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Geometry
Published in Leslie Hogben, Richard Brualdi, Anne Greenbaum, Roy Mathias, Handbook of Linear Algebra, 2006
This chapter begins by discussing affine geometry, which can be defined, very roughly, as Euclidean geometry without any mention of measurement. Thus, affine theorems concern such things as incidence and parallelism. An affine space is defined in terms of an action of a vector space V on a set X, pursuant to which a vector v acts on a point A of X by sending it to another element v(A) of X, and two points A and B of X define a unique vector AB→, which acts on A by sending it to B. It might be helpful for the reader to think of AB→ as an arrow starting at point A and ending at point B. This vector then acts on an arbitrary point C of X by placing the starting point of the arrow at C and letting the end point of the arrow be v(C).
Elements of topology and homology
Published in Rodrigo Rojas Moraleda, Nektarios A. Valous, Wei Xiong, Niels Halama, Computational Topology for Biomedical Image and Data Analysis, 2019
Rodrigo Rojas Moraleda, Nektarios A. Valous, Wei Xiong, Niels Halama
Every ordered pair of points P and Q in an affine space is associated with a vector PQ→ by applying ϕ(P,Q)=PQ→=Q−P. In the affine space, vectors and points are interchangeable terms.
Fundamentals to Geometric Modeling and Meshing
Published in Yongjie Jessica Zhang, Geometric Modeling and Mesh Generation from Scanned Images, 2018
There are three commonly used spaces: vector space, affine space and Euclidean space. The linear vector space consists of only two objects, scalars and vectors. The affine space includes an additional element, the point, to the vector space. The Euclidean space further adds distance to the affine space. In the following, let us review the definition and properties of scalars and these three spaces in detail.
Solving SDP completely with an interior point oracle
Published in Optimization Methods and Software, 2021
Bruno F. Lourenço, Masakazu Muramatsu, Takashi Tsuchiya
There is a growing body of research aimed at understanding SDPs and conic linear programs having pathological behaviours such as nonzero duality gaps and weak infeasibility. Here we will mention a few of them. A problem is called weakly infeasible if there is no feasible solution but the distance between the underlying affine space and the cone under consideration is zero. Weak infeasibility is known to be very hard to detect numerically, see for instance Pólik and Terlaky [38]. In [50], Waki showed that weakly infeasible problems sometimes arise from polynomial optimization. There is also a discussion on weak infeasibility semidefinite programming and second-order cone programming in [22,24], respectively. Some of the results in [22] were generalized to arbitrary closed convex cones by Liu and Pataki, see [18] for more details. See also [23], where some results of [18] on weakly infeasible problems are sharpened when the polyhedral faces of the underlying cone are taken into account.
A new wide neighbourhood primal-dual interior-point algorithm for semidefinite optimization
Published in Optimization, 2019
Interior-point methods (IPMs) are one of the efficient methods developed to solve semidefinite optimization (SDO), which optimizes a linear objective function over the intersection of the cone of positive semidefinite matrices with an affine space. The SDO problems have several applications in continuous optimization, combinatorial optimization [1], system and control theory [2] and eigenvalue optimization problems [3]. These two factors caused that the SDO has retreated a considerable attention in recent years. IPMs were originally and independently proposed by Nesterov and Nemirovskii [4] and Alizadeh [5] to the context of SDO. Several authors have proposed interior-point algorithms for solving SDO problems including Kojima, Shindoh and Hara [6], Nesterov and Todd [7,8], Monteiro [9–11] and Monteiro and Zhang [12].
Quadratic convergence to the optimal solution of second-order conic optimization without strict complementarity
Published in Optimization Methods and Software, 2019
Ali Mohammad-Nezhad, Tamás Terlaky
Second-order conic optimization (SOCO) problems minimize a linear objective function over the intersection of an affine space and Cartesian product of p second-order (Lorentz) cones of dimension , i.e. where The primal and dual SOCO problems in standard form are represented as where , , , , and , in which , , and for . Notice that x, s, and c are concatenation of the column vectors , , and , respectively. A wide range of applications in engineering, control, robust optimization, and combinatorial optimization can be modeled as SOCO problems, see e.g. [2,21] for the applications of SOCO.