# Questions on Mosek Fusion Power Cone example

Just trying to understand the example given in Mosek Fusion handbook as shown

I'm not exactly sure how to convert 7.3 to 7.4 in terms of objective function. I understand the power cone $$P_{3}^{0.2, 0.8}$$ constraint gives $$x_0^{0.2}x_1^{0.8} \ge |x_3|$$ => $$x^{0.2}y^{0.8} \ge |x_{3}|$$ Similarly, $$P_{3}^{0.4, 0.6}$$ constraint gives $$x_2^{0.4}x_5^{0.6} \ge |x_4|$$ => $$z^{0.4} \ge |x_{4}|$$.

However, I failed to see how to convert the new objective in 7.4 back to the objective in 7.3. I.e. $$\max x_3 + x4 - x_0$$ =>?? $$\max |x_3| + |x_4| - x_0$$ >=?? $$\max x^{0.2}y^{0.8} + z^{0.4} - x$$

Can someone explain?

Also, in the codes, it has:

Why is one Var.vstack and the other is Expr.vstack?

Thanks!

## 1 Answer

Maximizing $$f(x)$$ is equivalent to

$$\begin{array}{ll} \mathrm{maximize} & t \\ \mathrm{subject\ to} & f(x)\geq t \end{array}$$

where $$t$$ is a new variable that does not appear anywhere else, since it pays off to push $$t$$ as high as possible i.e to reach equality in the constraint. That is what happens with the extra variables in that model.

Var is used for a bit of efficiency because all argmuents are variables. You could use Expr throughout.

• Hi, thank you so much for your explanation!
– inf
Sep 3 '21 at 2:46
• Hope you don't mind a follow-up question. In the equivalent formulation, it has $f(x) \ge t$. In the example given by Mosek, it has $x^{0.2}y^{0.8} \ge |x_{3}|$. Notice the absolute value here. Is it true that we can remove the absolute value on $x_3$ because $x,y \ge 0$? Thanks again!
– inf
Sep 3 '21 at 2:53