Semi-infinite programming

In optimization theory, semi-infinite programming (SIP) is an optimization problem with a finite number of variables and an infinite number of constraints, or an infinite number of variables and a finite number of constraints. In the former case the constraints are typically parameterized.

Mathematical formulation of the problem

The problem can be stated simply as:

$\underset{x\in X}{\min }f(x)$

$\text{subject to: }$

$g(x,y)\le 0,\mathrm{\forall }y\in Y$

where

$f:R^n\to R$

$g:R^n\times R^m\to R$

$X\subseteq R^n$

$Y\subseteq R^m.$

SIP can be seen as a special case of bilevel programs in which the lower-level variables do not participate in the objective function.

Methods for solving the problem

In the meantime, see external links below for a complete tutorial.

Examples

In the meantime, see external links below for a complete tutorial.

See also

• Optimization
• Generalized semi-infinite programming (GSIP)

References

• Anderson, Edward J.; Nash, Peter (1987). Linear Programming in Infinite-Dimensional Spaces. Wiley. ISBN 0-471-91250-6. OCLC 15053031. 
• Bonnans, J. Frédéric; Shapiro, Alexander (2000). "5.4, 7.4.4 Semi-infinite programming". Perturbation analysis of optimization problems. Springer Series in Operations Research. Springer. pp. 496–526, 581. ISBN 978-0-387-98705-7. 
• Goberna, M.A.; López, M.A. (1998). Linear Semi-Infinite Optimization. Wiley. 
• Goberna, M.A.; López, M.A. (2014). Post-Optimal Analysis in Linear Semi-Infinite Optimization. SpringerBriefs in Optimization. Springer. doi:10.1007/978-1-4899-8044-1. ISBN 978-1-4899-8044-1. https://link.springer.com/book/10.1007/978-1-4899-8044-1. 
• Hettich, R.; Kortanek, K.O. (1993). "Semi-infinite programming: Theory, methods, and applications". SIAM Review 35 (3): 380–429. doi:10.1137/1035089. 
• Luenberger, David G. (1997). Optimization by Vector Space Methods. Wiley. ISBN 0-471-18117-X. OCLC 52405793. 
• Reemtsen and, Rembert; Rückmann, Jan-J., eds (1998). Semi-Infinite Programming. Nonconvex Optimization and Its Applications. 25. Springer. doi:10.1007/978-1-4757-2868-2. ISBN 978-1-4757-2868-2. https://link.springer.com/book/10.1007/978-1-4757-2868-2. 
• Guerra Vázquez, F.; Rückmann, J.-J.; Stein, O.; Still, G. (1 August 2008). "Generalized semi-infinite programming: A tutorial". Journal of Computational and Applied Mathematics 217 (2): 394–419. doi:10.1016/j.cam.2007.02.012. Bibcode: 2008JCoAM.217..394G. http://www.sciencedirect.com/science/article/pii/S0377042707000982. 

External links

• Description of semi-infinite programming from INFORMS (Institute for Operations Research and Management Science).