How to maximize the determinant of a N by N Matrix variable

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Yen-Chun Liu

2024年03月12日 01:21

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Hi,

Below are the notations of my problem:

• variable of interest: N*m*m binary array (B_1,...,B_N). Each B_i's is subjected to the constraint that row sum=1.
• the distance matrix of these N m*m arrays (D: N by N), each (i,j) element is the L1 distance between matrices B_i and B_j
• a given constant matrix (K: k by k), the distance matrix of some K*m*m arrays
• the distance matrix of the N arrays and the K arrays (R: N by K)

Now I hope to find B_1,..,B_N that maximizes det(D-RK^{-1}R), where D and R are functions of B's. I can code D, R, K. However, I have had a hard time calculating the determinant. This post (answered 2 years ago) provided a solution to calculate the determinant when N=2. But in general, my N is around 10. 

Does Gurobi now support general determinant calculation? or is there a way to do it without coding the 10-dim determinant formula?  Thank you.

 

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  Simranjit Kaur

  • Gurobi Staff

  2026年11月13日 01:28

  This post is more than two years old. Some information may not be up to date. For current information, please check the Gurobi Documentation or Knowledge Base. If you need more help, please create a new post in the community forum, or try Gurobot, our chatbot interface offering instant, expert-level support.
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  Riley Clement

  • Gurobi Staff

  2024年03月15日 01:04

  Hi Yen-Chun,

  If you have a matrix variable x, then you can calculate a determinant with the following code.  You are still encoding a formula for the determinant but using recursion to keep it tidy:

  import gurobipy as gp
  import numpy as np
  
  def det(A):
      v = m.addVar(lb=-float("inf"))
      if A.shape == (2,2):
          m.addConstr(v == A[0,0]*A[1,1]-A[1,0]*A[0,1])
          return v
      expr = gp.QuadExpr()
      cofactor = 1
      for i in range(A.shape[1]):
          cols = [c for c in range(A.shape[1]) if c != i]
          expr += cofactor*A[0,i]*det(A[1:][:,cols])
          cofactor = -cofactor
      m.addConstr(v == expr)
      return v
  
  n = 10
  m = gp.Model()
  x = m.addMVar((n,n), lb=-float("inf"))
  determinant = det(x) # determinant will be a gurobipy.Var
  

  - Riley

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  Yen-Chun Liu

  2024年03月15日 20:25

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  Thank you!

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