Minimization Alignment

Usage notes

For each of the following functions and for each minimization method described in class, determine which of the following cases applies:

method is not applicable
method is applicable and recommended
method is applicable, but not recommended
method converges to the minimum in one iteration

and choose which of the following comments (referenced as "Comment 1", ..., "Comment 7" in the table below) applies best for each function.

Note: Inverse parabolic interpolation was not covered by Prof. Gonnet this year, but you can find it in section 1.5.2.2 of the online script.

The method is applicable to one-dimensional problems only.
Brent includes Golden section search, but converges much faster for smooth functions.
Non-smooth function makes inverse parabolic interpolation of Brent's method useless.
For functions in several variables, random direction search is better than steepest descent or Newton only in case of non-smooth functions. All of f1, f4 and f5 are smooth enough.
Random directions and steepest descent are not applicable per se to one-dimensional problems, only the line search algorithm used in both can be aplied to one-dimensional problems.
Newton-Raphson converges in one step to the minimum when applied to quadratic functions.
The scalar Newton-Raphson is applicable (and quickly convergent) for functions smooth enough. The function f3 is not differentiable.

Function

Method: case and comment

f₃(x₁) = x₁ - |x₁-5| + |3x₁ + 10|

1. Golden section search
2. Brent's method
3. Random directions
4. Steepest descent and conjugate gradients
5. Newton and spectral method

f₄(x₁, x₂) = |x₁ - 3|² + |x₂ + 1|² - x₁x₂ + 3x₁ + 2

1. Golden section search
2. Brent's method
3. Random directions
4. Steepest descent and conjugate gradients
5. Newton and spectral method

f₅(x₁, x₂) = (x₁ - 3x₂) / (x₁² + x₂² + 10)

1. Golden section search
2. Brent's method
3. Random directions
4. Steepest descent and conjugate gradients
5. Newton and spectral method

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