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Concave and Convex Functions
Published in Prem K. Kythe, Elements of Concave Analysis and Applications, 2018
A set Y in X is called a cone with vertex at the origin if y∈Y $ y\in Y $ implies αy∈Y $ \alpha y\in Y $ for all α≥0 $ \alpha \ge 0 $ . Hence, if Y is a cone with vertex at the origin, then the set x0+Y $ x_0+Y $ , where x0∈X $ x_0\in X $ , is called a cone with vertex x0 $ x_0 $ . A convex cone is a set which is both convex and a cone. Some examples of cones are shown in Figure 3.3.
Gaussian Graphical Models
Published in Marloes Maathuis, Mathias Drton, Steffen Lauritzen, Martin Wainwright, Handbook of Graphical Models, 2018
Throughout this chapter, we denote by Sp $ \mathbb S ^p $ the vector space of real symmetric p×p $ p\times p $ matrices. This vector space is equipped with the trace inner product⟨A,B⟩:=tr(AB) $ \,\langle A, B \rangle := \mathrm{tr}(A B) $ . In addition, we denote by S⪰0p $ \mathbb S ^p_{\succeq 0} $ the convex cone of positive semidefinite matrices. Its interior is the open cone S≻0p $ \mathbb S ^p_{\succ 0} $ of positive definite matrices.
The minimal Orlicz mean width of convex bodies
Published in Applicable Analysis, 2022
Let A be a non-singular matrix. Then A can be represented in the form A = TQ, where T is symmetric and positive definite, and Q is an orthogonal matrix (see [63, p.112.]). Identify a symmetric matrix with the point (see [47]) The set of all symmetric positive definite matrices then is represented by an open convex cone with apex at the origin O.
On equivalent representations and properties of faces of the cone of copositive matrices
Published in Optimization, 2022
O. I. Kostyukova, T. V. Tchemisova
This work is motivated by our main challenge: the study of Copositive Programming (CoP) problems and their properties. CoP deals with a special class of conic problems and can be considered as an optimization over a convex cone of the so-called copositive matrices (i.e. matrices that are positive semidefinite on the non-negative orthant). Copositive problems attract the attention of researchers as they have many interesting uses (see, for example, Refs [1–4] and the references therein). The cone of the copositive matrices and its dual, the cone of completely positive matrices, have many applications in reformulation of hard optimization problems (for example, see Refs [2, 4–6]).
Complexity analysis and numerical implementation of large-update interior-point methods for SDLCP based on a new parametric barrier kernel function
Published in Optimization, 2018
Mohamed Achache, Nesrine Tabchouche
Let denotes the linear space of all real symmetric matrices and be the closed convex cone of symmetric positive semidefinite matrices. Given a linear transformation and a symmetric matrix , the semidefinite linear complementarity problem (SDLCP) is the problem of finding a pair matrices such that