Optimization

Visualizers for root finding, descent, and multipliers.

Parameters

Function f(x)

Start x₀

Learning rate α

Max iter

Steps taken 0

Current x —

f(x) —

|∇f| —

Function f(x)

Start x₀

Tolerance ε

Function f(x)

a (left)

b (right)

Tolerance

Iterations —

x* (min) —

f(x*) —

Objective f

Constraint g(x,y)=0

Lagrange Multipliers

Optimization Methods & Solvers Handbook

Mathematical Principles & Theorems

Solves multi-dimensional optimization \(\min_{\mathbf{x}} f(\mathbf{x})\). (1) Gradient Descent: updates along negative gradient \(\mathbf{x}_{k+1} = \mathbf{x}_k - \alpha \nabla f(\mathbf{x}_k)\); (2) Momentum Gradient Descent: \(\mathbf{v}_{k+1} = \beta \mathbf{v}_k + \alpha \nabla f(\mathbf{x}_k), \mathbf{x}_{k+1} = \mathbf{x}_k - \mathbf{v}_{k+1}\); (3) Simulated Annealing: probabilistic escape from local minima via Metropolis acceptance \(P(\text{accept}) = e^{-\Delta E / T}\); (4) Benchmark loss functions: Rosenbrock banana function, Rastrigin multimodal function, Ackley landscape.

Operating Instructions

Select benchmark 2D objective landscape (Rosenbrock, Rastrigin, Ackley, Sphere).
Choose optimization algorithm: Gradient Descent, Momentum, or Simulated Annealing.
Set learning rate \(\alpha\), momentum \(\beta\), and initial coordinates using numeric inputs.
Click Run to watch the optimizer navigate contour lines and track loss reduction over iteration steps.