Cone

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A set, LaTeX: C, with the property that if LaTeX: x \in C, then LaTeX: a x \in C, for all positive real LaTeX: a. A convex cone is a cone that is also a convex set. Equivalently, LaTeX: C is a convex cone if LaTeX: C = C + C. (An example of a cone that is not convex is the union of the axes.) A polyhedral cone is a cone that is also a polyhedron; equivalently, LaTeX: C is a polyhedral cone if there exists a matrix LaTeX: A such that LaTeX: C = \{ x : Ax \le 0\}. An example of a cone that is not polyhedral is LaTeX: \{(x,y,z): x^2+y^2-z^2=0\}.

A quadratic cone is of the form LaTeX: \{x: x^T Q x \le 0\}, where LaTeX: Q is any (real) matrix. If LaTeX: Q is negative semi-definite, the cone is all of LaTeX: \mathbb{R}^n space. If LaTeX: Q is positive definite, the cone is just the origin, LaTeX: \{0\}. So, quadratic cones usually arise when LaTeX: Q is indefinite. Example: LaTeX: \{x : x \ge 0, \; 2x_1x_2 \ge \|(x_3,...,x_n)||^2\}. See each of the following special cones:


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