---
# System prepended metadata

title: 1. Linear models
tags: [Optimization and Analytics]

---

 # 1. Linear models
 ###### tags: `Optimization and Analytics`

 
 It is worth remembering that linear models can have variables that are square or raised to any power, provided that the relationship between them through the coefficients is still a linear one.
 
 For example $$y = \beta_0 + \beta_1x + \beta_2x^2$$ is linear. For more information check our notes on [[Predictive Modelling]].

TODO: complete with internal notes.

## Feasible solutions

>Fundamental Theorem of Linear Programming (Carathéodory): if a LP is
nonempty and bounded, then there exists an optimal solution which is an
extreme point

This says that if our problem is bounded, our solution is going to be "one corner".

## Simplex algorithm
$A \cdot x_{new} = b$&nbsp;&nbsp;&nbsp;&nbsp; $\quad \quad P=(P_B\ \ P_N)$ &nbsp;&nbsp;&nbsp; $\quad \quad A=(B\ \ N)$
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;||
$Ax + A_p = b \implies A \cdot p = 0$
$$\boxed{P_B = B^{-1}N\cdot P_N} \quad \quad \boxed{x_{new} = x + p}$$
$C^{T} \cdot x_{new} = C^T \cdot x + C^T \cdot p$
$C^{T}p =C_B^T \cdot P_B \dots \ ??$

