Numerical Optimization Methods
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# Numerical Optimization Methods ## Line Search $$ x_{k+1} = x_k + \alpha_k d_k $$ Choose step size $\alpha_k$ via **Armijo condition**: $f(x_k + \alpha d_k) \leq f(x_k) + c_1 \alpha \nabla f_k^T d_k$ ## Trust Region Approximate $f$ near $x_k$ with model $m_k$, minimize within radius $\Delta_k$: $$ p_k = \arg\min_{|...
Numerical Optimization Methods
Line Search
Choose step size via Armijo condition:
Trust Region
Approximate near with model , minimize within radius :
pythonimport numpy as np from scipy.optimize import minimize # Example: Rosenbrock function def rosen(x): return sum(100*(x[1:]-x[:-1]**2)**2 + (1-x[:-1])**2) result = minimize(rosen, [0, 0], method='L-BFGS-B') print(f"Minimum at: {result.x}, Value: {result.fun}")