A forward-backward penalty scheme with inertial effects for montone inclusions. Applications to convex bilevel programming

We investigate forward-backward splitting algorithm of penalty type with inertial effects for finding the zeros of the sum of a maximally monotone operator and a cocoercive one and the convex normal cone to the set of zeroes of an another cocoercive operator. Weak ergodic convergence is obtained for the iterates, provided that a condition express … Read more

An inertial alternating direction method of multipliers

In the context of convex optimization problems in Hilbert spaces, we induce inertial effects into the classical ADMM numerical scheme and obtain in this way so-called inertial ADMM algorithms, the convergence properties of which we investigate into detail. To this aim we make use of the inertial version of the Douglas-Rachford splitting method for monotone … Read more

Forward-Backward and Tseng’s Type Penalty Schemes for Monotone Inclusion Problems

We deal with monotone inclusion problems of the form $0\in Ax+Dx+N_C(x)$ in real Hilbert spaces, where $A$ is a maximally monotone operator, $D$ a cocoercive operator and $C$ the nonempty set of zeros of another cocoercive operator. We propose a forward-backward penalty algorithm for solving this problem which extends the one proposed by H. Attouch, … Read more

On the convergence rate improvement of a primal-dual splitting algorithm for solving monotone inclusion problems

We present two modified versions of the primal-dual splitting algorithm relying on forward-backward splitting proposed in [21] for solving monotone inclusion problems. Under strong monotonicity assumptions for some of the operators involved we obtain for the sequences of iterates that approach the solution orders of convergence of ${\cal {O}}(\frac{1}{n})$ and ${\cal {O}}(\omega^n)$, for $\omega \in … Read more

A primal-dual splitting algorithm for finding zeros of sums of maximally monotone operators

We consider the primal problem of finding the zeros of the sum of a maximally monotone operator with the composition of another maximally monotone operator with a linear continuous operator and a corresponding dual problem formulated by means of the inverse operators. A primal-dual splitting algorithm which simultaneously solves the two problems in finite-dimensional spaces … Read more