Different ways of defining quadratic expressions

Answered

Permanently deleted user

July 28, 2021 20:00 Edited

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My question is very simple perhaps, but I'm new to Gurobi and I would like to be very sure about the answer.

So, assume we have a one-dimensional MVar in our model:

from gurobipy import *
model = Model()
x = model.addMVar(1000, vtype=GRB.INTEGER)

Do I figure it right that the quadratic expression

expr = x@x

is mathematically equivalent to

expr = quicksum(var*var for var in x.tolist())

?

There are certainly under-the-hood differences in the implementation since print(expr) results in completely different output for these two cases, but I presume the mathematical sense is the same.

Thanks.

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• Official comment

  

  Simranjit Kaur

  • Gurobi Staff

  June 23, 2025 02:50 Edited

  This post is more than three 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 why not try our AI Gurobot?.
• 

  Eli Towle

  • Gurobi Staff

  July 28, 2021 19:32

  Yes, those two expressions are mathematically equivalent. You can test this by adding both expressions to the model in separate constraints, writing out an LP file with Model.write(), then visually inspecting the LP file to compare the two constraints. For example, the following code

  import gurobipy as gp
  
  m = gp.Model()
  x = m.addMVar(5, name='x')
  
  # Add constraint in two different ways
  m.addConstr(x @ x == 1, name='approach1')
  m.addConstr(gp.quicksum(var*var for var in x.tolist()) == 1, name='approach2')
  
  # Write results to LP file
  m.write('foo.lp')

  creates a \( \texttt{foo.lp} \) model file with the exact same constraint repeated twice:

  Minimize
  
  Subject To
   approach1: [ x[0] ^2 + x[1] ^2 + x[2] ^2 + x[3] ^2 + x[4] ^2 ] = 1
   approach2: [ x[0] ^2 + x[1] ^2 + x[2] ^2 + x[3] ^2 + x[4] ^2 ] = 1
  Bounds
  End

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  Permanently deleted user

  July 28, 2021 20:04

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  Thanks. I have been looking for a way to transform these expressions to some "canonical form" to compare them but I haven't come up with the idea of exporting the model to file.

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