We present four examples.
Example 1
We define the model \(\max \{cx | Ax \leq b\}\) (could also be minimized) with three variables:
A <- matrix( c(
3, 2, 5,
2, 1, 1,
1, 1, 3,
5, 2, 4
), nc = 3, byrow = TRUE)
b <- c(55, 26, 30, 57)
obj <- c(20, 10, 15)
We load the preferred view angle for the RGL window:
view <- matrix( c(-0.412063330411911, -0.228006735444069, 0.882166087627411, 0, 0.910147845745087,
-0.0574885793030262, 0.410274744033813, 0, -0.042830865830183, 0.97196090221405,
0.231208890676498, 0, 0, 0, 0, 1), nc = 4)
The LP polytope:
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Note in the html file you can zoom/turn/twist the figure with your mouse (rglwidget).
The ILP model with LP and ILP faces:
loadView(v = view)
rgl::mfrow3d(nr = 1, nc = 2, sharedMouse = TRUE)
plotPolytope(A, b, faces = c("c","c","c"), type = c("i","i","i"), plotOptimum = TRUE, obj = obj,
argsTitle3d = list(main = "With LP faces"), argsPlot3d = list(box = F, axes = T) )
plotPolytope(A, b, faces = c("i","i","i"), type = c("i","i","i"), plotFeasible = FALSE, obj = obj,
argsTitle3d = list(main = "ILP faces") )
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Let us have a look at some MILP models (we use static images). MILP model with variable 1 and 3 integer:
loadView(v = view, close = T, zoom = 0.75)
plotPolytope(A, b, faces = c("c","c","c"), type = c("i","c","i"), plotOptimum = TRUE, obj = obj)
MILP model with variable 2 and 3 integer:
loadView(v = view, zoom = 0.75)
plotPolytope(A, b, faces = c("c","c","c"), type = c("c","i","i"), plotOptimum = TRUE, obj = obj)
MILP model with variable 1 and 2 integer:
loadView(v = view, zoom = 0.75)
plotPolytope(A, b, faces = c("c","c","c"), type = c("i","i","c"), plotOptimum = TRUE, obj = obj)
MILP model with variable 1 integer:
MILP model with variable 2 integer:
MILP model with variable 3 integer:
Example 2
We define the model:
A <- matrix( c(
1, 1, 1,
3, 0, 1
), nc = 3, byrow = TRUE)
b <- c(10, 24)
obj <- c(20, 10, 15)
We load the preferred view angle for the RGL window:
view <- matrix( c(-0.812462985515594, -0.029454167932272, 0.582268416881561, 0, 0.579295456409454,
-0.153386667370796, 0.800555109977722, 0, 0.0657325685024261, 0.987727105617523,
0.14168381690979, 0, 0, 0, 0, 1), nc = 4)
The LP polytope:
You must enable Javascript to view this page properly.
Note in the html file you can zoom/turn/twist the figure with your mouse (rglwidget).
The ILP model with LP and ILP faces:
loadView(v = view)
rgl::mfrow3d(nr = 1, nc = 2, sharedMouse = TRUE)
plotPolytope(A, b, faces = c("c","c","c"), type = c("i","i","i"), plotOptimum = TRUE, obj = obj,
argsTitle3d = list(main = "With LP faces"), argsPlot3d = list(box = F, axes = T) )
plotPolytope(A, b, faces = c("i","i","i"), type = c("i","i","i"), plotFeasible = FALSE, obj = obj,
argsTitle3d = list(main = "ILP faces") )
You must enable Javascript to view this page properly.
Let us have a look at some MILP models (we use static images). MILP model with variable 1 and 3 integer:
loadView(v = view, close = T, zoom = 0.75)
plotPolytope(A, b, faces = c("c","c","c"), type = c("i","c","i"), plotOptimum = TRUE, obj = obj)
MILP model with variable 2 and 3 integer:
loadView(v = view, zoom = 0.75)
plotPolytope(A, b, faces = c("c","c","c"), type = c("c","i","i"), plotOptimum = TRUE, obj = obj)
MILP model with variable 1 and 2 integer:
loadView(v = view, zoom = 0.75)
plotPolytope(A, b, faces = c("c","c","c"), type = c("i","i","c"), plotOptimum = TRUE, obj = obj)
MILP model with variable 1 integer:
MILP model with variable 2 integer:
MILP model with variable 3 integer:
Example 3
We define the model:
A <- matrix( c(
-1, 1, 0,
1, 4, 0,
2, 1, 0,
3, -4, 0,
0, 0, 4
), nc = 3, byrow = TRUE)
b <- c(5, 45, 27, 24, 10)
obj <- c(5, 45, 15)
We load the preferred view angle for the RGL window:
view <- matrix( c(0.976349174976349, -0.202332556247711, 0.0761845782399178, 0, 0.0903248339891434,
0.701892614364624, 0.706531345844269, 0, -0.196427255868912, -0.682940244674683,
0.703568696975708, 0, 0, 0, 0, 1), nc = 4)
The LP polytope:
You must enable Javascript to view this page properly.
Note in the html file you can zoom/turn/twist the figure with your mouse (rglwidget).
The ILP model with LP and ILP faces:
loadView(v = view)
rgl::mfrow3d(nr = 1, nc = 2, sharedMouse = TRUE)
plotPolytope(A, b, faces = c("c","c","c"), type = c("i","i","i"), plotOptimum = TRUE, obj = obj,
argsTitle3d = list(main = "With LP faces"), argsPlot3d = list(box = F, axes = T) )
plotPolytope(A, b, faces = c("i","i","i"), type = c("i","i","i"), plotFeasible = FALSE, obj = obj,
argsTitle3d = list(main = "ILP faces") )
You must enable Javascript to view this page properly.
Let us have a look at some MILP models (we use static images). MILP model with variable 1 and 3 integer:
loadView(v = view, close = T, zoom = 0.75)
plotPolytope(A, b, faces = c("c","c","c"), type = c("i","c","i"), plotOptimum = TRUE, obj = obj)
MILP model with variable 2 and 3 integer:
loadView(v = view, zoom = 0.75)
plotPolytope(A, b, faces = c("c","c","c"), type = c("c","i","i"), plotOptimum = TRUE, obj = obj)
MILP model with variable 1 and 2 integer:
loadView(v = view, zoom = 0.75)
plotPolytope(A, b, faces = c("c","c","c"), type = c("i","i","c"), plotOptimum = TRUE, obj = obj)
MILP model with variable 1 integer:
MILP model with variable 2 integer:
MILP model with variable 3 integer:
Example 4
We define the model:
Ab <- matrix( c(
1, 1, 2, 5,
2, -1, 0, 3,
-1, 2, 1, 3,
0, -3, 5, 2
# 0, 1, 0, 4,
# 1, 0, 0, 4
), nc = 4, byrow = TRUE)
A <- Ab[,1:3]
b <- Ab[,4]
obj = c(1,1,3)
We load the preferred view angle for the RGL window:
view <- matrix( c(-0.452365815639496, -0.446501553058624, 0.77201122045517, 0, 0.886364221572876,
-0.320795893669128, 0.333835482597351, 0, 0.0986008867621422, 0.835299551486969,
0.540881276130676, 0, 0, 0, 0, 1), nc = 4)
The LP polytope:
You must enable Javascript to view this page properly.
Note in the html file you can zoom/turn/twist the figure with your mouse (rglwidget).
The ILP model with LP and ILP faces:
loadView(v = view)
rgl::mfrow3d(nr = 1, nc = 2, sharedMouse = TRUE)
plotPolytope(A, b, faces = c("c","c","c"), type = c("i","i","i"), plotOptimum = TRUE, obj = obj,
argsTitle3d = list(main = "With LP faces"), argsPlot3d = list(box = F, axes = T) )
plotPolytope(A, b, faces = c("i","i","i"), type = c("i","i","i"), plotFeasible = FALSE, obj = obj,
argsTitle3d = list(main = "ILP faces") )
You must enable Javascript to view this page properly.
Let us have a look at some MILP models (we use static images). MILP model with variable 1 and 3 integer:
MILP model with variable 2 and 3 integer:
MILP model with variable 1 and 2 integer:
MILP model with variable 1 integer:
MILP model with variable 2 integer:
MILP model with variable 3 integer:
Making plots in LaTeX
It is possible to include the 3D plots in LaTeX using asymptote. First, save the plot as a LaTeX file:
loadView(v = view, close = F, zoom = 0.75)
plotPolytope(A, b, plotOptimum = TRUE, obj = obj, labels = "n",
argsTitle3d=list(xlab = '$x_1$', ylab='$y_1$', zlab='$y_2$'))
rgl::writeASY(outtype = "pdflatex", prc = TRUE, title = "polytope") # prc = T(interactive) / F(static)
Next, you include the file in you LaTeX document (test.tex):
\documentclass{article}
\usepackage{a4wide}
\usepackage[pdftex]{graphicx}
\usepackage{asymptote}
\begin{document}
Example of how to include a 3D plot in a \LaTeX document:
\input{polytope}
\end{document}
Now, you have to run a sequence of commands:
For some plots I have experienced an error when run asy and the light function. In this case I have just removed the lines defining the light function in polytope.tex.