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:

\min _{x\in X}\;\;f(x)

{\text{subject to: }}

g(x,y)\leq 0,\;\;\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.

gollark: I kind of prefer g, but I guess if it's a giveway I wouldn't mind h too.

gollark: "we like to look at things as if it all for us, as if something is so grand about us, in truth we can only be grand if we so choose, and can properly attain it. but if we can, then what ever IT is was never for us. and thus only a blip in time, our memory and all of action erased as if it was never there, what is so special about us? nothing really." sounds pretty nihilist.

gollark: No, seems like rebranded nihilism.

gollark: ...

gollark: That sounds like nihilism.

References

- 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 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): 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 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): 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

External links

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[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 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): 380–429. doi:10.1137/1035089. JSTOR 2132425. MR 1234637.

[2] 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 581. ISBN 978-0-387-98705-7. MR 1756264.

[3] M. A. Goberna and M. A. López, Linear Semi-Infinite Optimization, Wiley, 1998.

[4] Hettich, R.; Kortanek, K. O. (1993). "Semi-infinite programming: Theory, methods, and applications". SIAM Review. 35 (3): 380–429. doi:10.1137/1035089. JSTOR 2132425. MR 1234637.