Exploiting special structures in constructing a hierarchy of relaxations for \(0-1\) mixed integer problems (Q2770136)

A new hierarchy of relatations is presented that provides a unifying framework for constructing a spectrum of continuous relaxations spanning from the linear programming relaxation to the convex hull representation for linear mixed integer \(0-1\) problems. This hierarchy is an extension of the Reformulation-Linearization Technique (RLT) of [\textit{H. D. Sherali} and \textit{W. P. Adams} SIAM J. Discrete Math. 3, 411-430 (1990; Zbl 0712.90050); Discrete Appl. Math. 52, 83-106 (1994; Zbl 0819.90064)] and is particularly designed to exploit special structures. Specifically, inherent special structures are exploited by identifying particular classes of multiplicative factors that are applied to the original problem to reformulate it as an equivalent polynomial programming problem, and subsequently, this resulting problem is linearized to produce a tighter relaxation in a higher dimensional space. This general framework permits us to generate an explicit hierarchical sequence of tighter relaxations leading up to the convex hull representation. (A similar hierarchy can be constructed for polynomial mixed integer \(0-1\) problems.) Additional ideas for further strengthening RLT-based constraints by using conditional logical implications, as well as relationships with sequential lifting, are also explored. Several examples are presented to demonstrate how underlying special structures, including generalized and variable upper bounding, covering, partitioning, and packing constraints, as well as sparsity, can be exploited within this framework. For some types of structures, low level relaxations are exhibited to recover the convex hull of integer feasible solutions.