How to to define multivariate polynomials

Answered

Santanu Sarkar

• July 20, 2020 15:43

Hi,

 Is it possible to define multivariate polynomials in Gurobi?

I have y1=a1 l1+a2 l2 +a3 l3, y2= a1 l4 + a2 l5 + a3 l6, y3= a1 l7 + a2 l8 + a3 l9

where y1, y2, y3, a1, a2, a3 are binary variables.  I have three linear polynomials f1,f2 and f3 over

y1, y2 and y3. I have to assign l1,...,l9=0/1 with l1+l2+l3=l4+l5+l6=l7+l8+l9=1 such that the 

cardinality of the set {f1,f2,f3} is 2. Ofcourse this is toy example. If we can define multivariate 

polynomials, we can try like this

 

D1 : i1 = 1 -> f1-f2=0, D2: i2 = 1 -> f1-f3=0, D3: i3=1 -> f2-f3=0

D4: i1+i2+i3=2

 

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• Official comment

  

  Simranjit Kaur

  Gurobi Staff

  • December 06, 2024 00:43

  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?.
• 

  Jaromił Najman

  Gurobi Staff

  • July 30, 2020 10:22
  • Edited

  Hi,

  I am not sure if I understand fully understand your issue.

  It is currently not possible to directly define multivariate polynomials in Gurobi but it is certainly possible to model them.

  For example, let's consider the multivariate polynomial \( p:\mathbb{R}^2 \to \mathbb{R}, (x,y) \mapsto x^2\cdot y  + x \cdot y + x \cdot y^2 \). You can formulate a constraint such as \(p(x,y) = 0\) by the addition of auxiliary variables and the use of quadratic expressions as
  \(\begin{align}
  \omega_1 \cdot y + &x \cdot y + x \cdot \omega_2 = 0 \\
  \omega_1 &= x^2 \\
  \omega_2 &= y^2 \\
  x \in &[x^L, x^U] \\
  y \in &[y^L,y^U] \\
  \omega_1 \in &\begin{cases} [0, \max{(x^L)^2,(x^U)^2}], &\text{if } x^L < 0 < x^U \\ [(x^L)^2, (x^U)^2], &\text{if } x^L\geq 0  \\ [(x^U)^2, (x^L)^2], &\text{if } x^U\leq 0 \end{cases} \\
  \omega_2 \in &\begin{cases} [0, \max{(y^L)^2,(y^U)^2}], &\text{if } y^L < 0 < y^U \\ [(y^L)^2, (y^U)^2], &\text{if } y^L\geq 0  \\ [(y^U)^2, (y^L)^2], &\text{if } y^U\leq 0 \end{cases}
  \end{align}\)

  Note that I added finite bounds for \(x\) and \(y\) as this is recommended when working with nonlinear expressions.

  I hope the above could answer your questions. Feel free to ask further questions if not.

  Best regards,
  Jaromił

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