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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.
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.
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.
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Sep 22, 2022 at 15:57
1/2 x * H * xis just a fancy way to write $\sum_{j_1} \sum_{j_2} h_{j_1,j_2} x_{j_1} x_{j_2}$ $\endgroup$