joint work with Yair Censor, Aviv Gibali
The Douglas–Rachford (DR) algorithm is an iterative procedure that uses sequential reflections onto convex sets and which has become popular for convex feasibility problems. In this paper we propose a structural generalization that allows to use r-sets-DR operators in a cyclic fashion. We prove convergence and present numerical illustrations of the potential advantage of such operators with r>2 over the classical 2-sets-DR operators in a cyclic algorithm.
joint work with Oleg Wilfer
We approach via conjugate duality some optimization problems with intricate structure that cannot be directly solved by means of the existing proximal point type methods. A splitting scheme is employed on the dual problem and the optimal solutions of the original one are recovered by means of optimality conditions. We use this approach for minmax location and entropy constrained optimization problems, presenting also some computational results where our method is compared with some recent ones from the literature.
In this work we proposed a stochastic version of primal dual fixed point method(SPDFP) for solving a sum of two proper lower semi-continuous convex function one of which is composited with a linear operator. Under some assumptions we proved the convergence of the proposed algorithm and also provided a variant for the implementation. Convergence analysis shows that the expected error of iterate is of the order $O(k^{-\alpha})$ ) where $k$ is the iteration number and $\alpha\in (0, 1)$. Finally, two numerical examples are performed to demonstrate the effectiveness of the proposed algorithms.