Relaxation technique (mathematics)

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.

A relaxation of the minimization problem

is another minimization problem of the form

with these two properties

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.

If ${\displaystyle x^{*}}$ is an optimal solution of the original problem, then ${\displaystyle x^{*}\in X\subseteq X_{R}}$ and ${\displaystyle z=c(x^{*})\geq c_{R}(x^{*})\geq z_{R}}$. Therefore, ${\displaystyle x^{*}\in X_{R}}$ provides an upper bound on ${\displaystyle z_{R}}$.

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