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"Optimization Solution"
Multi-Newton Optimization Method
Minimum =
      Value =

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);
                 ListBox1.Items.Add("x = " + result.ToString());
                 ListBox2.Items.Add("f1(x) = " + f1(result).ToString());
              }

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



Other Implementations...


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Ore Extraction Optimization
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Higher Order Differential Equations
Nonlinear Systems
Numerical Integration
Numerical Differentiation
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