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'''Parametric programming''' is a type of [[mathematical optimization]], where the [[optimization problem]] is solved as a function of one or multiple [[parameters]].<ref>{{cite book |last1=Gal |first1=Tomas |year=1995 |title=Postoptimal Analyses, Parametric Programming, and Related Topics: Degeneracy, Multicriteria Decision Making, Redundancy |publisher=W. de Gruyter |___location=Berlin |isbn=978-3-11-087120-3 |edition=2nd}}</ref> Developed in parallel to [[sensitivity analysis]], its earliest mention can be found in a [[thesis]] from 1952.<ref>{{cite book |last1=Gal |first1=Tomas |last2=Greenberg |first2=Harvey J. |date=1997 |title=Advances in Sensitivity Analysis and Parametric Programming |publisher=Kluwer Academic Publishers |___location=Boston |doi=10.1007/978-1-4615-6103-3 |isbn=978-0-7923-9917-9 |series=International Series in Operations Research & Management Science |volume=6}}</ref> Since then, there have been considerable developments for the cases of multiple parameters, presence of [[integer]] variables as well as nonlinearities.
== Notation ==
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* If integer variables are present, then the problem is referred to as (multi)parametric mixed-integer programming problem<ref>{{cite journal|last1=Dua|first1=Vivek|last2=Pistikopoulos|first2=Efstratios N.|title=Algorithms for the Solution of Multiparametric Mixed-Integer Nonlinear Optimization Problems|journal=Industrial & Engineering Chemistry Research|date=October 1999|volume=38|issue=10|pages=3976–3987|doi=10.1021/ie980792u}}</ref>
* If constraints are [[Affine transformation|affine]], then additional classifications depending to nature of the objective function in (multi)parametric (mixed-integer) linear, quadratic and nonlinear programming problems is performed. Note that this generally assumes the constraints to be affine.<ref>{{cite book|last1=Pistikopoulos |first1=Efstratios N. |last2=Georgiadis |first2=Michael C. |last3=Dua |first3=Vivek |date=2007 |title=Multi-parametric Programming Theory, Algorithms and Applications |publisher=Wiley-VCH |___location=Weinheim |doi=10.1002/9783527631216 |isbn=9783527316915 }}</ref>
== Applications ==
The connection between parametric programming and [[model predictive control]] established in 2000 has contributed to an increased interest in the topic.<ref>{{cite book |last1=Bemporad |first1=Alberto |last2=Morari |first2=Manfred |last3=Dua |first3=Vivek |last4=Pistikopoulos |first4=Efstratios N. |year=2000 |chapter=The explicit solution of model predictive control via multiparametric quadratic programming |title=Proceedings of the 2000 American Control Conference |pages=872 |doi=10.1109/ACC.2000.876624 |isbn=0-7803-5519-9 }}</ref><ref>{{cite journal |last1=Bemporad |first1=Alberto |last2=Morari |first2=Manfred |last3=Dua |first3=Vivek |last4=Pistikopoulos |first4=Efstratios N. |title=The explicit linear quadratic regulator for constrained systems |journal=Automatica |date=January 2002 |volume=38 |issue=1 |pages=3–20 |citeseerx=10.1.1.67.2946 |doi=10.1016/S0005-1098(01)00174-1}}</ref>. Parametric programming supplies the idea that optimization problems can be parametrized as functions that can be (similar to a lookup table). This in turns allows the optimization algorithms in optimal controllers to be implemented as pre-computed (off-line) mathematical functions, which may in some cases be simpler and faster to evaluate than solving a full optimization problem on-line. This also opens up the possibility of creating optimal controllers on chips (MPC on chip<ref>https://www.researchgate.net/publication/223477621_MPC_on_a_chip-Recent_advances_on_the_application_of_multi-parametric_model-based_control</ref>). However, the off-line parametrization of optimal solutions runs into the curse of dimensionality as the number of possible solutions grows with the dimensionality and number of constraints in the problem.
==References==
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