Parametric programming: Difference between revisions

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{{orphan|date=February 2015}}

'''Parametric programming''' is a type of [[mathematical optimization]], where the [[optimization problem]] is solved as a function of one or multiple [[parameters]].<ref>Tomas Gal. Postoptimal analyses, parametric programming, and related topics: Degeneracy, multicriteria decision making, redundancy. Berlin : W. de Gruyter, 1995.</ref> Developed in parallel to [[sensitivity analysis]], its earliest mention can be found in a [[thesis]] from 1952.<ref>T Gal, H.J. Greenberg Advances in Sensitivity Analysis and Parametric Programming. Springer, 1997.</ref> 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.<ref>Bemporad,{{cite A.;journal Morari, M|doi=10.; Dua, V1109/ACC.; Pistikopoulos, E. N. (2000) The explicit solution of model predictive control via multiparametric quadratic programming. Proceedings of the American Control, vol. 2, 872–876.876624}}</ref><ref>{{cite journal |last1=Bemporad, |first1=Alberto; |last2=Morari, |first2=Manfred; |last3=Dua, |first3=Vivek; |last4=Pistikopoulos, |first4=Efstratios N. (2002) |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>

== Notation ==

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Depending on the nature of <math>f(x,\theta)</math> and <math>g(x,\theta)</math> and whether the optimization problem features integer variables, parametric programming problems are classified into different sub-classes:

* If more than one parameter is present, i.e. <math>m > 1</math>, then it is often referred to as multiparametric programming problem<ref>{{cite journal|last1=Gal, |first1=Tomas; |last2=Nedoma, |first2=Josef (1972) |title=Multiparametric Linear Programming. |journal=Management Science, |date=1972|volume=18 (|issue=7), |pages=406–422.|jstor=2629358}}</ref>

* 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. (1999) |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>Pistikopoulos, Efstratios N.; Georgiadis, Michael C.; Dua, Vivek Multi-parametric programming: Theory, algorithms and applications. Weinheim, Wiley-VCH, 2007.</ref>