Maximizing the volume of a box

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

In this example, we maximize the shape 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 \(w/d\). The optimization problem is

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

import cvxpy as cp

# Problem data.
A_wall = 100
A_flr = 10
alpha = 0.5
beta = 2
gamma = 0.5
delta = 2

h = cp.Variable(pos=True, name="h")
w = cp.Variable(pos=True, name="w")
d = cp.Variable(pos=True, name="d")

volume = h * w * d
wall_area = 2 * (h * w + h * d)
flr_area = w * d
hw_ratio = h/w
dw_ratio = d/w
constraints = [
    wall_area <= A_wall,
    flr_area <= A_flr,
    hw_ratio >= alpha,
    hw_ratio <= beta,
    dw_ratio >= gamma,
    dw_ratio <= delta
]
problem = cp.Problem(cp.Maximize(volume), constraints)
print(problem)

maximize h * w * d
subject to 2.0 * (h * w + h * d) <= 100.0
           w * d <= 10.0
           0.5 <= h / w
           h / w <= 2.0
           0.5 <= d / w
           d / w <= 2.0

assert not problem.is_dcp()
assert problem.is_dgp()
problem.solve(gp=True)
problem.value

77.45966630736292

h.value

7.7459666715289766

w.value

3.872983364643079

d.value

2.581988871583608

# A 1% increase in allowed wall space should yield approximately
# a 0.83% increase in maximum value.
constraints[0].dual_value

0.8333333206334043

# A 1% increase in allowed wall space should yield approximately
# a 0.66% increase in maximum value.
constraints[1].dual_value

0.6666666801983365