Continuity of functions

Linear programming problem statement

Non-linear program problem statement

Convexity and its application

A function is known as continuous when all of the points are connected

A function is known as discontinuous when all of the points are connected

A function can be both continuous and discontinuous.

A function can never be continuous and discontinuous in its derivatives.

$f=\left(x_1-3\right)^2+\left(x_2+10\right)^2$

$f=ax^3+9x-96$

$f=\frac{1}{x_1}-6x_{1_{ }}^2+8x_{2_{ }}$

$f=\frac{8}{x^2}-\frac{7}{x}+9$

 $x^THx>0\ for\ all\ x\ne0$  

 $x^THx<0\ for\ all\ x\ne0$  

 $x^THx\ge0\ for\ all\ x\ne0$  

 $x^THx<0\ for\ some\ x\ and\ >0for\ other\ x$  

Newton's method

Quasi-Newton's method

Analytical method

Finite different approximations to derivatives

Write down the f(x), f'(x) and f"(x)

Apply the equation: $x^{k+1}=x^k+\frac{f'\left(x^k\right)}{f"\left(x^k\right)}$

Check the values of $f\left(x^k\right)\$ and $f\left(x^{k+1}\right)$

Continue iteration until two x values are similar

The need to calculate both f'(x) and f"(x)

The method converges slowly if f"(x) is near zero

The method can converge if the initial point is not close to the minimum

The error introduced might be big due to finite differencing

The error introduced might be small due to finite differencing

The need to differentiate the function is not needed

The need to continue iteration is not needed

Identify $x^p$ and $x^q$

Find the gradient between these two points

Calculate the new $x^p$ or $x^q$

Write down the f(x), f'(x) and f"(x)