Optimization Algorithms- Bounding & Fibonacci Search

As far as these search are concerned, you can't just have solution of equation so simple than this.

So, here is the Algorithm for two of the best method use in Optimization

Bounding Phase Method

Algorithm

Step 1: Choose an initial guess x ⁽⁰⁾ and an increment Δ. Set k = 0.

Step 2: If f(x ⁽⁰⁾ - IΔI) > f(x ⁽⁰⁾ + IΔI), then Δ is positive;

Else if f (x ⁽⁰⁾ - IΔI) < f(x ⁽⁰⁾) < f(x ⁽⁰⁾ + IΔI), then Δ is negative;

Else go to Step 1.

            Step 3: Set x ^(k+1) = x ^(k) + 2^k Δ.

            Step 4: if f (x ^(k+1)) < f(x ^(k)), set k = k+1 and go to step 3;

Else the minimum lies in the interval (x ^(k-1) , x ^(k+1)) and

Terminate.

Fibonacci search method

Algorithm

Step 1: Choose a lower bound a and an upper bound b. Set L= b – a. Assume the desired number of function evaluations to be n. Set k = 2.

Step 2: Compute L_k^* = (F_n-k+1 / F _n+1) L. Set x₁ = a + L_k^* and x₂ = b - L_k^*.

Step 3: Compute one of f(x₁) or f(x₂), which was not evaluated earlier. Use the fundamental region elimination rule to eliminate a region. Set new a and b.

Step 4: is k = n? If no, set k = k + 1 and go to step 2;

Else Terminate.