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 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 Linear programming relaxation Lagrangian relaxation Semidefinite relaxation Surrogate relaxation and duality Notes References Translated by Steven Vajda from W. R. Pulleyblank, Polyhedral combinatorics (pp. 371–446); George L. Nemhauser and Laurence A. Wolsey, Integer programming (pp. 447–527); Claude Lemaréchal, Nondifferentiable optimization (pp. 529–572); Mathematical optimization Approximations