Solver Inputs
The page is stacked in one flow, while the calculator fields use a responsive 3-column, 2-column, and 1-column grid.
Example Data Table
These rows illustrate typical setups and plausible outcomes. Exact results can vary with tolerances, step rules, and derivative step sizes.
| Objective | Method | Start (x, y) | Estimated Solution | Final f(x,y) | Typical Iterations | Notes |
|---|---|---|---|---|---|---|
| Rosenbrock | BFGS | (-1.2, 1.0) | (1.0000, 1.0000) | Near 0 | 25–60 | Good benchmark for curved valleys. |
| Booth | Newton | (0.0, 0.0) | (1.0000, 3.0000) | 0 | 4–10 | Strongly convex near the solution. |
| Matyas | Gradient Descent | (3.5, -2.0) | (0.0000, 0.0000) | 0 | 20–80 | Smooth bowl with simple geometry. |
| Himmelblau | BFGS | (4.0, 0.0) | (3.0000, 2.0000) | 0 | 10–35 | Different starts can reach different minima. |
| Shifted Sphere | Gradient Descent | (-4.0, 5.0) | (2.0000, -1.0000) | 0 | 10–50 | Useful for sanity checks and tuning. |
Formula Used
1) Objective Model
The calculator minimizes a smooth two-variable objective function f(x, y). Available benchmark functions are:
- Rosenbrock: f(x,y) = 100(y - x²)² + (1 - x)²
- Himmelblau: f(x,y) = (x² + y - 11)² + (x + y² - 7)²
- Booth: f(x,y) = (x + 2y - 7)² + (2x + y - 5)²
- Matyas: f(x,y) = 0.26(x² + y²) - 0.48xy
- Shifted Sphere: f(x,y) = (x - 2)² + (y + 1)²
2) Numerical Gradient
Central differences estimate the gradient:
- ∂f/∂x ≈ [f(x+h, y) - f(x-h, y)] / (2h)
- ∂f/∂y ≈ [f(x, y+h) - f(x, y-h)] / (2h)
- Gradient norm: ||∇f|| = √[(∂f/∂x)² + (∂f/∂y)²]
3) Numerical Hessian
Second derivatives also use central differences:
- fxx ≈ [f(x+h, y) - 2f(x,y) + f(x-h, y)] / h²
- fyy ≈ [f(x, y+h) - 2f(x,y) + f(x, y-h)] / h²
- fxy ≈ [f(x+h,y+h) - f(x+h,y-h) - f(x-h,y+h) + f(x-h,y-h)] / (4h²)
4) Search Directions
- Gradient Descent: pk = -∇f(xk)
- Newton: pk = -H(xk)-1 ∇f(xk)
- BFGS: pk = -Bk-1 ∇f(xk) with inverse-Hessian updates
5) Backtracking Line Search
The step size α is reduced until the Armijo condition is satisfied: f(x + αp) ≤ f(x) + c1 α ∇f(x)ᵀp. This improves stability and prevents overly aggressive updates.
6) Hessian Classification
At the final point, the solver computes Hessian eigenvalues. Positive eigenvalues suggest a local minimum, mixed signs suggest a saddle point, and very small eigenvalues indicate a flat or poorly conditioned region.
How to Use This Calculator
- Choose an objective function. Start with Rosenbrock or Booth if you want a familiar benchmark.
- Select a method. Use BFGS for balanced performance, Newton for fast local convergence, and Gradient Descent for transparent behavior.
- Enter starting values for x and y. Different starts matter, especially for Himmelblau because it has multiple local minima.
- Set tolerance, maximum iterations, derivative step, and line-search controls.
- Click Solve Optimization Problem. The result panel appears above the form and under the page header.
- Review the minimum point, final objective value, gradient norm, Hessian classification, and iteration history.
- Inspect the convergence plot and contour path to understand how the method moved through the surface.
- Use the export buttons to save a CSV history file or a compact PDF report.
FAQs
1) What does this solver calculate?
It estimates a local minimum for a two-variable unconstrained objective function. The tool reports the final point, objective value, gradient norm, Hessian classification, step history, and convergence charts so you can judge both accuracy and method behavior.
2) Which method should I choose first?
BFGS is usually the safest default for smooth problems because it balances stability and speed. Newton can be faster near a solution but may struggle with poor Hessians. Gradient Descent is slower, yet it is great for learning and debugging solver behavior.
3) Why can Newton's method fail or slow down?
Newton relies on an invertible and informative Hessian. If the Hessian is singular, indefinite, or poorly conditioned, the computed direction may be unstable. This calculator adds regularization and can fall back to a descent-like step when needed.
4) What tolerance is reasonable?
A tolerance near 1e-6 is a practical default for many teaching and engineering examples. Tighter tolerances can improve accuracy, but they may require more iterations and can amplify numerical noise from finite-difference derivatives.
5) Why do different starting points matter?
Unconstrained nonlinear surfaces can have multiple local minima, curved valleys, and saddle regions. The chosen start influences the path and sometimes the final minimum. Himmelblau is a classic example where distinct starts converge to different valid minima.
6) Are the derivatives exact?
No. This version estimates derivatives numerically with central differences. That makes it flexible across several objective functions, but accuracy depends on the derivative step size. Extremely small values can increase rounding error, while large values reduce precision.
7) Can this tool solve constrained optimization problems?
No. It is designed for unconstrained problems only. Constraints require penalty methods, barrier terms, projected steps, KKT systems, or specialized solvers. Use this page when your decision variables are free from explicit equality or inequality restrictions.
8) What does Hessian classification tell me?
It describes local curvature near the final point. Positive eigenvalues suggest a local minimum, mixed signs suggest a saddle point, and near-zero eigenvalues indicate a flat or delicate region where the solution may be numerically sensitive.