Parametric programming: Difference between revisions

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'''Parametric programming''' denotesis 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, 872-876872–876.</ref><ref>Bemporad, Alberto; Morari, Manfred; Dua, Vivek; Pistikopoulos, Efstratios N. (2002) The explicit linear quadratic regulator for constrained systems. Automatica, 38 (1), 3-203–20.</ref>

== Notation ==

In general, the following optimization problem is considered

: <math>

\begin{align}

J^*(\theta) = & \min_{x\in\mathbb R^n}\; f(x,\theta) \\

& g(x,\theta)\leq 0, \\

& \theta \in \Theta \subset \mathbb R^m

\end{align}

</math>

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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>Gal, Tomas; Nedoma, Josef (1972) Multiparametric Linear Programming. Management Science, 18 (7), 406-422406–422.</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), 3976-39873976–3987.</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>