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 "Optimization Solution" Multi-Newton Optimization Method
 Minimum = x = (0.5, -1.2)
 Value = f1(x) = 2.5

 f(x,y) = 1.5 (x - 0.5)2 + 3.4(y + 1.2)2 + 2.5 Initial-Array = { , } [ Tolerance: 1.0e-5]

IMPLEMENTATION
Multi-Newton Method Optimization

We showed optimization methods applicable to functions with a single variable (i.e functions are defined in one-dimensional space). The Newton-Multi optimization method extends this concept to find the minimum of a function with multiple variables.

Algorithm Creation

The basic idea is simple:

• Start with an initial array, which represents initial points in n-dimensional space.

• For each variable, i.e xn, minimize the multi-variable function f(x), where x is a n-dimensional vector.

• Loop over all the variables.

The minimization along a line with a single variable can be accomplished with one-dimensional optimization algorithm. We applied the Newton optimization.

Testing the Multi-Newton Method

To test it out, we find the minimum of a function with multiple variables, given by:

 f(x,y) = 1.5 (x - 0.5)2 + 3.4(y + 1.2)2 + 2.5

static void TestMultiNewton();
{
ListBox1.Items.Clear();
ListBox2.Items.Clear();
double result = Optimization.multiNewton(f1, xarray, 1.0e-5);
}

To test the Multi-Newton method, we used the function defined above. The user can manipulate initial-array as desired.

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