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- Title
On the Existence of Saddle Points for ℓ1-Minimization Problems.
- Authors
Yuefang Lian; Jinchuan Zhou; Jingyong Tang; Xia Liu
- Abstract
The sparse optimization problem has a wide range of applications in image processing, compressed sensing, and machine learning, etc. It is well known that l1-minimization problem plays an important role in studying sparse optimization problem from theoretical and algorithm aspects. In this paper, we mainly study the existence theory on saddle points for l1- minimization problem. Firstly, to overcome the nonsmoothness of l1-norm, we translate l1-minimization problem to an optimization programming with linear cost function by introducing new variable. Secondly, based on a new augmented Lagrangian function, the relationship on saddle points between the primal problem and the translated problems, associated with their duality problems, is established. It allows us to establish local saddle points by taking into account of second-order sufficient conditions. Finally, global saddle points is established by using two different approaches. One is requiring that the optimal solution is unique. This assumption can be further removed in our another approach by using the perturbation analysis of primal problem.
- Subjects
LINEAR programming; LAGRANGIAN functions; SADDLERY; IMAGE processing; MACHINE learning
- Publication
IAENG International Journal of Applied Mathematics, 2021, Vol 51, Issue 1, p48
- ISSN
1992-9978
- Publication type
Article