Abstract—Many image processing problems are ill posed and must be regularized. Usually, a roughness penalty is imposed on the solution. The difficulty is to avoid the smoothing of edges, which are very important attributes of the image. In this paper, we first give conditions for the design of such an edge-preserving regularization. Under these conditions, we show that it is possible to introduce an auxiliary variable whose role is twofold. First, it marks the discontinuities and ensures their preservation from smoothing. Second, it makes the criterion half-quadratic. The optimization is then easier. We propose a deterministic strategy, based on alternate minimizations on the image and the auxiliary variable. This leads to the definition of an original reconstruction algorithm, called ARTUR. Some theoretical properties of ARTUR are discussed. Experimental results illustrate the behavior of the algorithm. These results are shown in the field of tomography, but this method can be applied in a large number of applications in image processing. I.

. We introduce an abstract definition of pattern search methods for solving nonlinear unconstrained optimization problems. Our definition unifies an important collection of optimization methods that neither computenor explicitly approximate derivatives. We exploit our characterization of pattern search methods to establish a global convergence theory that does not enforce a notion of sufficient decrease. Our analysis is possible because the iterates of a pattern search method lie on a scaled, translated integer lattice. This allows us to relax the classical requirements on the acceptance of the step, at the expense of stronger conditions on the form of the step, and still guarantee global convergence. Key words. unconstrained optimization, convergence analysis, direct search methods, globalization strategies, alternating variable search, axial relaxation, local variation, coordinate search, evolutionary operation, pattern search, multidirectional search, downhill simplex search AMS(M...

In this paper we discuss the development and use of low-rank approximate nonnegative matrix factorization (NMF) algorithms for feature extraction and identification in the fields of text mining and spectral data analysis. The evolution and convergence properties of hybrid methods based on both sparsity and smoothness constraints for the resulting nonnegative matrix factors are discussed. The interpretability of NMF outputs in specific contexts are provided along with opportunities for future work in the modification of NMF algorithms for large-scale and time-varying datasets. Key words: nonnegative matrix factorization, text mining, spectral data analysis, email surveillance, conjugate gradient, constrained least squares.

Abstract. Direct search methods are best known as unconstrained optimization techniques that do not explicitly use derivatives. Direct search methods were formally proposed and widely applied in the 1960s but fell out of favor with the mathematical optimization community by the early 1970s because they lacked coherent mathematical analysis. Nonetheless, users remained loyal to these methods, most of which were easy to program, some of which were reliable. In the past fifteen years, these methods have seen a revival due, in part, to the appearance of mathematical analysis, as well as to interest in parallel and distributed computing. This review begins by briefly summarizing the history of direct search methods and considering the special properties of problems for which they are well suited. Our focus then turns to a broad class of methods for which we provide a unifying framework that lends itself to a variety of convergence results. The underlying principles allow generalization to handle bound constraints and linear constraints. We also discuss extensions to problems with nonlinear constraints.

An algorithm for solving large-scale nonlinear ' programs with linear constraints is presented. The method combines efficient sparse-matrix techniques as in the revised simplex method with stable quasi-Newton methods for handling the nonlinearities. A general-purpose production code (MINOS) is described, along with computational experience on a wide variety of problems.

Abstract. We study the convergence properties of a (block) coordinate descent method applied to minimize a nondifferentiable (nonconvex) function f(x1,...,xN) with certain separability and regularity properties. Assuming that f is continuous on a compact level set, the subsequence convergence of the iterates to a stationary point is shown when either f is pseudoconvex in every pair of coordinate blocks from among NA1 coordinate blocks or f has at most one minimum in each of NA2 coordinate blocks. If f is quasiconvex and hemivariate in every coordinate block, then the assumptions of continuity of f and compactness of the level set may be relaxed further. These results are applied to derive new (and old) convergence results for the proximal minimization algorithm, an algorithm of Arimoto and Blahut, and an algorithm of Han. They are applied also to a problem of blind source separation. Key Words. Block coordinate descent, nondifferentiable minimization, stationary point, Gauss–Seidel method, convergence, quasiconvex functions,

Asynchronous parallel pattern search (APPS) is a nonlinear optimization algorithm that dynamically initiates actions in response to events, rather than cycling through a fixed set of search directions, as is the case for synchronous pattern search. This gives us a versatile concurrent strategy that allows us to effectively balance the computational load across all available processors. However, the semi-autonomous nature of the search complicates the analysis. We concentrate on elucidating the concepts and notation required to track the iterates produced by APPS across all participating processes. To do so, we consider APPS and its synchronous counterpart (PPS) applied to a simple problem. This allows us both to introduce the bookkeeping we found necessary for the analysis and to highlight some of the fundamental differences between APPS and PPS.

We discuss direct search methods for unconstrained optimization. We give a modern perspective on this classical family of derivative-free algorithms, focusing on the development of direct search methods during their golden age from 1960 to 1971. We discuss how direct search methods are characterized by the absence of the construction of a model of the objective. We then consider a number of the classical direct search methods and discuss what research in the intervening years has uncovered about these algorithms. In particular, while the original direct search methods were consciously based on straightforward heuristics, more recent analysis has shown that in most — but not all — cases these heuristics actually

We present an algorithmic model for distribnted computation of fixed points whereby several processors participate simultaneously in the calculations while exchanging information via communication links. We place essentially no assumptions on the ordering of computation and communication between processors thereby allowing for completely uncoordinated execution. We provide a general convergence theorem for algorithms of this type, and demonstrate its applicability to several classes of problems including the calculation of fixed points of contraction and monotone mappings arising in linear and nonlinear systems of equations, optimization problems, shortest path problems, and dynamic programming.

In this paper a family of trust-region interior-point SQP algorithms for the solution of a class of minimization problems with nonlinear equality constraints and simple bounds on some of the variables is described and analyzed. Such nonlinear programs arise e.g. from the discretization of optimal control problems. The algorithms treat states and controls as independent variables. They are designed to take advantage of the structure of the problem. In particular they do not rely on matrix factorizations of the linearized constraints, but use solutions of the linearized state equation and the adjoint equation. They are well suited for large scale problems arising from optimal control problems governed by partial differential equations. The algorithms keep strict feasibility with respect to the bound constraints by using an affine scaling method proposed for a different class of problems by Coleman and Li and they exploit trust--region techniques for equality-constrained optimizatio...