6
$\begingroup$

I am a bit new to programming and am currently working with solving optimization problems in JuMP in Julia.

I got a tip from the JuMP page that I should also use @constraint if a term is quadratic. However, how do I know if a constraint is quadratic?

For instance, should I use @constraint or @NLconstraint for $z - 2 + 4 z y = 0$ and $x - z^2 \geq 0$?

All I know is that it should have the form $\frac{1}2 x H x + q x$ where $x$ is the vector with all the variables.

$\endgroup$
2
  • $\begingroup$ 1/2 x * H * x is just a fancy way to write $\sum_{j_1} \sum_{j_2} h_{j_1,j_2} x_{j_1} x_{j_2}$ $\endgroup$
    – fontanf
    Sep 23, 2022 at 13:48
  • $\begingroup$ If you're looking for solvers, note that quadratic constraints mean that your model is not a quadratic programming problem. You need a QPQC solver, or nonlinear programming solver $\endgroup$ Sep 23, 2022 at 15:41

1 Answer 1

11
$\begingroup$

A "quadratic constraint" is a constraint of the form $f(x) \leq 0$, where $f(x)$ is a quadratic function, i.e., as you wrote, $$ f(x) = \frac{1}{2}x^{T}Hx + q^{T}x + b $$ for some square matrix $H \in \mathbb{R}^{n \times n}$, vector $q \in \mathbb{R}^{n}$, and some scalar $b \in \mathbb{R}$.

In JuMP, you can declare such quadratic constraints with the @constraint syntax directly. The @NLconstraint macro is designed for more general nonlinear that require special handling of function evaluation and derivatives. For instance, if your constraints have high-order polynomial functions ($x^{3}$ and above), or terms like sin, cos, log or exp, you should use @NLconstraint.

You can have a look at the quadratic portfolio optimization and quadratically constrained programming examples from the JuMP documentation.

PS: there is more JuMP-centric traffic on the Julia Discourse forum. You are more likely to get expert feedback there, especially on code-related questions.

$\endgroup$
1
  • $\begingroup$ You can build higher order polynomials via quadratic constraints though or use PolyJuMP compared to the general purpose @NLConstraints. This can be useful to make some solvers like Gurobi solve problems they won't otherwise as @NLConstraints is not supported by some, however quadratic terms are. $\endgroup$ Sep 22, 2022 at 15:57

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.