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Since $ \epsilon $ is a parameter you need to set why not trade it with another parameter you need to set to create an easily solvable problem (Relaxation of the Problem)? You can transform the problem into the following form ($ {L}_{1} $ Regularized Least Squares): $$ \arg \min_{x} \frac{1}{2} \left\| A x - z \right\|^{2} + \lambda \left\| x \right\|_{1} $...


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