Question

For an integer programming problem, the linear relaxation refers to:

Group of answer choices
The same optimization problem but with binary constraints on the decision variables
A different optimization problem but with shadow prices for constraints set to 0
The same optimization problem but with the constraints linearly scaled by a factor of SQRT(2)
The same optimization problem but without the integer constraints

Answer 1

For an integer programming problem, the linear relaxation refers to:

The same optimization problem but without the integer constraints.

In integer programming, the decision variables are required to take integer values. However, in the linear relaxation of an integer programming problem, the same optimization problem is solved, but the integer constraints are relaxed, allowing the decision variables to take on fractional values. This means that the linear relaxation solves the problem as a linear programming (LP) problem, where the decision variables can be non-integer values.

By relaxing the integer constraints, the linear relaxation provides a lower bound on the optimal objective value of the original integer programming problem. The solution to the linear relaxation can be used to obtain insights into the problem, determine the quality of heuristics or algorithms, and provide a starting point for finding good integer solutions.