```markdown
### Newton Decrement

The slide presents the concept of Newton decrement, denoted as \(\lambda(x)\), with the following expressions:

\[
\lambda(x) = \left( \Delta x_{nt}^T \nabla^2 f(x) \Delta x_{nt} \right)^{1/2} = \left( -\nabla f(x)^T \Delta x_{nt} \right)^{1/2}
\]

#### Properties

- **Quadratic Approximation Estimate**: It provides an estimate of \(f(x) - p^\star\) using the quadratic approximation \(\hat{f}\):

  \[
  f(x) - \inf_{Ay=b} \hat{f}(y) = \frac{1}{2} \lambda(x)^2
  \]

- **Directional Derivative in Newton Direction**: 

  \[
  \frac{d}{dt} f(x + t \Delta x_{nt}) \bigg|_{t=0} = -\lambda(x)^2
  \]

- **General Condition**: 

  \[
  \lambda(x) \neq \left( \nabla f(x)^T \nabla^2 f(x)^{-1} \nabla f(x) \right)^{1/2}
  \]

*Source: A. d’Aspremont, Convex Optimization M2, Slide 40/53.*
```