Two Splitting Methods For Two-block Convex Programming

Two-block convex optimization problems have been widely appeared in many fields,such as signal and image processing,data mining and classifi-cation,machine and statistical learning,principal component analysis,asset allocation and so on.Therefore,it is significant to study algorithms for solv-ing two-block convex optimization problems.Alternating directions method of multipliers(ADMM)and Douglas-Rachford splitting method(PRSM)are two kinds of effective algorithms for solving two-block convex optimization problems.This thesis mainly studies the convergence rate of proximal ver-sion of ADMM and the theory of proximal Peaceman-Rachford splitting al-gorithm with non-positive definite regularization matrix.We show the main ideas in the following.Firstly,for solving the two-block convex optimization problems,we put forward a proximal strictly contractive Peaceman-Rachford splitting method(PSC-PRSM).We show that the new algorithm is convergent and has a worst-case O(1/k)convergence rate in ergodic sence measured by the itera-tion complexity.Numerical experiments in Lasso demonstrate the efficiency of the new algorithm.Secondly,we introduce the regularized structures into both x and y subproblems,we focus on anlysing the O(1/k)convergence rate of proximal alternating directions method of multipliers(PADMM)for solv-ing convex two-block optimization problems.Numerical experiments show the PADMM is stable and efficient.