: When computing the infimal convolution of a convex function f with the squared norm, one obtains the so-called Moreau-Yosida regularization of f . Among other things, this function has a Lipschitzian gradient. We investigate some more of its properties, relevant for optimization. Our main result concerns second-order differentiability and is as follows. Under assumptions that are quite reasonable in optimization, the Moreau-Yosida is twice diffferentiable if and only if f is twice differentiable as well. In the course of our development, we give some results of general interest in convex analysis. In particular, we establish primaldual relationship between the remainder terms in the first-order development of a convex function and its conjugate. Key-words: Convex optimization, mathematical programming, proximal point, secondorder differentiability. (R'esum'e : tsvp) Unite de recherche INRIA Rocquencourt Domaine de Voluceau, Rocquencourt, BP 105, 78153 LE CHESNAY Cedex (France) Telep...

A unified fixed point theoretic framework is proposed to investigate the asymptotic behavior of algorithms for finding solutions to monotone inclusion problems. The basic iterative scheme under consideration involves nonstationary compositions of perturbed averaged nonexpansive operators. The analysis covers proximal methods for common zero problems as well as various splitting methods for finding a zero of the sum of monotone operators.

To minimize a convex function, we combine Moreau-Yosida regularizations, quasiNewton matrices and bundling mechanisms. First we develop conceptual forms using "reversal " quasi-Newton formulae and we state their global and local convergence. Then, to produce implementable versions, we incorporate a bundle strategy together with a "curve-search". No convergence results are given for the implementable versions; however some numerical illustrations show their good behaviour even for large-scale problems.

Abstract. Image inpainting is a fundamental problem in image processing and has many applications. Motivated by the recent tight frame based methods on image restoration in either the image or the transform domain, we propose an iterative tight frame algorithm for image inpainting. We consider the convergence of this framelet-based algorithm by interpreting it as an iteration for minimizing a special functional. The proof of the convergence is under the framework of convex analysis and optimization theory. We also discuss the relationship of our method with other wavelet-based methods. Numerical experiments are given to illustrate the performance of the proposed algorithm. Key words. Tight frame, inpainting, convex analysis 1. Introduction. The problem of inpainting [2] occurs when part of the pixel data in a picture is missing or over-written by other means. This arises for example in restoring ancient drawings, where a portion of the picture is missing or damaged due to aging or scratch; or when an image is transmitted through a noisy channel. The task of inpainting is to recover the missing region from the incomplete data observed. Ideally, the restored image should possess shapes and patterns consistent

We present a new Bregman-function-based algorithm which is a modification of the generalized proximal point method for solving the variational inequality problem with a maximal monotone operator. The principal advantage of the presented algorithm is that it allows a more constructive error tolerance criterion in solving the proximal point subproblems. Furthermore, we eliminate the assumption of pseudomonotonicity which was, until now, standard in proving convergence for paramonotone operators. Thus we obtain a convergence result which is new even for exact generalized proximal point methods. Finally, we present some new results on the theory of Bregman functions. For example, we show that the standard assumption of convergence consistency is a consequence of the other properties of Bregman functions, and is therefore superfluous.

We propose a modification of the classical proximal point algorithm for finding zeroes of a maximal monotone operator in a Hilbert space. In particular, an approximate proximal point iteration is used to construct a hyperplane which strictly separates the current iterate from the solution set of the problem. This step is then followed by a projection of the current iterate onto the separating hyperplane. All information required for this projection operation is readily available at the end of the approximate proximal step, and therefore this projection entails no additional computational cost. The new algorithm allows significant relaxation of tolerance requirements imposed on the solution of proximal point subproblems, which yields a more practical framework. Weak global convergence and local linear rate of convergence are established under suitable assumptions. Additionally, presented analysis yields an alternative proof of convergence for the exact proximal point method, which allow...

Abstract — Under consideration is the large body of signal recovery problems that can be formulated as the problem of minimizing the sum of two (not necessarily smooth) lower semicontinuous convex functions in a real Hilbert space. This generic problem is analyzed and a decomposition method is proposed to solve it. The convergence of the method, which is based on the Douglas-Rachford algorithm for monotone operator-splitting, is obtained under general conditions. Applications to non-Gaussian image denoising in a tight frame are also demonstrated. Index Terms — Convex optimization, denoising, Douglas-Rachford, frame, nondifferentiable optimization, Poisson noise,

A convex variational framework is proposed for solving inverse problems in Hilbert spaces with a priori information on the representation of the target solution in a frame. The objective function to be minimized consists of a separable term penalizing each frame coefficient individually and of a smooth term modeling the data formation model as well as other constraints. Sparsity-constrained and Bayesian formulations are examined as special cases. A splitting algorithm is presented to solve this problem and its convergence is established in infinite-dimensional spaces under mild conditions on the penalization functions, which need not be differentiable. Numerical simulations demonstrate applications to frame-based image restoration. 1

. At a given point p, a convex function f is differentiable in a certain subspace U (the subspace along which @f(p) has 0-breadth). This property opens the way to defining a second derivative of f at p, along U . We do this via an intermediate function, convex on U . We call this function the U-Lagrangian; it coincides with the ordinary Lagrangian in composite cases: exact penalty, semidefinite programming. Also, we use this new theory to design a conceptual pattern for superlinearly convergent minimization algorithms. Finally, we establish a connection with the Moreau-Yosida regularization. 1. Introduction This paper deals with higher-order expansions of a nonsmooth function, a problem addressed in [4], [5], [7], [13], [29] among others. The initial motivation for our present work lies in the following facts. When trying to generalize the classical second-order Taylor expansion of a function f at a nondifferentiability point p, the major difficulty is by far the nonlinear...

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