We consider projection algorithms for solving (nonconvex) feasibility problems in Euclidean spaces. Of special interest are the Method of Alternating Projections (MAP) and the Douglas-Rachford algorithm (DR). In the case of convex feasibility, firm nonexpansiveness of projection mappings is a global property that yields global convergence of MAP and for consistent problems DR. A notion of local sub-firm nonexpansiveness with respect to the intersection is introduced for consistent feasibility problems. This, together with a coercivity condition that relates to the regularity of the collection of sets at points in the intersection, yields local linear convergence of MAP for a wide class of nonconvex problems, and even local linear convergence of nonconvex instances of the Douglas-Rachford algorithm.2010 Mathematics Subject Classification: Primary 65K10; Secondary 47H04, 49J52, 49M20, 49M37, 65K05, 90C26, 90C30

We propose a general alternating minimization algorithm for nonconvex optimization problems with separable structure and nonconvex coupling between blocks of variables. To fix our ideas, we apply the methodology to the problem of blind ptychographic imaging. Compared to other schemes in the literature, our approach differs in two ways: (i) it is posed within a clear mathematical framework with practically verifiable assumptions, and (ii) under the given assumptions, it is provably convergent to critical points. A numerical comparison of our proposed algorithm with the current state-of-the-art on simulated and experimental data validates our approach and points toward directions for further improvement.

The problem of finding a vector with the fewest nonzero elements that satisfies an underdetermined system of linear equations is an NP-complete problem that is typically solved numerically via convex heuristics or nicely-behaved nonconvex relaxations. In this work we consider elementary methods based on projections for solving a sparse feasibility problem without employing convex heuristics. It has been shown recently that, locally, the fundamental method of alternating projections must converge linearly to a solution to the sparse feasibility problem with an affine constraint. In this paper we apply different analytical tools that allow us to show global linear convergence of alternating projections under familiar constraint qualifications. These analytical tools can also be applied to other algorithms. This is demonstrated with the prominent Douglas-Rachford algorithm where we establish local linear convergence of this method applied to the sparse affine feasibility problem.

We investigate mild solutions for stochastic evolution equations driven by a fractional Brownian motion (fBm) with Hurst parameter H ∈ (1/3, 1/2] in infinite-dimensional Banach spaces. Using elements from rough paths theory we introduce an appropriate integral with respect to the fBm. This allows us to solve pathwise our stochastic evolution equation in a suitable function space. We are grateful to M. J. Garrido-Atienza and B. Schmalfuß for helpful comments. We thank the referee for carefully reading the manuscript and for the valuable suggestions.AN acknowledges support by a DFG grant in the D-A-CH framework (KU 3333/2-1). 1 index H ∈ (1/3, 1/2]. In order to solve (1.1) we need to give a meaning of the rough integral t 0 S(t − r)G(y r )dω r . (1.2) Results in this context are available in [10] via fractional calculus and in Gubinelli et al [11], [4] [12], [13] using rough paths techniques. In this work, we combine Gubinelli's approach with the arguments employed by [10] to solve (1.1). This theory should hopefully be more simple in order to investigate the long-time behavior of such equations using a random dynamical systems approach such in [1], [8] or [2]. In this work we establish only the existence of a local mild solution. We investigate in a forthcoming paper global solutions and random dynamical systems for (1.1) as in [8]. This work should be seen as a first step in order to close the gap between rough paths and random dynamical systems in infinite-dimensional spaces. Since the fractional Brownian motion is not a semi-martingale, the construction of an appropriate integral represents a challenging problem. This has been intensively investigated and numerous results and various techniques are available, see [26], [8], [25], [16], [19] and the references specified therein. There is a huge literature where certain tools from fractional calculus (i.e. fractional/compensated fractional derivative/integral) are employed to give a pathwise meaning of the stochastic integral with respect to the fractional Brownian motion with Hurst parameter H ∈ (1/2, 1) or H ∈ (1/3, 1/2]. A different method which has been recently introduced and explored is given by the rough path approach of Gubinelli et. al. [11], [13], [12]. This goes through if H ≤ 1/2. Moreover it is suitable to define (1.2) not only with respect to the fractional Brownian motion but also to Gaussian processes for which the covariance function satisfies certain structure, see [6] or [5, Chapter 10]. An overview on the connection between rough paths and fractional calculus can be looked up in [16]. Of course, in some situations, various other techniques for H ≤ 1/3 are available. After using an appropriate integration theory with respect to the fractional Brownian motion, the next step is to analyze SDEs/SPDEs driven by this kind of noise. There is a growing interest in establishing suitable properties of the solution under several assumptions on the coefficients, consult [19], [20], [14], [7], [8], [26] and the references specified therein. To our aims we ...

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