Abstract
In this paper, we propose two descent alternating direction methods based on a logarithmic-quadratic proximal method for structured variational inequalities. The first method can be viewed as an extension of the method proposed by Bnouhachem and Xu (Comput. Math. Appl. 67:671-680, 2014) by performing an additional step at each iteration. The second method generates the new iterate by searching the optimal step size along a new descent direction, which can be viewed as a refinement and improvement of the first one. Under certain conditions, the global convergence of the both methods is proved.
Highlights
We consider the constrained convex programming problem with the following separate structure: min θ (x) + θ (y)|Ax + By = b, x ∈ Rn+, y ∈ Rm+, ( . )where θ : Rn+ → R and θ : Rm+ → R are closed proper convex functions, A ∈ Rl×n, B ∈ Rl×m are given matrices, and b ∈ Rl is a given vector.A large number of problems can be modeled as problem ( . )
The first one can be viewed as an extension of the method proposed in [ ] by performing an additional step at each iteration
2 Iterative methods and convergence results we suggest and analyze two new modified logarithmic-quadratic proximal alternating direction methods for solving structured variational inequalities
Summary
We consider the constrained convex programming problem with the following separate structure: min θ (x) + θ (y)|Ax + By = b, x ∈ Rn+, y ∈ Rm+ , ). where θ : Rn+ → R and θ : Rm+ → R are closed proper convex functions, A ∈ Rl×n, B ∈ Rl×m are given matrices, and b ∈ Rl is a given vector. ). A large number of problems can be modeled as problem These classes of problems have very large size and, due to their practical importance, they have received a great deal of attention from many researchers. ). A popular approach is the alternating direction method (ADM) which was proposed by Gabay and Mercier [ ] and Gabay [ ]. To make the ADM more efficient and practical, some strategies have been studied; For further details, we refer to [ – ] and the references therein
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
