`latmath.optimize` — Optimization

Root-finding, gradient-based, and discrete optimization for scalar and multivariate functions. Pure stdlib, zero dependencies.

Root-Finding

from latpy.latmath.optimize import bisection, newton

Bisection

Signature	Description
`bisection(f, a, b, tol=1e-12, max_iter=100) -> float`	Bracketed root via interval halving

Robust but slow (linear convergence). Requires f(a) and f(b) to have opposite signs.

from latpy.latmath.optimize import bisection
root = bisection(lambda x: x * x - 4.0, 0.0, 5.0)
print(root)  # 2.0

Edge cases: Raises ValueError if f(a) and f(b) have the same sign.

Newton

Signature	Description
`newton(f, df, x0, tol=1e-12, max_iter=100) -> float`	Newton-Raphson iteration

Fast (quadratic convergence) but requires the derivative df and a good initial guess.

from latpy.latmath.optimize import newton
root = newton(lambda x: x * x - 4.0, lambda x: 2.0 * x, 3.0)
print(root)  # 2.0

Edge cases: Raises ValueError if the derivative is zero at any step.

Gradient Descent

from latpy.latmath.optimize import gradient_descent

Signature	Description
`gradient_descent(f, df, x0, lr=0.01, tol=1e-6, max_iter=1000) -> (x_min, f_min)`	First-order scalar optimization

Steps in the negative gradient direction with fixed learning rate lr. Returns the minimizing point and function value.

from latpy.latmath.optimize import gradient_descent
x_min, f_min = gradient_descent(
    lambda x: x * x,
    lambda x: 2.0 * x,
    x0=5.0, lr=0.1,
)
print(x_min)  # ≈ 0.0
print(f_min)  # ≈ 0.0

Edge cases: May oscillate if lr is too large; may converge slowly if lr is too small.

Discrete Optimization

from latpy.latmath.optimize import grid_search, random_search

Grid Search

Signature	Description
`grid_search(f, bounds, n_points=100) -> (x_min, f_min)`	Exhaustive grid over bounded domain

Evaluates f at every point on a regular grid with n_points per dimension. Guaranteed to find the global minimum on the grid, but scales exponentially with dimension.

from latpy.latmath.optimize import grid_search
best_x, best_f = grid_search(
    lambda x, y: (x - 1.0) ** 2 + (y + 1.0) ** 2,
    [(-5.0, 5.0), (-5.0, 5.0)],
    n_points=200,
)
print(best_x)  # ≈ (1.0, -1.0)

Edge cases: Raises ValueError if bounds is empty.

Random Search

Signature	Description
`random_search(f, bounds, n_iter=1000, seed=None) -> (x_min, f_min)`	Random sampling over bounded domain

Samples uniformly from the bounded domain. Does not scale with dimension. Pass seed for reproducibility.

from latpy.latmath.optimize import random_search
best_x, best_f = random_search(
    lambda x: (x - 3.0) ** 2,
    [(0.0, 6.0)],
    n_iter=5000, seed=42,
)
print(best_x[0])  # ≈ 3.0

Edge cases: seed=None produces non-deterministic results; same seed produces identical results.

latmath.optimize — Optimization