dual-feasible function. We
explored the theoretical properties of these functions in
depth, and we proposed general schemes for generating some
instances of these functions. Additionally, we proposed and
analyzed different new families of vector packing
dual-feasible functions. All the proposed approaches were
tested extensively using benchmark instances of the
2-dimensional vector packing problem. Our computational
results showed that these functions can approximate very
efficiently the best lower bounds for this problem.
In a second paper, currently submitted to a journal, we show that extending these functions to negative arguments raises many issues. Additionally, we describe different construction principles to obtain dual-feasible functions with domain and range