Convex Optimization - Conic Combination


A point of the form $\alpha_1x_1+\alpha_2x_2+....+\alpha_nx_n$ with $\alpha_1, \alpha_2,...,\alpha_n\geq 0$ 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, $coni\left ( S \right )=\left \{ \displaystyle\sum\limits_{i=1}^k \lambda_ix_i:x_i \in S,\lambda_i\in \mathbb{R}, \lambda_i\geq 0,i=1,2,...\right \}$

  • The conic hull is a convex set.
  • The origin always belong to the conic hull.