In this work we consider the problem $\min_x \{ f(x) + P(x) \}$, where $f$ is $\mathcal{C}^2$ and $P$ is nonsmooth, but contains an underlying smooth substructure. Specifically, we assume the function $P$ is prox-regular partly smooth with respect to a active manifold $\M$. Recent work by Tseng and Yun \cite{tseng-yun-2009}, showed that such a problem can be approached by minimizing the sum of a quadratic approximation of $f$ and the prox-regular partly smooth function $P$ $$x_{k+1} = \min_x \{ \langle \nabla f(x_k), x - x_k \rangle + (x - x_k)' H_k (x - x_k) + P(p)\}.$$ When $P$ is well structured this provides a practical approach to solving the original problem. In this work we examine Tseng and Yun's approach in terms of active set identification. In particular, we show that this method will correctly identify the active manifold $\M$ in a finite number of iteration. Furthermore, we confirm a conjecture of Tseng that, regardless of what technique is used to solve the original problem, the subproblem $p_k = \argmin_p \{ \langle \nabla f(x_k), p \rangle + \frac{r}{2}|x_k - p|^2 + P(p)\}$ would correctly identify the active manifold in a finite number of iterations.

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View Identifying Active Manifolds in Regularization Problems