Relaxation (approximation)


In mathematical optimization and related fields, relaxation is a modeling strategy. A relaxation is an approximation of a difficult problem by a nearby problem that is easier to solve. A solution of the relaxed problem provides information about the original problem.
For example, a linear programming relaxation of an integer programming problem removes the integrality constraint and so allows non-integer rational solutions. A Lagrangian relaxation of a complicated problem in combinatorial optimization penalizes violations of some constraints, allowing an easier relaxed problem to be solved. Relaxation techniques complement or supplement branch and bound algorithms of combinatorial optimization; linear programming and Lagrangian relaxations are used to obtain bounds in branch-and-bound algorithms for integer programming.
The modeling strategy of relaxation should not be confused with iterative methods of relaxation, such as successive over-relaxation ; iterative methods of relaxation are used in solving problems in differential equations, linear least-squares, and linear programming. However, iterative methods of relaxation have been used to solve Lagrangian relaxations.

Definition

A relaxation of the minimization problem
is another minimization problem of the form
with these two properties
  2. for all.
The first property states that the original problem's feasible domain is a subset of the relaxed problem's feasible domain. The second property states that the original problem's objective-function is greater than or equal to the relaxed problem's objective-function.

Properties

If is an optimal solution of the original problem, then and. Therefore, provides an upper bound on.
If in addition to the previous assumptions,,, the following holds: If an optimal solution for the relaxed problem is feasible for the original problem, then it is optimal for the original problem.

Some relaxation techniques