Liftandprojectkutt

Liftandprojectkutt is a framework in combinatorial optimization that merges lifting and projection techniques to tighten relaxations of discrete problems. The concept is described in theoretical optimization literature as a method to derive stronger polyhedral descriptions of the feasible integral solutions for binary and mixed-integer programs. It is typically presented as a family of lift-and-project operations intended to produce progressively tighter relaxations.

Conceptually, lifting introduces auxiliary variables to capture higher-order relations among the original variables; projection then removes these auxiliaries while preserving the implications of the lifted constraints. The central idea is to generate a hierarchy of convex relaxations by alternately lifting the problem into a higher-dimensional space and projecting back to the original variable set, producing valid inequalities (cuts) that exclude fractional solutions but retain feasibility for integral solutions.

Procedure often follows an iterative pattern. Begin with a base relaxation, such as the linear programming relaxation of a binary program. Apply a lift operation to introduce new variables representing joint configurations or higher-order interactions. Enforce lifted constraints to encode combinatorial structure, then apply a projection to derive linear inequalities in the original variables that are valid for the integral hull. Add these cuts to the model and resolve. Repeat until convergence criteria are met or computational resources are exhausted.

Applications include mixed-integer programming, graph partitioning, scheduling, facility location, and network design. The method is particularly useful when problem structure suggests strong higher-order interactions that ordinary cuts fail to capture. Challenges include increased problem size from lifting and the need for problem-specific guidance to select effective lifts and projections.

See also: lifting, projection, polyhedral combinatorics, cutting planes, lift-and-project cuts.