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Chapter 4 Unconstrained optimization

Why are numerical methods ?

Attack to minimization problemdirectly by the numerical methods

For analytical method

Step size

Gradient

What are numerical methods ?

Numerical methods that search for an extremum by usingfunction and sometimes derivative values of 𝑓 𝑋 at asequence of trial points 𝑋1, 𝑋2… until 𝑓 𝑋𝑛+1 − 𝑓 𝑋𝑛 < 𝜀

Procedure for numerical methods

• Starting at 𝑋𝑜 in a certain direction Ԧ𝑠𝑜Step 1

• Move to 𝑋𝑛 with a step size ∆𝑋 = 𝛼𝑜 Ԧ𝑠𝑜Step 2

• Change the direction (if any) and move to 𝑋𝑛+1Step 3

• Stop searching until 𝑓 𝑋𝑛+1 − 𝑓 𝑋𝑛 < 𝜀Step 4

Minimize the following function:

Starting at 𝑥𝑜 = 0 and 𝑥𝑛+1 = 𝑥𝑛 + 𝛼2𝑛−1

Classification of numerical methods

Bracketing methods

𝑋𝑛+1 = 𝑋𝑛 + ∆𝑋

Golden section search

2 ≅ 1.618

Golden section search

Golden section search

Choose the best

For single variable problem, 𝑥5 is determined by vector 𝐴

Scanning methods

6 points and center

Starting at a center point and move

to given point in certain direction

Simplex search

𝑋1

3 points and 3 reflection

Starting at set of 3 points and move

to another set in certain direction

Derivative methods

The first approach for single variable

Based on “necessary condition”, a local extremum isalso the stationary point

Extrema searching directly is replaced by rootsolution of first order indirectly

𝑓′ 𝑥 = 0

Regula falsi method

For quadratic function, Newton

method needs only one step

The second approach for multivariable

Scalar value 𝛼𝑛 and search direction Ԧ𝑠 are calculated

based on the derivatives

𝑋𝑛+1 = 𝑋𝑛 + 𝛼𝑛 Ԧ𝑠

The second approach for multivariable

Scalar value 𝛼𝑛 can be given or calculatedbased on “necessary” condition

Conjugate search

Starting at a point and move to another

point in two conjugate directions

𝑛 variables, 𝑛 axes, only 2 directions needed

𝑓 𝑋 = 2𝑥12 + 𝑥22 − 3

Minimize the following function:

Using conjugate search and starting at 𝑋𝑜 = 1

1 withinitial direction being Ԧ𝑠𝑜 = −4

−2

𝑓 𝑋 = 2𝑥12 + 𝑥22 − 3

Determine the conjugate direction Ԧ𝑠1 of Ԧ𝑠𝑜

For the objective function:

𝑋𝑜′ = 𝑋𝑜 + 𝛼𝑜𝑠𝑜 = −0.1111

0.4444

Steepest descent

Starting at a point and move to another

point in gradient direction

𝑓 𝑋 = 𝑥14 − 2𝑥2𝑥12 + 𝑥22 + 𝑥12 − 2𝑥1 + 5

Minimize the following function:

Using Steepest descent method, starting at 𝑋𝑜 = 1

2and scalar 𝛼𝑜 = 0.05

Determine the gradient

Confirm position at starting point

Calculate the next point 𝑋1 = 𝑋𝑜 − 𝛼𝑜𝛻𝑓 𝑋𝑜 = 1.2

1.9

𝑓 𝑋1 = 4.25

Conjugate gradient search

• Conjugate search: moving on linear combination of Ԧ𝑠𝑖 and Ԧ𝑠𝑗

• Steepest descent: moving on gradient direction 𝛻𝑓 𝑋𝑛

• Conjugate gradient: moving on linear combination of

𝛻𝑓 𝑋𝑛−1 and 𝛻𝑓 𝑋𝑛

Conjugate gradient search

𝑋𝑛+1 = 𝑋𝑛 + 𝛼𝑛𝑠𝑛

Scalar 𝛼𝑛 is calculated by:

Direction 𝑠𝑛 is determined by:

𝑓 𝑋 = 4𝑥12 + 𝑥22 − 2𝑥1𝑥2

Minimize the following function:

Using Newton method and starting at 𝑋𝑜 = 1

1

Secant: 𝑘 = 1+ 5

2 ≅ 1.618Newton: 𝑘 = 2

The function value methods:

• The rate of convergence is low

• Suitable to the complicated or discontinuous functions

• Easily for coding