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

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: constraint

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

Thanks!

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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.

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  • $\begingroup$ Hi, thank you so much for your explanation! $\endgroup$
    – inf
    Sep 3 at 2:46
  • $\begingroup$ 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! $\endgroup$
    – inf
    Sep 3 at 2:53

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