How to formulate the inequality constraint $\sqrt{x^2+y^2} \leq z$ using gurobipy?

Question

How to formulate the following constraint using gurobipy

$$ \sqrt{x^2 + y^2} \le z$$

where $x, y, z$ are continuous optimization variables?

I saw how to formulate it using CVXPY:

cp.norm(cp.hstack([x, y])) <= z

I am now wondering how to formulate it in gurobipy. I did not find any information on how to formulate the norm convex function in the gurobipy documentation.

If take the square of both sides,

$$ x^2 + y^2 \le z^2 $$

gurobipy solves it but doesn't this constraint violate DCP rules? I thought the constraint will be following the DCP rules if convex $\leq$ constant, convex $\leq$ concave, but here it is convex $\leq$ convex.

$x^2+y^2\le z^2, z\ge 0$ is recognized by Gurobi as a second-order cone. That is a convex constraint. — Erwin Kalvelagen, Nov 29 '22 at 15:13
Thank you so much, @Erwin Kalvelagen. Gurobi accepts this constraint, but I was concerned about what follows: $ x^2+y^2$ is convex, and $ z^2 $ is another convex function. I know that convex<= convex is not a disciplined convex problem. Thanks! — Hussein Sharadga, Nov 29 '22 at 18:12
I guess gurobi will handle this as quadratic constraints: z^2 is linearized by McCormick envelope; in that case: x^2+^2<= linear is a convex constraint. Thanks again to @ Erwin Kalvelagen — Hussein Sharadga, Dec 26 '22 at 11:46

score 3 · Accepted Answer · edited Dec 01 '22 at 21:06

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Thanks to the Gurobi staff, this is how to formulate it using the norm constraint:

import gurobipy as gp
m=gp.Model()
m.addGenConstrNorm(aux, [x, y], 2, "normconstr")
m.addConstr(aux<=z)

Ref: https://support.gurobi.com/hc/en-us/community/posts/10762215603345-SOCP

edited Dec 01 '22 at 21:06

Rodrigo de Azevedo

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answered Dec 01 '22 at 19:01

Hussein Sharadga

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Congrats! Does it appear in any of their documentation/manual? It used to appear then I myself wasn't able to find it. – Sutanu Majumdar Dec 01 '22 at 19:04
Thank you! You can the following: https://support.gurobi.com/hc/en-us/community/posts/10762215603345-SOCP – Hussein Sharadga Dec 01 '22 at 19:10

How to formulate the inequality constraint $\sqrt{x^2+y^2} \leq z$ using gurobipy?

1 Answers1