Plotting polytopes in 2D

With gMOIP you can make 2D plots of the the polytope/feasible region/solution space of a linear programming (LP), integer linear programming (ILP) model, or mixed integer linear programming (MILP) model. This vignette gives examples on how to make plots given a model with two variables.

First we load the package:

library(gMOIP)

We define the model \(\max \{cx | Ax \leq b\}\) (could also be minimized) with 2 variables:

A <- matrix(c(-3,2,2,4,9,10), ncol = 2, byrow = TRUE)
b <- c(3,27,90)
obj <- c(7.75, 10)  # coefficients c

Plots are created using function plotPolytope which outputs a ggplot2 object.

LP model

Let us consider different plots of the polytope of the LP model with non-negative variables (\(x \in \mathbb{R}_0, x \geq 0\)):

# The polytope with the corner points
p1 <- plotPolytope(
   A,
   b,
   obj,
   type = rep("c", ncol(A)),
   crit = "max",
   faces = rep("c", ncol(A)),
   plotFaces = TRUE,
   plotFeasible = TRUE,
   plotOptimum = FALSE,
   labels = NULL
) + ggplot2::ggtitle("Feasible region only")

p2 <- plotPolytope(
   A,
   b,
   obj,
   type = rep("c", ncol(A)),
   crit = "max",
   faces = rep("c", ncol(A)),
   plotFaces = TRUE,
   plotFeasible = TRUE,
   plotOptimum = TRUE,
   labels = "coord"
) + ggplot2::ggtitle("Solution LP max")

p3 <- plotPolytope(
   A,
   b,
   obj,
   type = rep("c", ncol(A)),
   crit = "min",
   faces = rep("c", ncol(A)),
   plotFaces = TRUE,
   plotFeasible = TRUE,
   plotOptimum = TRUE,
   labels = "n"
) + ggplot2::ggtitle("Solution LP min")

p4 <- plotPolytope(
   A,
   b,
   obj,
   type = rep("c", ncol(A)),
   crit = "max",
   faces = rep("c", ncol(A)),
   plotFaces = TRUE,
   plotFeasible = TRUE,
   plotOptimum = TRUE,
   labels = "coord"
) + ggplot2::xlab("x") + ggplot2::ylab("y") + ggplot2::ggtitle("Solution (max) with other axis labels")

gridExtra::grid.arrange(p1, p2, p3, p4, nrow = 2)

You may also consider a LP model with no non-negativity constraints:

A <- matrix(c(-3, 2, 2, 4, 9, 10, 1, -2), ncol = 2, byrow = TRUE)
b <- c(3, 27, 90, 2)
obj <- c(7.75, 10)
plotPolytope(
   A,
   b,
   obj,
   type = rep("c", ncol(A)),
   nonneg = rep(FALSE, ncol(A)),
   crit = "max",
   faces = rep("c", ncol(A)),
   plotFaces = TRUE,
   plotFeasible = TRUE,
   plotOptimum = FALSE,
   labels = NULL
)

Note The package don’t plot feasible regions that are unbounded e.g if we drop the second and third constraint we get the wrong plot:

A <- matrix(c(-3,2), ncol = 2, byrow = TRUE)
b <- c(3)
obj <- c(7.75, 10)
# Wrong plot
plotPolytope(
   A,
   b,
   obj,
   type = rep("c", ncol(A)),
   crit = "max",
   faces = rep("c", ncol(A)),
   plotFaces = TRUE,
   plotFeasible = TRUE,
   plotOptimum = FALSE,
   labels = NULL
)

One solution is to add a bounding box and check if the bounding box is binding

A <- rbind(A, c(1,0), c(0,1))
b <- c(b, 10, 10)
plotPolytope(
   A,
   b,
   obj,
   type = rep("c", ncol(A)),
   crit = "max",
   faces = rep("c", ncol(A)),
   plotFaces = TRUE,
   plotFeasible = TRUE,
   plotOptimum = FALSE,
   labels = NULL
)

You may also use e.g lpsolve to check if the solution is unbounded.

ILP model

If we add integer constraints to the model (\(x\in\mathbb{Z}\)) you may view the feasible region different ways:

A <- matrix(c(-3,2,2,4,9,10), ncol = 2, byrow = TRUE)
b <- c(3,27,90)
obj <- c(7.75, 10)

p1 <- plotPolytope(
   A,
   b,
   obj,
   type = rep("i", ncol(A)),
   crit = "max",
   faces = rep("c", ncol(A)),
   plotFaces = TRUE,
   plotFeasible = TRUE,
   plotOptimum = FALSE,
   labels = "n"
) + ggplot2::ggtitle("Relaxed region with LP faces")

p2 <- plotPolytope(
   A,
   b,
   obj,
   type = rep("i", ncol(A)),
   crit = "max",
   faces = rep("i", ncol(A)),
   plotFaces = TRUE,
   plotFeasible = TRUE,
   plotOptimum = FALSE,
   labels = "n"
) + ggplot2::ggtitle("Relaxed region with IP faces")

p3 <- plotPolytope(
   A,
   b,
   obj,
   type = rep("i", ncol(A)),
   crit = "max",
   faces = rep("c", ncol(A)),
   plotFaces = TRUE,
   plotFeasible = TRUE,
   plotOptimum = TRUE,
   labels = "n"
) + ggplot2::ggtitle("Optimal solution (max)")

p4 <- plotPolytope(
   A,
   b,
   obj = c(-3, 3),
   type = rep("i", ncol(A)),
   crit = "max",
   faces = rep("i", ncol(A)),
   plotFaces = TRUE,
   plotFeasible = TRUE,
   plotOptimum = TRUE,
   labels = "n"
) + ggplot2::ggtitle("Other objective (min)")

gridExtra::grid.arrange(p1, p2, p3, p4, nrow = 2)

MILP model

Finally, let us have a look at a MILP model:

A <- matrix(c(-3,2,2,4,9,10), ncol = 2, byrow = TRUE)
b <- c(3,27,90)
obj <- c(7.75, 10)

p1 <- plotPolytope(
   A,
   b,
   obj,
   type = c("c", "i"),
   crit = "max",
   faces = c("c", "c"),
   plotFaces =  TRUE,
   plotFeasible = TRUE,
   plotOptimum = TRUE,
   labels = "n"
) + ggplot2::ggtitle("Second coordinate integer (LP faces)")

p2 <- plotPolytope(
   A,
   b,
   obj,
   type = c("c", "i"),
   crit = "max",
   faces = c("c", "i"),
   plotFaces =  TRUE,
   plotFeasible = TRUE,
   plotOptimum = TRUE,
   labels = "coord"
) + ggplot2::ggtitle("Second coordinate integer (MILP faces)")

p3 <- plotPolytope(
   A,
   b,
   obj,
   type = c("i", "c"),
   crit = "max",
   faces = c("c", "c"),
   plotFaces = TRUE,
   plotFeasible = TRUE,
   plotOptimum = TRUE,
   labels = "n"
) + ggplot2::ggtitle("First coordinate integer (LP faces)")

p4 <- plotPolytope(
   A,
   b,
   obj,
   type = c("i", "c"),
   crit = "max",
   faces = c("i", "c"),
   plotFaces = TRUE,
   plotFeasible = TRUE,
   plotOptimum = TRUE,
   labels = "coord"
) + ggplot2::ggtitle("First coordinate integer (MILP faces)")

gridExtra::grid.arrange(p1, p2, p3, p4, nrow = 2)

Saving plots to LaTeX

If you write a paper using LaTeX, you may create a TikZ file of the plot for LaTeX using

library(tikzDevice)
tikz(file = "plot_polytope.tex", standAlone=F, width = 7, height = 6)
plotPolytope(
   A,
   b,
   obj,
   type = rep("i", ncol(A)),
   crit = "max",
   faces = rep("c", ncol(A)),
   plotFaces = TRUE,
   plotFeasible = TRUE,
   plotOptimum = TRUE,
   labels = "n",
   latex = TRUE
)
dev.off()