Convex Optimization Tutorial
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Conic Combination
Convex Optimization - Conic Combination
A point of the form $alpha_1x_1+alpha_2x_2+....+alpha_nx_n$ with $alpha_1, alpha_2,...,alpha_ngeq 0$ is called conic combination of $x_1, x_2,...,x_n.$
If $x_i$ are in convex cone C, then every conic combination of $x_i$ is also in C.
A set C is a convex cone if it contains all the conic combination of its elements.
Conic Hull
A conic hull is defined as a set of all conic combinations of a given set S and is denoted by coni(S).
Thus, $conileft ( S ight )=left { displaystylesumpmits_{i=1}^k lambda_ix_i:x_i in S,lambda_iin mathbb{R}, lambda_igeq 0,i=1,2,... ight }$
The conic hull is a convex set.
The origin always belong to the conic hull.