Regularity of collections of sets and convergence of inexact alternating projections

We study the usage of regularity properties of collections of sets in convergence analysis of alternating projection methods for solving feasibility problems. Several equivalent characterizations of these properties are provided. Two settings of inexact alternating projections are considered and the corresponding convergence estimates are established and discussed. Article Download View Regularity of collections of sets … Read more

On a new class of matrix support functionals with applications

A new class of matrix support functionals is presented which establish a connection between optimal value functions for quadratic optimization problems, the matrix-fractional function, the pseudo matrix-fractional function, and the nuclear norm. The support function is based on the graph of the product of a matrix with its transpose. Closed form expressions for the support … Read more

Zero-Convex Functions, Perturbation Resilience, and Subgradient Projections for Feasibility-Seeking Methods

The convex feasibility problem (CFP) is at the core of the modeling of many problems in various areas of science. Subgradient projection methods are important tools for solving the CFP because they enable the use of subgradient calculations instead of orthogonal projections onto the individual sets of the problem. Working in a real Hilbert space, … 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

Convergence Analysis of DC Algorithm for DC programming with subanalytic data

DC Programming and DCA have been introduced by Pham Dinh Tao in 1986 and extensively developed by Le Thi Hoai An and Pham Dinh Tao since 1993. These approaches have been successfully applied to solving real life problems in their large scale setting. In this paper, by using the Lojasiewicz inequality for nonsmooth subanalytic functions, … 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

Trajectories of Descent

Steepest descent drives both theory and practice of nonsmooth optimization. We study slight relaxations of two influential notions of steepest descent curves — curves of maximal slope and solutions to evolution equations. In particular, we provide a simple proof showing that lower-semicontinuous functions that are locally Lipschitz continuous on their domains — functions playing a … Read more

Some criteria for error bounds in set optimization

We obtain sufficient and/or necessary conditions for global/local error bounds for the distances to some sets appeared in set optimization studied with both the set approach and vector approach (sublevel sets, constraint sets, sets of {\it all } Pareto efficient/ Henig proper efficient/super efficient solutions, sets of solutions {\it corresponding to one} Pareto efficient/Henig proper … 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