Golden section method

Professor:Dr. Sahand Daneshvar

Presented by: Mahdi banar

Problem Statement

Basic DefinitionsThe function f (x) is usually called the objective

function.gi (x) ≤ 0, I = 1;:::; n g, is called an inequality

constraint.hi(x) = 0, i = 1; : : : ; n h, is called an equality

constraint.

TheoremLet θ : R R be strictly quasi convex over

the interval [a, b]. Let λ, μ ε [a, b] be such that λ < μ.

If θ (λ) > θ (μ), then θ (z) ≥ θ (μ) for all z ε [a, λ).

If θ (λ) ≤ θ (μ), then θ (z) ≤ θ (λ) for all z ε (μ, b].

The Golden Section MethodThe golden section search is a technique

for finding the minimum or maximum of a strictly unimodal function by successively narrowing the range of values inside which the min or max is known to exist. The technique derives its name from the fact that the algorithm maintains the function values for triples of points whose distances form a golden ratio.

Cont.At a general iteration k of the golden section

method, let the interval of uncertainty be [ak,

book].By Theorem the new interval of uncertainty [ak+1, bk+1] is given by [λk, bk] if θ (λ) > θ (μ) and by [ak, μk] if θ (λ) ≤ θ (μ). The points λk and μk are selected such that the following hold true.

λk = ak +(1-α)(bk - ak) if α ε (0, 1) μk = ak +α (bk - ak)

Cont.As λk+1 and μk+1 are selected for the purpose

of a new iteration, either λk+1 coincides with μk or coincides with λk . If this can be realized, then during iteration k + 1, only one extra observation is needed.

Golden section ruleCase 1 : if θ(λk) > θ(μk) ak+1= λk

bk+1 = bk

λk+1 =μk

μk+1 = ?

Cont.Case 2: if θ (λk) ≤ θ (μk) ak+1= ak

bk+1 = μk

μ k+1 =λk

λ k+1 =?

Golden Number (α)Consider case 1 or 2 Case 1: μk = λk+1 = ak+1 + (1-α) (bk+1 - ak+1)

= λk + (1- α)(bk - λk) we know λk = ak +(1-α)(bk - ak) and μk = ak +α (bk

- ak) (1)After Substituting the expressions λk and μk from

(1) into the above equation, we get The roots are α≈0. 618 and α≈-1.618For α must be in the interval (0,1) then α≈0. 618 This number is golden.

Algorithm golden section methodThe golden section method for minimizing a

strictly quasiconvex function over the interval [a1, b1].

Initialization Step:Choose an allowable final length of

uncertainty ℓ>0[a1, b1] is the initial interval of uncertainty: We computed λ1 = a1 + (1-α) (b1 - a1) and μ1 = a1 +α (b1 - a1) let α≈0. 618 Evaluate θ (λ1) and θ (μ1), let k=1 and go to main

step

MAIN STEP1. If bk – ak < ℓ stop. The optimal solution lies in the

interval [ak, book]. Otherwise, if θ (λk) > θ (μk), go to step 2; and if θ (λk) ≤ θ (μk) go to step 3.

2. ak+1= λk, bk+1 = bk , λk+1 =μk , μk+1 = ?

3. ak+1= ak , bk+1 = μk , μ k+1 =λk , λ k+1

= ?

4 . Replace k by k+1 and go to step 1

Example Consider the following problem:

Solution: design a table with below components:

k ak bk λk μk θ(λk) θ(μk)1 -3.000 5.000 0.056 1.944 0.115 7.667

Cont.

After eight iterations involving nine observations, the interval of uncertainty is [-1.112 , -0.9361], so that the minimum can be estimated to be the midpoint -1.024.Note that the true minimum is in fact -1.0.