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 $ℝ$. Specific instances obtained from these principles are proposed and analyzed.