Quiz 2
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Numerical Optimization Methods

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Numerical Optimization Methods

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xk+1=xk+αkdkx_{k+1} = x_k + \alpha_k d_k
Choose step size αk\alpha_k via Armijo condition: f(xk+αdk)≤f(xk)+c1α∇fkTdkf(x_k + \alpha d_k) \leq f(x_k) + c_1 \alpha \nabla f_k^T d_k

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Approximate ff near xkx_k with model mkm_k, minimize within radius Δk\Delta_k:
pk=arg⁡min⁡∣∣p∣∣≤Δkmk(xk+p)p_k = \arg\min_{||p|| \leq \Delta_k} m_k(x_k + p) 
python
import 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}")
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