using convex in amidpoint method

[Rock]

Hello,


How can we derived the midpoint method using the convex set
( Hint A set C in S is said to be convex if, for all x and y in C and all t in the interval [0,1], the point (1 − t ) x + t y is in C.)


Regards

 

[pka]

How can we derived the midpoint method using the convex set ( Hint A set C in S is said to be convex if, for all x and y in C and all t in the interval [0,1], the point (1 − t ) x + t y is in C.)

What if \(\displaystyle t=\frac{1}{2}~?\)

 

[Rock]

What if \(\displaystyle t=\frac{1}{2}~?\)

But how we choose the value of t?

 

[pka]

But how we choose the value of t?

Well actually I had to guess that you meant to find the midpoint of the line segment \(\displaystyle \overline{xy}\).
Now look at the hint: \(\displaystyle (1-t)x+ty\) is a continuous mapping of \(\displaystyle [0,1]\) onto \(\displaystyle \overline{xy}\). Half way, the midpoint, occurs if \(\displaystyle t=\frac{1}{2}~.\) That is \(\displaystyle (1-\frac{1}{2})x+\frac{1}{2}y=\frac{x+y}{2}~.\)