In this paper we propose a primal-dual proximal extragradient algorithm to solve the generalized Dantzig selector (GDS) estimation problem, based on a new convex-concave saddle-point (SP) reformulation. Our new formulation makes it possible to adopt recent developments in saddle-point optimization, to achieve the optimal O(1/k) rate of convergence. Compared to the optimal non-SP algorithms, our...

Several strong convergence results involving two distinct four parameter proximal point algorithms are proved under different sets of assumptions on these parameters and the general condition that the error sequence converges to zero in norm. Thus our results address the two important problems related to the proximal point algorithm − one being of strong convergence (instead of weak convergence...

It is known, by Rockafellar [SIAM J. Control Optim., 14 (1976), 877898], that the proximal point algorithm (PPA) converges weakly to a zero of a maximal monotone operator in a Hilbert space, but it fails to converge strongly. Lehdili and Moudafi [Optimization, 37(1996), 239-252] introduced the new proxTikhonov regularization method for PPA to generate a strongly convergent sequence and establis...

We in this paper study the nonexpansive operators equipped with arbitrary metric and investigate connections between firm nonexpansiveness, cocoerciveness averagedness. The convergence of associated fixed-point iterations is discussed particular focus on case degenerate metric, since degeneracy often encountered when reformulating many existing first-order operator splitting algorithms as a res...

In this paper, we introduce and study a system of mixed variational-like inclusions and a system of proximal operator equations in Banach spaces which contains variational inequalities, variational inclusions, resolvent equations, system of variational inequalities and system of variational inclusions in the literature as special cases. It is established that the system of mixed variational-lik...

The theory of maximal set-valued monotone mappings provide a powerful framework to the study of convex programming and variational inequalities. Based on the notion of relatively maximal relaxed monotonicity, the approximation solvability of a general class of inclusion problems is discussed, while generalizing most of investigations on weak convergence using the proximal point algorithm in a r...

Abstract In a recent paper, Bauschke et al. study $$\rho $$ ρ -comonotonicity as generalized notion of monotonicity set-valued operators A in Hilbert space and characterize this condition on terms the averagedness its resolvent $$J_A.$$ J A <...

This paper establishes convergence of generalized Bregman-function-based proximal point algorithms when the iterates are computed only approximately. The problem being solved is modeled as a general maximal monotone operator, and need not reduce to minimization of a function. The accuracy conditions on the iterates resemble those required for the classical "linear" proximal point algorithm, but...

In this paper, we consider completely generalized nonlinear quasi-variational-like inclusions in Banach spaces and propose an Ishikawa type iterative algorithm for computing their approximate solutions by applying the new notion of Jη-proximal mapping given in [R. Ahmad, A.H. Siddiqi, Z. Khan, Proximal point algorithm for generalized multi-valued nonlinear quasi-variational-like inclusions in B...