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 (SOR); 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


 * $$z = \min \{c(x) : x \in X \subseteq \mathbf{R}^{n}\}$$

is another minimization problem of the form


 * $$z_R = \min \{c_R(x) : x \in X_R \subseteq \mathbf{R}^{n}\}$$

with these two properties


 * 1) $$X_R \supseteq X$$
 * 2) $$c_R(x) \leq c(x)$$ for all $$x \in X$$.

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 $$x^*$$ is an optimal solution of the original problem, then $$x^* \in X \subseteq X_R$$ and $$z = c(x^*) \geq c_R(x^*)\geq z_R$$. Therefore, $$x^* \in X_R$$ provides an upper bound on $$z_R$$.

If in addition to the previous assumptions, $$c_R(x)=c(x)$$, $$\forall x\in X$$, 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

 * Linear programming relaxation
 * Lagrangian relaxation


 * Semidefinite relaxation
 * Surrogate relaxation and duality