Parametric programming: Difference between revisions

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'''Parametric Programming''' denotes 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, 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.</ref><ref>Bemporad, Alberto; Morari, Manfred; Dua, Vivek; Pistikopoulos, Efstratios N. (2002) The explicit linear quadratic regulator for constrained systems. Automatica, 38 (1), 3-20.</ref>

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* If more than one parameter is present, i.e. <math>m > 1</math>, then it is often referred to as multiparametric programming problem<ref>Gal, Tomas; Nedoma, Josef (1972) Multiparametric Linear Programming. Management Science, 18 (7), 406-422.</ref>

* If integer variables are present, then the problem is referred to as (multi)parametric mixed-integer programming problem<ref>Dua, Vivek; Pistikopoulos, Efstratios N. (1999) Algorithms for the Solution of Multiparametric Mixed-Integer Nonlinear Optimization Problems. Industrial & Engineering Chemistry Research, 38 (10), 3976-3987.</ref>

* If constraints are [[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>

==References==

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== Parametric Programming ==

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