Direct Search Methods - Numerical Methods - Lecture Slides, Slides for Numerical Analysis. Central University of Kerala

Numerical Analysis

Description: Main points are: Direct Search Methods, Multi Dimensional, Objective Function, Coordinate Cycling Method, Determine Optimal Solution, Golden Search Method, Two-Dimensional Search Problem, First Iteration, Coordinate Direction
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Multi Dimensional Direct Search
Methods
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Multi Dimensional Direct Search
Methods Method-Overview
Obvious approach is to enumerate all possible
solutions and find the min or the max.
Very generally applicable but computationally
complex
Direct search methods are open
A good initial estimate of the solution is
required
The objective function need not be
differentiable
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Coordinate Cycling Method
Starts from an initial point and looks for an optimal
solution along each coordinate direction iteratively.
For a function with two independent variables x and y,
starting at an initial point (x0,y0), the first iteration will
first move along direction (1, 0) until an optimal
solution is found for the function .
The next search involves searching along the direction
(0,1) to determine the optimal value for the function.
Once searches in all directions are completed, the
process is repeated in the next iteration and iterations
continue until convergence occurs.
The search along each coordinate direction can be
conducted using anyone of the one-dimensional search
techniques previously covered.
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Example
The cross-sectional area A of a gutter with base length
b
and edge length of
l
is given by
θθ
sin
)cos2(
2
1llb
bA ++=
.
Assuming that the width of material to be bent into the gutter shape is 6, find the angle
θ
and edge length
l
which maximizes the cross-sectional area of the gutter.
l
b
l
θ θ
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