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 x_n, 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:

           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());
              }