- Convex Optimization Tutorial
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- Introduction
- Linear Programming
- Norm
- Inner Product
- Minima and Maxima
- Convex Set
- Affine Set
- Convex Hull
- Caratheodory Theorem
- Weierstrass Theorem
- Closest Point Theorem
- Fundamental Separation Theorem
- Convex Cones
- Polar Cone
- Conic Combination
- Polyhedral Set
- Extreme point of a convex set
- Direction
- Convex & Concave Function
- Jensen's Inequality
- Differentiable Convex Function
- Sufficient & Necessary Conditions for Global Optima
- Quasiconvex & Quasiconcave functions
- Differentiable Quasiconvex Function
- Strictly Quasiconvex Function
- Strongly Quasiconvex Function
- Pseudoconvex Function
- Convex Programming Problem
- Fritz-John Conditions
- Karush-Kuhn-Tucker Optimality Necessary Conditions
- Algorithms for Convex Problems
- Convex Optimization Resources
- Convex Optimization - Quick Guide
- Convex Optimization - Resources
- Convex Optimization - Discussion
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.
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