LP Centering Task:

We want to find x solves the following LP centering problem, which arises in standard form LP 
with nonnegativity constraint. 

   maximize    c^Tx - \sum_i log x_i
   subject to  Ax = b

c is a n-dimensional real-valued vector defining the objective,
A is a (m x n)-dimensional real-valued matrix for the equality constraint,
b is a m-dimensional real-valued vector for the equality constraint.

Given input parameters (c, A, b), compute and return the optimal x.

Input: A dictionary with keys:
   - "c": A list of n numbers representing the vector c.
   - "A": A list of m lists of numbers representing the matrix A.
   - "b": A list of m numbers representing the vector b.

Example input:
{
  "c": [1,2],
  "A": [[0,2], [1, 0],[3,4]],
  "b": [2,1,7]
}

Output: A dictionary with keys:
  - "solution": A list of n numbers representing the optimal (primal) solution.

Example output:
{
  "solution": [1, 1]
}

Category: convex_optimization