Code for init_algo using a lp

[tschm]

I have used this fragment.

def init_algo_lp(
    mean: MATRIX,
    lower_bounds: MATRIX,
    upper_bounds: MATRIX,
    A_eq: MATRIX | None = None,
    b_eq: MATRIX | None = None,
    A_ub: MATRIX | None = None,
    b_ub: MATRIX | None = None,
) -> Next:
    if A_eq is None:
        A_eq = np.atleast_2d(np.ones_like(mean))

    if b_eq is None:
        b_eq = np.array([1.0])

    if A_ub is None:
        A_ub = np.atleast_2d(np.zeros_like(mean))

    if b_ub is None:
        b_ub = np.array([0.0])

    w = cp.Variable(mean.shape[0], "weights")

    objective = cp.Maximize(mean.T @ w)
    constraints = [
        A_eq @ w == b_eq,
        A_ub @ w <= b_ub,
        lower_bounds <= w,
        w <= upper_bounds,
    ]

    cp.Problem(objective, constraints).solve()

    w = w.value

    # compute the distance from the closest bound
    distance = np.min(
        np.array([np.abs(w - lower_bounds), np.abs(upper_bounds - w)]), axis=0
    )

    # which element has the largest distance to any bound?
    # Even if all assets are at their bounds,
    # we get a (somewhat random) free asset.
    index = np.argmax(distance)

    free = np.full_like(mean, False, dtype=np.bool_)
    free[index] = True

    return Next(free=free, weights=w, mean=mean.T @ w)