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 4.2.1 (2022-06-23)
## Platform: x86_64-apple-darwin21.6.0 (64-bit)
## Running under: macOS Ventura 13.0
## 
## Matrix products: default
## BLAS:   /usr/local/Cellar/openblas/0.3.21/lib/libopenblasp-r0.3.21.dylib
## LAPACK: /usr/local/Cellar/r/4.2.1_4/lib/R/lib/libRlapack.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_0.4.1 ggplot2_3.3.6 CVXR_1.0-11  
## 
## loaded via a namespace (and not attached):
##  [1] tidyselect_1.2.0 xfun_0.34        bslib_0.4.0      slam_0.1-50     
##  [5] lattice_0.20-45  Rmosek_10.0.25   colorspace_2.0-3 vctrs_0.5.0     
##  [9] generics_0.1.3   htmltools_0.5.3  yaml_2.3.6       gmp_0.6-6       
## [13] utf8_1.2.2       rlang_1.0.6      jquerylib_0.1.4  pillar_1.8.1    
## [17] glue_1.6.2       Rmpfr_0.8-9      withr_2.5.0      DBI_1.1.3       
## [21] Rcplex_0.3-5     tweenr_2.0.2     bit64_4.0.5      lifecycle_1.0.3 
## [25] stringr_1.4.1    munsell_0.5.0    blogdown_1.13    gtable_0.3.1    
## [29] gurobi_9.5-2     codetools_0.2-18 evaluate_0.17    labeling_0.4.2  
## [33] knitr_1.40       fastmap_1.1.0    cccp_0.2-9       fansi_1.0.3     
## [37] highr_0.9        Rcpp_1.0.9       scales_1.2.1     cachem_1.0.6    
## [41] osqp_0.6.0.5     jsonlite_1.8.3   farver_2.1.1     bit_4.0.4       
## [45] digest_0.6.30    stringi_1.7.8    bookdown_0.29    dplyr_1.0.10    
## [49] Rglpk_0.6-4      polyclip_1.10-4  grid_4.2.1       cli_3.4.1       
## [53] tools_4.2.1      magrittr_2.0.3   sass_0.4.2       tibble_3.1.8    
## [57] pkgconfig_2.0.3  rcbc_0.1.0.9001  MASS_7.3-58.1    Matrix_1.5-1    
## [61] assertthat_0.2.1 rmarkdown_2.17   R6_2.5.1         compiler_4.2.1

Source

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