# Blog posts

Under construction....

## Why is RRT efficient?

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Trajectory optimization is a local method, it only explores the neighborhood of an initial seed, thus it scales well in high dimensional space. But RRT is a global method, what makes it efficient in exploring a collision-free path in high dimensional space?

## Why is convex optimization important?

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### Definition of convex function

For all ${ 0\leq t \leq 1}$ and all ${ x_{1},x_{2}\in X,}$ ${ f\left(tx_{1}+(1-t)x_{2}\right)~\leq ~tf\left(x_{1}\right)+(1-t)f\left(x_{2}\right).}$

Quadratic form in variables ${ x_1,x_2…, x_n}$ is a polynomial function $Q$, where all the terms in $Q(x_1, x_2,…, x_n)$ have order two. Quadratic functions $\neq$ convex functions.

Let $Q(x_1, x_2,…, x_n)=\boldsymbol{x}^TA\boldsymbol{x}$ be a quadratic form in n variables, $A$ is an sysmetric matrix, then:

• $Q$ is convex/concave $\iff$ $A$ is positive/negative semidefinite
• $Q$ is strictly convex/concave $\iff$ $A$ is positive/negative definite

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