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I've asked this question. Essentially I have the same question for problem $\arg\min |x|^{1.5}$. We can consider next variant $\arg\min t^{1.5}$ subject to $x\le t$ and $-x \le t$. But $t^{1.5}$ is not twice continuously differentiable.

Intuitively $|x|^{1.5}$ is "between" $|x|$ and $x^2$, so there should be nice formulation of optimization problem for it.

Community
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ashim
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1 Answers1

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$\min_x \ |x|^{3/2}$ may be written $$ \min_{x,t} \ t^{3/2} \quad \text{s.t.} \ -t \leq x \leq t. $$ Now this constraint is equivalent to $-t^3 \leq x^3 \leq t^3$. Define $s := t^{3/2}$. You have $t^3 = s^2$, so now you can rewrite your problem as $$ \min_{x,s} \ s \quad \text{s.t.} \ -s^2 \leq x^3 \leq s^2. $$

Dominique
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