Largest Ball in a Polyhedron in 2D
Problem
The following is a problem from Boyd and Vandenberghe (2004), section 4.3.1.
Find the largest Euclidean ball (i.e. its center and radius) that lies in a polyhedron described by affine inequalites:
\[ P = {x : a_i'*x <= b_i, i=1,...,m} \]
where x is in \({\mathbf R}^2\).
We define variables that determine the polyhedron.
a1 <- matrix(c(2,1))
a2 <- matrix(c(2,-1))
a3 <- matrix(c(-1,2))
a4 <- matrix(c(-1,-2))
b <- rep(1,4)
Next, we formulate the CVXR
problem.
r <- Variable(name = "radius")
x_c <- Variable(2, name = "center")
obj <- Maximize(r)
constraints <- list(
t(a1) %*% x_c + p_norm(a1, 2) * r <= b[1],
t(a2) %*% x_c + p_norm(a2, 2) * r <= b[2],
t(a3) %*% x_c + p_norm(a3, 2) * r <= b[3],
t(a4) %*% x_c + p_norm(a4, 2) * r <= b[4]
)
p <- Problem(obj, constraints)
All that remains is to solve the problem and read off the solution.
result <- solve(p)
radius <- result$getValue(r)
center <- result$getValue(x_c)
cat(sprintf("The radius is %0.5f for an area %0.5f\n", radius, pi * radius^2))
## The radius is 0.44721 for an area 0.62832
A Plot
ggplot() +
geom_abline(slope = -a1[1] / a1[2], intercept = b[1] / a1[2]) +
geom_abline(slope = -a2[1] / a2[2], intercept = b[2] / a2[2]) +
geom_abline(slope = -a3[1] / a3[2], intercept = b[3] / a3[2]) +
geom_abline(slope = -a4[1] / a4[2], intercept = b[4] / a4[2]) +
geom_circle(mapping = aes(x0 = center[1], y0 = center[2], r = radius), color = "blue") +
geom_point(mapping = aes(x = center[1], y = center[2]), color = "red", size = 2) +
geom_line(mapping = aes(x = c(center[1], center[1] - radius), y = c(center[2], center[2])),
arrow = arrow(length = unit(0.03, "npc"), ends = "first", type = "closed"),
color = "brown") +
annotate("text", x = -0.2, y = 0.04, label = sprintf("r = %0.5f", radius)) +
labs(x = "x", y = "y") +
xlim(-1, 1) + ylim(-1, 1)
Session Info
sessionInfo()
## R version 4.0.2 (2020-06-22)
## Platform: x86_64-apple-darwin19.5.0 (64-bit)
## Running under: macOS Catalina 10.15.7
##
## Matrix products: default
## BLAS/LAPACK: /usr/local/Cellar/openblas/0.3.10_1/lib/libopenblasp-r0.3.10.dylib
##
## locale:
## [1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
##
## attached base packages:
## [1] stats graphics grDevices datasets utils methods base
##
## other attached packages:
## [1] ggforce_0.3.2 ggplot2_3.3.2 CVXR_1.0-9
##
## loaded via a namespace (and not attached):
## [1] tidyselect_1.1.0 xfun_0.15 slam_0.1-47 purrr_0.3.4
## [5] lattice_0.20-41 Rmosek_9.2.3 colorspace_1.4-1 vctrs_0.3.2
## [9] generics_0.0.2 htmltools_0.5.0 yaml_2.2.1 gmp_0.6-0
## [13] rlang_0.4.7 pillar_1.4.6 glue_1.4.1 Rmpfr_0.8-1
## [17] withr_2.2.0 Rcplex_0.3-3 tweenr_1.0.1 bit64_0.9-7
## [21] lifecycle_0.2.0 stringr_1.4.0 munsell_0.5.0 blogdown_0.19
## [25] gtable_0.3.0 gurobi_9.0.3.1 codetools_0.2-16 evaluate_0.14
## [29] labeling_0.3 knitr_1.28 cccp_0.2-4 Rcpp_1.0.5
## [33] scales_1.1.1 osqp_0.6.0.3 farver_2.0.3 bit_1.1-15.2
## [37] digest_0.6.25 stringi_1.4.6 bookdown_0.19 dplyr_1.0.0
## [41] polyclip_1.10-0 grid_4.0.2 Rglpk_0.6-4 tools_4.0.2
## [45] magrittr_1.5 tibble_3.0.3 crayon_1.3.4 pkgconfig_2.0.3
## [49] ellipsis_0.3.1 MASS_7.3-51.6 rcbc_0.1.0.9001 Matrix_1.2-18
## [53] assertthat_0.2.1 rmarkdown_2.3 R6_2.4.1 compiler_4.0.2
Source
References
Boyd, S., and L. Vandenberghe. 2004. Convex Optimization. Cambridge University Press.