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How would you approach a standard convex quadratic problem with convex constraints but one non-convex term ? Say $|x|^{0.4}$.

$$\min_x \frac{1}{2} x^{T}Qx + g^Tx + c^T \mathrm{sign}(x) |x|^{0.4} $$ subject to $$Ax\leq b$$

Where $|\cdot|, \mathrm{sign}()$ are element-wise.

Is there any other way to solve the problem within convex framework ? What`s the best way to approximate if the dimensions are large ?

This is similar to Reformulate a strictly convex QP problem containing absolute value term but here there is a power term.

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