# 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.[1]

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.[2][3][4] However, iterative methods of relaxation have been used to solve Lagrangian relaxations.[5]

## 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.[1]

### 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.[1]

## Some relaxation techniques

- Linear programming relaxation
- Lagrangian relaxation
- Semidefinite relaxation
- Surrogate relaxation and duality

## Notes

- Geoffrion (1971)
- Murty, Katta G. (1983). "16 Iterative methods for linear inequalities and linear programs (especially 16.2 Relaxation methods, and 16.4 Sparsity-preserving iterative SOR algorithms for linear programming)".
*Linear programming*. New York: John Wiley & Sons, Inc. pp. 453–464. ISBN 978-0-471-09725-9. MR 0720547. - Goffin, J.-L. (1980). "The relaxation method for solving systems of linear inequalities".
*Math. Oper. Res*.**5**(3): 388–414. doi:10.1287/moor.5.3.388. JSTOR 3689446. MR 0594854. - Minoux, M. (1986).
*Mathematical programming: Theory and algorithms*. Egon Balas (foreword) (Translated by Steven Vajda from the (1983 Paris: Dunod) French ed.). Chichester: A Wiley-Interscience Publication. John Wiley & Sons, Ltd. pp. xxviii+489. ISBN 978-0-471-90170-9. MR 0868279. (2008 Second ed., in French:*Programmation mathématique: Théorie et algorithmes*. Editions Tec & Doc, Paris, 2008. xxx+711 pp.. MR2571910) - Relaxation methods for finding feasible solutions to linear inequality systems arise in linear programming and in Lagrangian relaxation. Goffin (1980) and Minoux (1986)|loc=Section 4.3.7, pp. 120–123 cite Shmuel Agmon (1954), and Theodore Motzkin and Isaac Schoenberg (1954), and L. T. Gubin, Boris T. Polyak, and E. V. Raik (1969).

## References

- G.Buttazzo (1989).
*Semicontinuity, Relaxation and Integral Representation in the Calculus of Variations*. Pitman Res. Notes in Math. 207. Harlow: Longmann.

- Geoffrion, A. M. (1971). "Duality in Nonlinear Programming: A Simplified Applications-Oriented Development".
*SIAM Review*.**13**(1). pp. 1–37. JSTOR 2028848.. - Goffin, J.-L. (1980). "The relaxation method for solving systems of linear inequalities".
*Math. Oper. Res*.**5**(3): 388–414. doi:10.1287/moor.5.3.388. JSTOR 3689446. MR 0594854. - Minoux, M. (1986).
*Mathematical programming: Theory and algorithms*((With a foreword by Egon Balas) Translated by Steven Vajda from the (1983 Paris: Dunod) French ed.). Chichester: A Wiley-Interscience Publication. John Wiley & Sons, Ltd. pp. xxviii+489. ISBN 978-0-471-90170-9. MR 0868279. (2008 Second ed., in French:*Programmation mathématique: Théorie et algorithmes*. Editions Tec & Doc, Paris, 2008. xxx+711 pp.. MR2571910)|

- Nemhauser, G. L.; Rinnooy Kan, A. H. G.; Todd, M. J., eds. (1989).
*Optimization*. Handbooks in Operations Research and Management Science.**1**. Amsterdam: North-Holland Publishing Co. pp. xiv+709. ISBN 978-0-444-87284-5. MR 1105099.- 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);

- Rardin, Ronald L. (1998).
*Optimization in operations research*. Prentice Hall. ISBN 978-0-02-398415-0. - Roubíček, T. (1997).
*Relaxation in Optimization Theory and Variational Calculus*. Berlin: Walter de Gruyter. ISBN 978-3-11-014542-7.