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 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. Properties. If formula_6 is an optimal solution of the original problem, then formula_7 and formula_8. Therefore, formula_9 provides an upper bound on formula_10. If in addition to the previous assumptions, formula_11, formula_12, 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.