Maximizing the Volume of a Box

Introduction

This example is adapted from Boyd, Kim, Vandenberghe, and Hassibi, “A Tutorial on Geometric Programming”.

In this example, we maximize the volume of a box with height $h$, width $w$, and depth $d$, with limits on the wall area $2(hw+hd)$ and the floor area $wd$, subject to bounds on the aspect ratios $h/w$ and $d/w$. The optimization problem is

$$\begin{array}{ll}\text{maximize}& hwd\\ \text{subject to}& 2(hw+hd)\le A_{\text{wall}},\\ & wd\le A_{\text{flr}},\\ & \alpha \le h/w\le \beta ,\\ & \gamma \le d/w\le \delta .\end{array}$$

Problem Formulation

# Problem data
A_wall <- 100
A_flr <- 10
alpha <- 0.5
beta <- 2
gamma <- 0.5
delta <- 2

h <- Variable(pos = TRUE)
w <- Variable(pos = TRUE)
d <- Variable(pos = TRUE)

volume <- h * w * d
wall_area <- 2 * (h * w + h * d)
flr_area <- w * d
hw_ratio <- h / w
dw_ratio <- d / w

constraints <- list(
  wall_area <= A_wall,
  flr_area <= A_flr,
  hw_ratio >= alpha,
  hw_ratio <= beta,
  dw_ratio >= gamma,
  dw_ratio <= delta
)

problem <- Problem(Maximize(volume), constraints)
cat("Is problem DGP?", is_dgp(problem), "\n")

Is problem DGP? TRUE 

Solution

result <- psolve(problem, gp = TRUE)
check_solver_status(problem)
cat("Optimal value (volume):", result, "\n")
cat("h:", value(h), "\n")
cat("w:", value(w), "\n")
cat("d:", value(d), "\n")

Optimal value (volume): 77.45967 
h: 7.745967 
w: 3.872983 
d: 2.581989 

Sensitivity Analysis via Dual Values

The dual values provide sensitivity information. A 1% increase in the allowed wall area should yield approximately a proportional increase in the maximum volume value.

cat("Dual value for wall area constraint:", dual_value(constraints[[1]]), "\n")
cat("Dual value for floor area constraint:", dual_value(constraints[[2]]), "\n")

Dual value for wall area constraint: 0.8333587 
Dual value for floor area constraint: 0.6666413 

The dual value for the wall area constraint is approximately 0.83, meaning a 1% increase in allowed wall space yields approximately a 0.83% increase in the maximum volume. The dual value for the floor area constraint is approximately 0.67, meaning a 1% increase in allowed floor space yields approximately a 0.67% increase in maximum volume.

Session Info

R version 4.5.2 (2025-10-31)
Platform: aarch64-apple-darwin20
Running under: macOS Tahoe 26.3

Matrix products: default
BLAS:   /Library/Frameworks/R.framework/Versions/4.5-arm64/Resources/lib/libRblas.0.dylib 
LAPACK: /Library/Frameworks/R.framework/Versions/4.5-arm64/Resources/lib/libRlapack.dylib;  LAPACK version 3.12.1

locale:
[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8

time zone: America/Los_Angeles
tzcode source: internal

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base     

other attached packages:
[1] CVXR_1.8.1

loaded via a namespace (and not attached):
 [1] slam_0.1-55       cli_3.6.5         knitr_1.51        ECOSolveR_0.6.1  
 [5] rlang_1.1.7       xfun_0.56         clarabel_0.11.2   otel_0.2.0       
 [9] gurobi_13.0-1     Rglpk_0.6-5.1     highs_1.12.0-3    cccp_0.3-3       
[13] scs_3.2.7         S7_0.2.1          jsonlite_2.0.0    Rcplex_0.3-8     
[17] backports_1.5.0   rprojroot_2.1.1   htmltools_0.5.9   Rmosek_11.1.1    
[21] gmp_0.7-5.1       piqp_0.6.2        rmarkdown_2.30    grid_4.5.2       
[25] evaluate_1.0.5    fastmap_1.2.0     yaml_2.3.12       compiler_4.5.2   
[29] codetools_0.2-20  htmlwidgets_1.6.4 Rcpp_1.1.1        here_1.0.2       
[33] osqp_1.0.0        lattice_0.22-9    digest_0.6.39     checkmate_2.3.4  
[37] Matrix_1.7-4      tools_4.5.2      

References

• Boyd, S., Kim, S.-J., Vandenberghe, L., Hassibi, A. (2007). A Tutorial on Geometric Programming. Optimization and Engineering, 8(1), 67–127.