Abstract
In 2009, Tseng and Yun [Math. Programming (Ser. B) 117, 387–423 (2009)], showed that the regularization problem of minimizing f(x) + | | x | | 1, where f is a \({\mathcal{C}}^{2}\) function and | | x | | 1 is the l 1 norm of x, can be approached by minimizing the sum of a quadratic approximation of f and the l 1 norm. We consider a generalization of this problem, in which the l 1 norm is replaced by a more general nonsmooth function that contains an underlying smooth substructure. In particular, we consider the problem
where f is \({\mathcal{C}}^{2}\) and P is prox-regular and partly smooth with respect to an active manifold \(\mathcal{M}\) (the l 1 norm satisfies these conditions.) We reexamine Tseng and Yun’s algorithm in terms of active set identification, showing that their method will correctly identify the active manifold in a finite number of iterations. That is, after a finite number of iterations, all future iterates x k will satisfy \({x}^{k} \in \mathcal{M}\). Furthermore, we confirm a conjecture of Tseng that, regardless of what technique is used to solve the original problem, the subproblem \({p}^{k} =\mathrm{{ argmin}}_{p}\{\langle \nabla f({x}^{k}),p\rangle + \frac{r} {2}\vert {x}^{k} - p{\vert }^{2} + P(p)\}\) will correctly identify the active manifold in a finite number of iterations.
AMS 2010 Subject Classification: Primary: 49K40, 65K05; Secondary: 52A30, 52A41, 90C53
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Hare, W.L.: A proximal method for identifying active manifolds. Comput. Optim. Appl. 43, 295–306 (2009)
Hare, W.L., Lewis, A.S.: Identifying active constraints via partial smoothness and prox-regularity. J. Convex Anal. 11, 251–266 (2004)
Hare, W.L., Poliquin, R.A.: Prox-regularity and stability of the proximal mapping. J. Convex Anal 14, 589–606 (2007)
Lewis, A.S.: Active sets, nonsmoothness, and sensitivity. SIAM J. Optim. 13, 702–725 (2003)
Poliquin, R.A., Rockafellar, R.T.: Prox-regular functions in variational analysis. Trans. Amer. Math. Soc. 348, 1805–1838 (1996)
Rockafellar, R.T., Wets, R.J-B: Variational Analysis. Springer, Berlin (1998)
Tseng, P., Yun, S.: A coordinate gradient descent method for nonsmooth separable minimization. Math. Programming (Ser. B) 117, 387–423 (2009)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer Science+Business Media, LLC
About this chapter
Cite this chapter
Hare, W.L. (2011). Identifying Active Manifolds in Regularization Problems. In: Bauschke, H., Burachik, R., Combettes, P., Elser, V., Luke, D., Wolkowicz, H. (eds) Fixed-Point Algorithms for Inverse Problems in Science and Engineering. Springer Optimization and Its Applications(), vol 49. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-9569-8_13
Download citation
DOI: https://doi.org/10.1007/978-1-4419-9569-8_13
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4419-9568-1
Online ISBN: 978-1-4419-9569-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)