# 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.[1]

## Mathematical formulation of the problem

The problem can be stated simply as:

${\displaystyle \min _{x\in X}\;\;f(x)}$
${\displaystyle {\text{subject to: }}}$
${\displaystyle g(x,y)\leq 0,\;\;\forall y\in Y}$

where

${\displaystyle f:R^{n}\to R}$
${\displaystyle g:R^{n}\times R^{m}\to R}$
${\displaystyle X\subseteq R^{n}}$
${\displaystyle Y\subseteq R^{m}.}$

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.

## References

1. ^
• Bonnans, J. Frédéric; Shapiro, Alexander (2000). "5.4 and 7.4.4 Semi-infinite programming". Perturbation analysis of optimization problems. Springer Series in Operations Research. New York: Springer-Verlag. pp. 496–526 and&nbsp, 581. ISBN 978-0-387-98705-7. MR 1756264.
• M. A. Goberna and M. A. López, Linear Semi-Infinite Optimization, Wiley, 1998.
• Hettich, R.; Kortanek, K. O. (1993). "Semi-infinite programming: Theory, methods, and applications". SIAM Review. 35 (3). pp. 380–429. doi:10.1137/1035089. JSTOR 2132425. MR 1234637.
• Edward J. Anderson and Peter Nash, Linear Programming in Infinite-Dimensional Spaces, Wiley, 1987.
• Bonnans, J. Frédéric; Shapiro, Alexander (2000). "5.4 and 7.4.4 Semi-infinite programming". Perturbation analysis of optimization problems. Springer Series in Operations Research. New York: Springer-Verlag. pp. 496–526 and&nbsp, 581. ISBN 978-0-387-98705-7. MR 1756264.
• M. A. Goberna and M. A. López, Linear Semi-Infinite Optimization, Wiley, 1998.
• Hettich, R.; Kortanek, K. O. (1993). "Semi-infinite programming: Theory, methods, and applications". SIAM Review. 35 (3). pp. 380–429. doi:10.1137/1035089. JSTOR 2132425. MR 1234637.
• David Luenberger (1997). Optimization by Vector Space Methods. John Wiley & Sons. ISBN 0-471-18117-X.
• Rembert Reemtsen and Jan-J. Rückmann (Editors), Semi-Infinite Programming (Nonconvex Optimization and Its Applications). Springer, 1998, ISBN 0-7923-5054-5, 1998