Nonlinear Optimization Solver
Nonlinear Optimization Solver uses gradient descent to minimize user-defined functions, showing convergence plots and iteration steps for analysis.
Result:
Formulas Used
For an objective function f(x), gradient descent updates variables to minimize f:
- Gradient:
∇f(x) = [∂f/∂x₁, ∂f/∂x₂, ...], approximated numerically. - Update rule:
x[i+1] = x[i] - α * ∇f(x[i]), whereαis the learning rate. - Stop after max iterations or when
|∇f(x)| < 1e-6.
Examples and Solutions
- Example 1: Function:
x^2 + y^2, Initial:[1,1], Steps:100, α:0.1
Solution: Minimum at[0,0], Value:0 - Example 2: Function:
(x-1)^2 + (y-2)^2, Initial:[0,0], Steps:100, α:0.05
Solution: Minimum at[1,2], Value:0 - Example 3: Function:
x^2 + 2*y^2, Initial:[2,2], Steps:100, α:0.1
Solution: Minimum at[0,0], Value:0 - Example 4: Function:
x^2 + y^2 + x*y, Initial:[1,1], Steps:100, α:0.05
Solution: Minimum at[0,0], Value:0 - Example 5: Function:
(x-3)^2 + (y-4)^2, Initial:[0,0], Steps:100, α:0.1
Solution: Minimum at[3,4], Value:0