bv_cvxbook_extra_exercises

2 2 810 functions with bounded newton decrement let f

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Unformatted text preview: , φtv (x) = i=1 |xi+1 − xi |. Here xcor ∈ Rn is the (given) corrupted signal, x ∈ Rn is the de-noised signal to be computed, and φtv is the total variation function. This bi-criterion problem can be formulated as an SOCP, or, by squaring the first objective, as a QP. In this problem we consider a method used to approximately formulate the total variation de-noising problem as an unconstrained problem with twice differentiable objective, for which Newton’s method can be used. We first observe that the Pareto optimal points for the bi-criterion total variation de-noising problem can found as the minimizers of the function x − xcor 2 2 + µφtv (x), where µ ≥ 0 is parameter. (Note that the Euclidean norm term has been squared here, and so is twice differentiable.) In approximate total variation de-noising, we substitute a twice differentiable approximation of the total variation function, n −1 φatv (x) = i=1 ǫ2 + (xi+1 − xi )2 − ǫ , for the total variation function φtv . Here ǫ > 0 is parameter that controls the level of approxi...
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