parametric programming

In general, the following optimization problem is considered

: $$

\begin{align}

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

& \text{subject to } g(x,\theta)\leq 0.\\

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

\end{align}

where $x$ is the optimization variable, $\backslash theta$ are the parameters, $f(x,\backslash theta)$ is the objective function and $g(x,\backslash theta)$ denote the constraints. $J^*$ denotes a function whose output is the optimal value of the objective function $f$. The set $\backslash Theta$ is generally referred to as parameter space.

The optimal value (i.e. result of solving the optimization problem) is obtained by evaluating the function with an argument $\backslash theta$.

Depending on the nature of $f(x,\backslash theta)$ and $g(x,\backslash theta)$ 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. $m\; >\; 1$, then it is often referred to as multiparametric programming problem{{cite journal|last1=Gal|first1=Tomas|last2=Nedoma|first2=Josef|title=Multiparametric Linear Programming|journal=Management Science|date=1972|volume=18|issue=7|pages=406–422|jstor=2629358|doi=10.1287/mnsc.18.7.406}}
• If integer variables are present, then the problem is referred to as (multi)parametric mixed-integer programming problem{{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}}
• 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.{{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 }}

== Applications ==

The connection between parametric programming and model predictive control for process manufacturing, established in 2000, has contributed to an increased interest in the topic.{{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 |s2cid=1068816 }}{{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}} Parametric programming supplies the idea that optimization problems can be parametrized as functions that can be evaluated (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[https://www.researchgate.net/publication/223477621_MPC_on_a_chip-Recent_advances_on_the_application_of_multi-parametric_model-based_control MPC on a chip—Recent advances on the application of multi-parametric model-based control | Request PDF]). 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.{{Citation needed|date=February 2023}}

Parametric programming in the context of CNC (computer numerical control) is defining part-cutting cycles in terms of variables with reassignable values rather than via hardcoded/hardwired instances. An archetypically simple example is writing a G-code program to machine a family of washers: there is often no need to write 15 programs for 15 members of the family with various hole diameters, outer diameters, thicknesses, and materials, when it is practical instead to write 1 program that calls various variables and reads their current values from a table of assignments. The program then instructs the machine slides and spindles to move to various positions at various velocities, accordingly, addressing not only the sizes of the part (i.e., OD, ID, thickness) but also even the speeds and feeds needed for any given material (e.g., low-carbon steel, high-carbon steel; stainless steel of whichever grade; bronze, brass, or aluminum of whichever grade; polymer of whichever type). Custom Macros are often used in such programming.{{cite web |last1=Lynch |first1=Mike |date=2023-05-12 |title=5 Reasons Why You Should Know How to Write Custom Macros. Custom macros enhance what can be done in G-code programs, giving users the ability to code operations that were previously not possible |url=https://www.mmsonline.com/articles/5-reasons-why-you-should-know-how-to-write-custom-macros |website=www.mmsonline.com |access-date=2023-05-20 |language=en }}

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Category:Optimization algorithms and methods