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

# # Perron-Frobenius matrix completion
# 
# The DGP atom library has several functions of positive matrices, including the trace, (matrix) product, sum, Perron-Frobenius eigenvalue, and $(I - X)^{-1}$ (eye-minus-inverse). In this notebook, we use some of these atoms to formulate and solve an interesting matrix completion problem.
# 
# In this problem, we are given some entries of an elementwise positive matrix $A$, and the goal is to choose the missing entries so as to minimize the Perron-Frobenius eigenvalue or spectral
# radius. Letting $\Omega$ denote the set of indices $(i, j)$ for which $A_{ij}$ is known, the optimization problem is
# 
# $$
# \begin{equation}
# \begin{array}{ll}
# \mbox{minimize} & \lambda_{\text{pf}}(X) \\
# \mbox{subject to} & \prod_{(i, j) \not\in \Omega} X_{ij} = 1 \\
# & X_{ij} = A_{ij}, \, (i, j) \in \Omega,
# \end{array}
# \end{equation}
# $$
# 
# which is a log-log convex program.  Below is an implementation of this problem, with specific problem data
# 
# $$
# A = \begin{bmatrix}
# 1.0 & ? &  1.9 \\
# ? & 0.8 &  ? \\
# 3.2 & 5.9&  ?
# \end{bmatrix},
# $$
# 
# where the question marks denote the missing entries.

# In[1]:


import cvxpy as cp

n = 3
known_value_indices = tuple(zip(*[[0, 0], [0, 2], [1, 1], [2, 0], [2, 1]]))
known_values = [1.0, 1.9, 0.8, 3.2, 5.9]
X = cp.Variable((n, n), pos=True)
objective_fn = cp.pf_eigenvalue(X)
constraints = [
  X[known_value_indices] == known_values,
  X[0, 1] * X[1, 0] * X[1, 2] * X[2, 2] == 1.0,
]
problem = cp.Problem(cp.Minimize(objective_fn), constraints)
problem.solve(gp=True)
print("Optimal value: ", problem.value)
print("X:\n", X.value)