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.^{[lower-alpha 1]}

A relaxation of the minimization problem

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

is another minimization problem of the form

${\displaystyle z_R=\min \{c_R(x):x\in X_R\subseteq {\mathbf{𝐑}}^n\}}$

with these two properties

1. ${\displaystyle X_R\supseteq X}$
2. ${\displaystyle c_R(x)\le c(x)}$ for all ${\displaystyle x\in X}$.

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.^[1]

If ${\displaystyle x^{*}}$ is an optimal solution of the original problem, then ${\displaystyle x^{*}\in X\subseteq X_R}$ and ${\displaystyle z=c(x^{*})\ge c_R(x^{*})\ge z_R}$. Therefore, ${\displaystyle x^{*}\in X_R}$ provides an upper bound on ${\displaystyle z_R}$.

If in addition to the previous assumptions, ${\displaystyle c_R(x)=c(x)}$, ${\displaystyle \mathrm{\forall }x\in X}$, 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]

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

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• Geoffrion, A. M. (1971). "Duality in Nonlinear Programming: A Simplified Applications-Oriented Development". SIAM Review 13 (1): 1–37. 
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• 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.. ISBN 978-0-444-87284-5. 
  • 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. 

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