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A crucial issue in digital topology is to ensure topology preservation for reductions acting on binary pictures (i.e., operators that never change a white point to black one). Some sufficient conditions for topology-preserving reductions have been proposed for pictures on the three possible regular partitionings of the plane (i.e., the triangular, the square, and the hexagonal grids). In this paper, the relationships among these conditions are stated.

We study the set , where , j =  1, ..., n , and B 1 | ... | B n . The set S generalizes the mixed-integer rounding (MIR) set of Nemhauser and Wolsey and the mixing-MIR set of Günlük and Pochet. In addition, it arises as a substructure in general mixed-integer programming (MIP), such as in lot-sizing. Despite its importance, a number of basic questions about S remain unanswered, including the tractability of optimization over S and how to efficiently find a most violated cutting plane valid for P =  conv ( S ). We address these questions by analyzing the extreme points and extreme rays of P . We give all extreme points and extreme rays of P . In the worst case, the number of extreme points grows exponentially with n . However, we show that, in some interesting cases, it is bounded by a polynomial of n . In such cases, it is possible to derive strong cutting planes for P efficiently. Finally, we use our results on the extreme points of P to give a polynomial-time algorithm for solving optimization over S .

Robust optimization is still a relatively new approach to optimization problems affected by uncertainty, but it has already proved so useful in real applications that it is difficult to tackle such problems today without considering this powerful methodology. Written by the principal developers of robust optimization, and describing the main achievements of a decade of research, this is the first book to provide a comprehensive and up-to-date account of the subject. Robust optimization is designed to meet some major challenges associated with uncertainty-affected optimization problems: to operate under lack of full information on the nature of uncertainty; to model the problem in a form that can be solved efficiently; and to provide guarantees about the performance of the solution. The book starts with a relatively simple treatment of uncertain linear programming, proceeding with a deep analysis of the interconnections between the construction of appropriate uncertainty sets and the classical chance constraints (probabilistic) approach. It then develops the robust optimization theory for uncertain conic quadratic and semidefinite optimization problems and dynamic (multistage) problems. The theory is supported by numerous examples and computational illustrations. An essential book for anyone working on optimization and decision making under uncertainty,Robust Optimizationalso makes an ideal graduate textbook on the subject.

Critical kernels constitute a general framework in the category of abstract complexes for the study of parallel thinning in any dimension. The most fundamental result in this framework is that, if a subset Y of X contains the critical kernel of  X , then Y is guaranteed to have “the same topology as X ”. Here, we focus on 2D structures in spaces of two and three dimensions. We introduce the notion of crucial pixel, which permits to link this work with the framework of digital topology. We prove simple local characterizations, which allow us to express thinning algorithms by way of sets of masks. We propose several new parallel algorithms, which are both fast and simple to implement, that yield symmetrical or non-symmetrical skeletons of 2D objects in 2D or 3D grids. We prove some properties of these skeletons, related to topology preservation, to minimality, and to the inclusion of the topological axis. The latter may be seen as a generalization of the medial axis. We also show how to use critical kernels in order to provide simple proofs of the topological soundness of existing thinning schemes. Finally, we clarify the link between critical kernels, minimal non-simple sets, and P-simple points.

Critical kernels constitute a general framework in the category of abstract complexes for the study of parallel homotopic thinning in any dimension. In this article, we present new results linking critical kernels to minimal non-simple sets (MNS) and P-simple points, which are notions conceived to study parallel thinning in discrete grids. We show that these two previously introduced notions can be retrieved, better understood and enriched in the framework of critical kernels. In particular, we propose new characterizations which hold in dimensions 2, 3 and 4, and which lead to efficient algorithms for detecting P-simple points and minimal non-simple sets.

We consider two-level fractional factorial designs under a baseline parametrization that arises naturally when each factor has a control or baseline level. While the criterion of minimum aberration can be formulated as usual on the basis of the bias that interactions can cause in the estimation of main effects, its study is hindered by the fact that level permutation of any factor can impact such bias. This poses a serious challenge especially in the practically important highly fractionated situations where the number of factors is large. We address this problem for regular designs via explicit consideration of the principal fraction and its cosets, and obtain certain rank conditions which, in conjunction with the idea of minimum moment aberration, are seen to work well. The role of simple recursive sets is also examined with a view to achieving further simplification. Details on highly fractionated minimum aberration designs having up to 256 runs are provided.

Preserving topological properties of objects during thinning procedures is an important issue in the field of image analysis. In the case of 2-D digital images (i.e. images defined on ℤ 2 ) such procedures are usually based on the notion of simple point. In contrast to the situation in ℤ n , n ≥3, it was proved in the 80s that the exclusive use of simple points in ℤ 2 was indeed sufficient to develop thinning procedures providing an output that is minimal with respect to the topological characteristics of the object. Based on the recently introduced notion of minimal simple set (generalising the notion of simple point), we establish new properties related to topology-preserving thinning in 2-D spaces which extend, in particular, this classical result to cubical complexes in 2-D pseudomanifolds.

In this paper, we discuss the notion of an m-polar fuzzy (normal) subalgebra in B-algebras and its related properties. We consider characterizations of an m-polar fuzzy (normal) subalgebra. We define the concept of an m-polar intuitionistic fuzzy (normal) subalgebra in a B-algebra, and we characterize the m-polar intuitionistic fuzzy (normal) subalgebra. Given an m-polar fuzzy set, we construct a simple m-polar fuzzy set and discuss on m-polar intuitionistic fuzzy subalgebras of B-algebras.

Preserving topological properties of objects during reduction procedures is an important issue in the field of discrete image analysis. Such procedures are generally based on the notion of simple point , the exclusive use of which may result in the appearance of “topological artifacts.” This limitation leads to consider a more general category of objects, the simple sets , which also enable topology-preserving image reduction. A study of two-dimensional simple sets in two-dimensional spaces has been proposed recently. This article is devoted to the study of two-dimensional simple sets in spaces of higher dimension (i.e., n -dimensional spaces, n ≥3). In particular, several properties of minimal simple sets (i.e., which do not strictly include any other simple sets) are proposed, leading to a characterisation theorem. It is also proved that the removal of a two-dimensional simple set from an object can be performed by only considering the minimal ones, thus authorising the development of efficient thinning algorithms.

We are concerned with concave programming or the convex maximization problem. In this paper, we propose a method and algorithm for solving the problem which are based on the global optimality conditions first obtained by Strekalovsky (Soviet Mathematical Doklady, 8(1987)). The method continues approaches given in (Journal of global optimization, 8(1996); Journal of Nolinear and convex Analyses 4(1)(2003)). Under certain assumptions a convergence property of the proposed method has been established. Some computational results are reported. Also, it has been shown that the problem of finding the largest eigenvalue can be found by the proposed method. [PUBLICATION ABSTRACT]