কনস্ট্রেইনড অপ্টিমাইজেশন — লাগ্রাঞ্জ মাল্টিপ্লায়ার

def gaussian_elimination(A, b):
    n = len(A)
    M = [row[:] + [b[i]] for i, row in enumerate(A)]
    for col in range(n):
        pivot_row = max(range(col, n), key=lambda r: abs(M[r][col]))
        M[col], M[pivot_row] = M[pivot_row], M[col]
        pivot = M[col][col]
        for r in range(col + 1, n):
            factor = M[r][col] / pivot
            for c in range(col, n + 1):
                M[r][c] -= factor * M[col][c]
    x = [0.0] * n
    for i in range(n - 1, -1, -1):
        s = M[i][n] - sum(M[i][j] * x[j] for j in range(i + 1, n))
        x[i] = s / M[i][i]
    return x

# minimize f(x,y) = (x-1)^2 + (y-2)^2  subject to  g(x,y) = x + y - 6 = 0
# KKT system:  2x - lambda = 2 ; 2y - lambda = 4 ; x + y = 6
A = [
    [2, 0, -1],
    [0, 2, -1],
    [1, 1,  0],
]
b = [2, 4, 6]

x, y, lam = gaussian_elimination(A, b)
print(f"x = {x:.6f}, y = {y:.6f}, lambda = {lam:.6f}")

def f(x, y):
    return (x - 1)**2 + (y - 2)**2

print(f"f(x, y) = {f(x, y):.6f}")

constraint_value = x + y - 6
print(f"কনস্ট্রেইন্ট চেক: x + y - 6 = {constraint_value:.10f}  (শূন্যের কাছাকাছি হওয়া উচিত)")

r1 = 2*x - lam - 2
r2 = 2*y - lam - 4
r3 = x + y - 6
print(f"রেসিডুয়াল চেক: r1={r1:.2e}, r2={r2:.2e}, r3={r3:.2e}")