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I am trying to reformulate this optimization problem in order to get a DCP-complaint expression on Julia (I am using the Conjex.jl package)

$$ \text{Minimize}\;\;\; \mid\log(x)\mid $$

The code

p = Variable()
f = abs(log(p))

Prompts to a warning

WARNING: Expression not DCP compliant. Trying to solve non-DCP compliant problems can lead to unexpected behavior.

I am beginner regarding convex optimization. But, as far as I read

CVX-family does not consider a function to be convex or concave if it is so only over a portion of its domain, even if the argument is constrained to lie in one of these portions

Hence, the function $\mid x\mid$ is nonincreasing for $x < 0$, and nondecreasing for $x>0$. I suppose that the composite property fails because of it.

How to reformulate it in order to get DCP-compliant expresison?

PS: solutions using other scientific programming languages (Matlab, R, Python...) is welcome too.

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  • $\begingroup$ log(x) is not convex anywhere (it is concave). So |log(x)| is also not convex. $\endgroup$ Dec 9, 2022 at 3:41
  • $\begingroup$ @ErwinKalvelagen nice, so how to reformulate this expression? $\endgroup$ Dec 9, 2022 at 11:51
  • $\begingroup$ Not all problems are convex or can be convexified. (After all, why would global solvers exist). $\endgroup$ Dec 9, 2022 at 11:55
  • $\begingroup$ @ErwinKalvelagen and can this problem be convexified? $\endgroup$ Dec 9, 2022 at 11:58
  • $\begingroup$ @ErwinKalvelagen yeah, that is my question, can I state that this problem is equivalent to? $$ \text{Minimize}\;\;\; \mid x - 1 \mid $$ $\endgroup$ Dec 9, 2022 at 12:00

1 Answer 1

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I agree that the problem is nonconvex, but since the function is piecewise monotone, you could separately solve

$\min x$

subject to

$x\geq 1$

plus other constraints

and

$\max x$

subject to

$ x <= 1$

plus other constraints

and pick the better of the two solutions.

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    $\begingroup$ Or use binary to solve as a single MIDCP formulation. $\endgroup$ Dec 10, 2022 at 1:25

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