Nonconvex problem considered convex by Pyomo/ Gurobi solver ! Weird?

Awaiting user input

Ros1_imperial

July 28, 2023 04:51 Edited

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QUESTION 1: 

We have an optimization problem with objective function

f(x,y,z,p) = 

0.12 * x^2 + 14.8 * x + 89 +
0.17 * y^2 + 16.57 * y + 83 +
0.15 * z^2 + 15.55 * z + 100 +
0.19 * p^2 + 16.21 * p + 70

And the Hessian of this Objective function is :
(0.24 | 0 | 0 | 0
0 | 0.34 | 0 | 0
0 | 0 | 0.3 | 0
0 | 0 | 0 | 0.38)

Therefore the eigenvalues are : 0.24, 0.34, 0.3 and 0.38 i.e all are positive and so the Hessian is positive semidefinite.

As a result, the objective function is convex. 

All other constraints are convex / linear in this optimization problem, and so the gurobi solver will solve it without issues.
Agree?

QUESTION 2: 
Now assume that in the above problem we add the following equality constraint: 

1.2 * x^2   - 5  * x + 3 +
2.3 * y^2   - 4.24 * y + 6.09 +
1.1 * z^2  - 2.15  * z + 5.69 +
1.1 * p^2  - 3.99 * p + 6.2 = 0

The above is a NONCONVEX constraint and I see that Gurobi can only solve it by having :
solver.options['NonConvex'] = 2.
Do you agree with the above observations?

 QUESTION 3

Is there a way by pyomo / gurobi to tell us quickly if an objective function or if a constraint is convex or non convex? 

Related to

• Nonconvex

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  Jaromił Najman

  • Gurobi Staff

  August 01, 2023 07:29

  All other constraints are convex / linear in this optimization problem, and so the gurobi solver will solve it without issues.
  Agree?

  Yes.

  The above is a NONCONVEX constraint and I see that Gurobi can only solve it by having :
  solver.options['NonConvex'] = 2.
  Do you agree with the above observations?

  Yes.

  Is there a way by pyomo / gurobi to tell us quickly if an objective function or if a constraint is convex or non convex? 

  I don't know about Pyomo but there is currently no way to ask Gurobi whether an objective function or a constraint is convex or nonconvex. One way would be to solve a model \(\min f(x)\) without constraints and check whether the NonConvex parameter has to be set. If not, then function \(f\) is convex or Gurobi has found a way to turn the optimization problem into a convex one. This can happen if, e.g., you have binary variables and one can add diagonal terms to make the \(Q\) matrix PSD by exploiting the property that \(b = b^2\) if \(b\) is binary.

  The title of you post states

  Nonconvex problem considered convex by Pyomo / Gurobi. Weird?

  Could you please elaborate what exactly is weird? Do you have a case where you think that a function is nonconvex but then you don't have to set the NonConvex parameter?

  Best regards, 
  Jaromił

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