Numerical on bisection method

OPTIMIZATION ALGORITHMS

NUMERICAL ON BISECTION METHOD

BY Sumita Das

Bisection Method It is a Derivative Based Method for Optimization

Requirements for Bisection Method f -> c’ i.e. f is continuous for the first derivative.There exists a minima in the level of uncertainty [a

b]Function must be unimodal.

PowerPoint Presentation by Sumita Das, GHRCE

Algorithm Initialize Level of uncertainty [a b]k=1ak =abk =bϵ > 0l : Allowable level of uncertainty such that (1/2) n <= (1/(b`-a`))


While k<= nck= (ak +bk )/2if f(ck)=0

Stop with ck as the solution

if f(ck)>0ak+1 = ak

bk+1 = ck else

ak+1 = ck

bk+1 = bk end if

k=k+1end while

Find midpoint

c is now b.a remains same

c is now a. b remains same

Midpoint is the minima


In Simple words

Midpoint

ma b

Example: f(x)=3x2 – 2x f’(x)=6x-2

Put midpoint value in derivative.f’(x)=6*50-2=298

1. if f(m)=0, Midpoint is minima

2. if f(m)>0, Level of uncertainty will be [a, m]

3. if f(m)<0, Level of uncertainty will be [m, b]

50 9010

So, 298>0, level of uncertainty will be [10, 50] ,Follow the procedure.


ExampleQue: Find Minima f(x)=(x-2)2 [0 6]Solution: f ‘(x)=2x-4k ak bk ck f’(ck )1 0 6 3 22 0 3 1.5 -1

3 1.5 3 2.25 0.5

4 1.5 2.25 1.875 -0.25

5 1.875 2.25 2.062 0.123

6 1.875 2.062 1.9685 -0.063

7 1.9685 2.062 2.015 0.0305

8 1.9685 2.015 1.99175 -0.016

9 1.99175 2.015 2.003 0.006


References[1] Singiresu S. Rao, “Engineering Optimization, Chapter 5: Nonlinear Programming I: One-Dimensional Minimization Methods”, 4th Edition


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Numerical on bisection method