LPrelaxaation

LPrelaxaation is a term used in some optimization discussions to describe the use of linear programming relaxation applied to a discrete optimization problem as a preprocessing or approximation step. In this context, the problem is initially modeled as an integer linear program or mixed-integer program, and the integrality constraints are relaxed to obtain a linear program. The approach yields a fractional optimal solution and a bound on the original problem's objective value.

Practitioners use the LP relaxation results to guide solution methods. The fractional solution provides bounds, and the values can inform heuristic decisions such as fixing variables with extreme fractional values, prioritizing certain branches in a branch-and-bound framework, or generating cutting planes. It can also be used to evaluate the quality of proposed solutions or as a starting point for more exact algorithms.

Limitations include possibly weak bounds for certain problem classes, especially when the LP relaxation is far from integral; the added step may not reduce overall complexity in some cases; and the term LPrelaxaation is not standard in literature, with LP relaxation or linear relaxation being the conventional terminology. Nonetheless, the concept shares common ground with standard LP relaxation and is sometimes discussed as part of pre-processing or approximation strategies for complex optimization problems.

See also LP relaxation, linear programming, integer programming, branch-and-bound, and cutting planes.