#!/usr/bin/env python
# coding: utf-8

# # Maximizing the volume of a box
# 
# *This example is adapted from Boyd, Kim, Vandenberghe, and Hassibi,* "[A Tutorial on Geometric Programming](https://web.stanford.edu/~boyd/papers/pdf/gp_tutorial.pdf)."
# 
# 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
# 
# $$
# \begin{array}{ll}
# \mbox{maximize} & hwd \\
# \mbox{subject to} & 2(hw + hd) \leq A_{\text wall}, \\
# & wd \leq A_{\text flr}, \\
# & \alpha \leq h/w \leq \beta, \\
# & \gamma \leq d/w \leq \delta.
# \end{array}
# $$

# In[1]:


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)


# In[2]:


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


# In[3]:


h.value


# In[4]:


w.value


# In[5]:


d.value


# In[6]:


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


# In[7]:


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