Gradient Descent - Solution of A Non-linear System

Solution of A Non-linear System

Gradient descent can also be used to solve a system of nonlinear equations. Below is an example that shows how to use the gradient descent to solve for three unknown variables, x1, x2, and x3. This example shows one iteration of the gradient descent.

Consider a nonlinear system of equations:

\begin{cases} 3x_1-\cos(x_2x_3)-\tfrac{3}{2}=0 \\ 4x_1^2-625x_2^2+2x_2-1=0 \\ \exp(-x_1x_2)+20x_3+\tfrac{10\pi-3}{3}=0 \end{cases}

suppose we have the function

G(\mathbf{x}) = \begin{bmatrix} 3x_1-\cos(x_2x_3)-\tfrac{3}{2} \\ 4x_1^2-625x_2^2+2x_2-1 \\ \exp(-x_1x_2)+20x_3+\tfrac{10\pi-3}{3} \\ \end{bmatrix}

where

\mathbf{x} =\begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ \end{bmatrix}

and the objective function

With initial guess

\mathbf{x}^{(0)}=\begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ \end{bmatrix} =\begin{bmatrix} 0 \\ 0 \\ 0 \\ \end{bmatrix}

We know that

where

The Jacobian matrix

J_G = \begin{bmatrix} 3 & \sin(x_2x_3)x_3 & \sin(x_2x_3)x_2 \\ 8x_1 & -1250x_2+2 & 0 \\ -x_2\exp{(-x_1x_2)} & -x_1\exp(-x_1x_2) & 20\\ \end{bmatrix}

Then evaluating these terms at

J_G \left(\mathbf{x}^{(0)}\right) = \begin{bmatrix} 3 & 0 & 0\\ 0 & 2 & 0\\ 0 & 0 & 20 \end{bmatrix}

and

G(\mathbf{x}^{(0)}) = \begin{bmatrix} -2.5\\ -1\\ 10.472 \end{bmatrix}

So that

\mathbf{x}^{(1)}=0-\gamma_0 \begin{bmatrix} -7.5\\ -2\\ 209.44 \end{bmatrix}.

and

F \left(\mathbf{x}^{(0)}\right) = 0.5((-2.5)^2 + (-1)^2 + (10.472)^2) = 58.456

Now a suitable must be found such that . This can be done with any of a variety of line search algorithms. One might also simply guess which gives

\mathbf{x}^{(1)}=\begin{bmatrix} 0.0075 \\ 0.002 \\ -0.20944 \\ \end{bmatrix}

evaluating at this value,

F \left(\mathbf{x}^{(1)}\right) = 0.5((-2.48)^2 + (-1.00)^2 + (6.28)^2) = 23.306

The decrease from to the next step's value of is a sizable decrease in the objective function. Further steps would reduce its value until a solution to the system was found.

Read more about this topic:  Gradient Descent

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