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- 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.^{[1]} ^{[2]} ^{[3]} However, iterative methods of relaxation have been used to solve Lagrangian relaxations.^{[4]}

A *relaxation* of the minimization problem

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

is another minimization problem of the form

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

with these two properties

*X*_{R}*\supseteq**X*

*c*_{R(x)}*\leq**c(x)*

*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.

If

*x*^{*}

*x*^{*}*\in**X**\subseteq**X*_{R}

*z*=*c(x*^{*)}*\geq*

*)\geq | |

c | |

R(x |

*z*_{R}

*x*^{*}*\in**X*_{R}

*z*_{R}

If in addition to the previous assumptions,

*c*_{R(x)=c(x)}

*\forall**x\in**X*

- Semidefinite relaxation
- Surrogate relaxation and duality

- Book: Murty, Katta G.. Katta G. Murty
. Katta G. Murty. 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. John Wiley & Sons, Inc.. New York. 1983. 453–464. 978-0-471-09725-9. 720547.

- Goffin. J.-L.. The relaxation method for solving systems of linear inequalities. Math. Oper. Res.. 5. 1980. 3. 388–414. 3689446. 10.1287/moor.5.3.388. 594854.
- Book: Minoux, M.. Michel Minoux
. Michel Minoux. Mathematical programming: Theory and algorithms. Egon Balas (foreword). Translated by Steven Vajda from the (1983 Paris: Dunod) French. A Wiley-Interscience Publication. John Wiley & Sons, Ltd.. Chichester. 1986. xxviii+489. 978-0-471-90170-9. 868279. (2008 Second ed., in French:

*Programmation mathématique: Théorie et algorithmes*. Editions Tec & Doc, Paris, 2008. xxx+711 pp. . .) - Relaxation methods for finding feasible solutions to linear inequality systems arise in linear programming and in Lagrangian relaxation. and |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).

- Book: G.Buttazzo. Semicontinuity, Relaxation and Integral Representation in the Calculus of Variations . Pitman Res. Notes in Math. 207. Longmann. Harlow. 1989.
- News: Geoffrion . A. M. . Duality in Nonlinear Programming: A Simplified Applications-Oriented Development. SIAM Review . 13 . 1971 . 1 . 1–37. 2028848. .
- Goffin. J.-L.. The relaxation method for solving systems of linear inequalities. Math. Oper. Res.. 5. 1980. 3. 388–414. 3689446. 10.1287/moor.5.3.388. 594854.
- Book: Minoux, M.. Michel Minoux

. Michel Minoux. Mathematical programming: Theory and algorithms . (With a foreword by Egon Balas) Translated by Steven Vajda from the (1983 Paris: Dunod) French. A Wiley-Interscience Publication. John Wiley & Sons, Ltd.. Chichester. 1986. xxviii+489. 978-0-471-90170-9. 868279. (2008 Second ed., in French: *Programmation mathématique: Théorie et algorithmes*. Editions Tec & Doc, Paris, 2008. xxx+711 pp. . .)|

- Book: Optimization. G. L.. Nemhauser. George L. Nemhauser. A. H. G.. Rinnooy Kan. M. J.. Todd. Michael J. Todd (mathematician). Handbooks in Operations Research and Management Science. 1. North-Holland Publishing Co.. Amsterdam. 1989. xiv+709. 978-0-444-87284-5. 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);

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