Parametric programming

This sandbox is in the article namespace. Either move this page into your userspace, or remove the {{User sandbox}} template. Parametric Programming denotes a type of mathematical optimization, where the optimization problem is solved as a function of one or multiple parameters.^[1] Developed in parallel to sensitivity analysis, its earliest mention can be found in a thesis from 1952.^[2] Since then, there have been considerable developments for the cases of multiple parameters, presence of integer variables as well as nonlinearities. In particular the connection between parametric programming and model predictive control established in 2000 has contributed to an increased interest in the topic.^[3]^[4]

Notation

In general, the following optimization problem is considered

${\begin{aligned}J^{*}(\theta )=&\min _{x\in \mathbb {R} ^{n}}\;f(x,\theta )\\&g(x,\theta )\leq 0,\end{aligned}}$

where $x$ is the optimization variable, $\theta \in \mathbb {R} ^{m}$ are the paramters, $f(x,\theta )$ is the objective function and $g(x,\theta )$ denote the constraints.

References

^ Tomas Gal. Postoptimal analyses, parametric programming, and related topics: Degeneracy, multicriteria decision making, redundancy. Berlin : W. de Gruyter, 1995.
^ T Gal, H.J. Greenberg Advances in Sensitivity Analysis and Parametric Programming. Springer, 1997.
^ Bemporad, A.; Morari, M.; Dua, V.; Pistikopoulos, E. N. (2000) The explicit solution of model predictive control via multiparametric quadratic programming. Proceedings of the American Control, vol. 2, 872-876.
^ Bemporad, Alberto; Morari, Manfred; Dua, Vivek; Pistikopoulos, Efstratios N. (2002) The explicit linear quadratic regulator for constrained systems. Automatica, 38 (1), 3-20.

Parametric Programming

[1] Tomas Gal. Postoptimal analyses, parametric programming, and related topics: Degeneracy, multicriteria decision making, redundancy. Berlin : W. de Gruyter, 1995.

[2] T Gal, H.J. Greenberg Advances in Sensitivity Analysis and Parametric Programming. Springer, 1997.

[3] Bemporad, A.; Morari, M.; Dua, V.; Pistikopoulos, E. N. (2000) The explicit solution of model predictive control via multiparametric quadratic programming. Proceedings of the American Control, vol. 2, 872-876.

[4] Bemporad, Alberto; Morari, Manfred; Dua, Vivek; Pistikopoulos, Efstratios N. (2002) The explicit linear quadratic regulator for constrained systems. Automatica, 38 (1), 3-20.

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