Vehicle-cpp/hkpc/software/scipy/optimize/_lsq/trf.py

"""Trust Region Reflective algorithm for least-squares optimization.

The algorithm is based on ideas from paper [STIR]_. The main idea is to
account for the presence of the bounds by appropriate scaling of the variables (or,
equivalently, changing a trust-region shape). Let's introduce a vector v:

           | ub[i] - x[i], if g[i] < 0 and ub[i] < np.inf
    v[i] = | x[i] - lb[i], if g[i] > 0 and lb[i] > -np.inf
           | 1,           otherwise

where g is the gradient of a cost function and lb, ub are the bounds. Its
components are distances to the bounds at which the anti-gradient points (if
this distance is finite). Define a scaling matrix D = diag(v**0.5).
First-order optimality conditions can be stated as

    D^2 g(x) = 0.

Meaning that components of the gradient should be zero for strictly interior
variables, and components must point inside the feasible region for variables
on the bound.

Now consider this system of equations as a new optimization problem. If the
point x is strictly interior (not on the bound), then the left-hand side is
differentiable and the Newton step for it satisfies

    (D^2 H + diag(g) Jv) p = -D^2 g

where H is the Hessian matrix (or its J^T J approximation in least squares),
Jv is the Jacobian matrix of v with components -1, 1 or 0, such that all
elements of matrix C = diag(g) Jv are non-negative. Introduce the change
of the variables x = D x_h (_h would be "hat" in LaTeX). In the new variables,
we have a Newton step satisfying

    B_h p_h = -g_h,

where B_h = D H D + C, g_h = D g. In least squares B_h = J_h^T J_h, where
J_h = J D. Note that J_h and g_h are proper Jacobian and gradient with respect
to "hat" variables. To guarantee global convergence we formulate a
trust-region problem based on the Newton step in the new variables:

    0.5 * p_h^T B_h p + g_h^T p_h -> min, ||p_h|| <= Delta

In the original space B = H + D^{-1} C D^{-1}, and the equivalent trust-region
problem is

    0.5 * p^T B p + g^T p -> min, ||D^{-1} p|| <= Delta

Here, the meaning of the matrix D becomes more clear: it alters the shape
of a trust-region, such that large steps towards the bounds are not allowed.
In the implementation, the trust-region problem is solved in "hat" space,
but handling of the bounds is done in the original space (see below and read
the code).

The introduction of the matrix D doesn't allow to ignore bounds, the algorithm
must keep iterates strictly feasible (to satisfy aforementioned
differentiability), the parameter theta controls step back from the boundary
(see the code for details).

The algorithm does another important trick. If the trust-region solution
doesn't fit into the bounds, then a reflected (from a firstly encountered
bound) search direction is considered. For motivation and analysis refer to
[STIR]_ paper (and other papers of the authors). In practice, it doesn't need
a lot of justifications, the algorithm simply chooses the best step among
three: a constrained trust-region step, a reflected step and a constrained
Cauchy step (a minimizer along -g_h in "hat" space, or -D^2 g in the original
space).

Another feature is that a trust-region radius control strategy is modified to
account for appearance of the diagonal C matrix (called diag_h in the code).

Note that all described peculiarities are completely gone as we consider
problems without bounds (the algorithm becomes a standard trust-region type
algorithm very similar to ones implemented in MINPACK).

The implementation supports two methods of solving the trust-region problem.
The first, called 'exact', applies SVD on Jacobian and then solves the problem
very accurately using the algorithm described in [JJMore]_. It is not
applicable to large problem. The second, called 'lsmr', uses the 2-D subspace
approach (sometimes called "indefinite dogleg"), where the problem is solved
in a subspace spanned by the gradient and the approximate Gauss-Newton step
found by ``scipy.sparse.linalg.lsmr``. A 2-D trust-region problem is
reformulated as a 4th order algebraic equation and solved very accurately by
``numpy.roots``. The subspace approach allows to solve very large problems
(up to couple of millions of residuals on a regular PC), provided the Jacobian
matrix is sufficiently sparse.

References
----------
.. [STIR] Branch, M.A., T.F. Coleman, and Y. Li, "A Subspace, Interior,
      and Conjugate Gradient Method for Large-Scale Bound-Constrained
      Minimization Problems," SIAM Journal on Scientific Computing,
      Vol. 21, Number 1, pp 1-23, 1999.
.. [JJMore] More, J. J., "The Levenberg-Marquardt Algorithm: Implementation
    and Theory," Numerical Analysis, ed. G. A. Watson, Lecture
"""
import numpy as np
from numpy.linalg import norm
from scipy.linalg import svd, qr
from scipy.sparse.linalg import lsmr
from scipy.optimize import OptimizeResult

from .common import (
    step_size_to_bound, find_active_constraints, in_bounds,
    make_strictly_feasible, intersect_trust_region, solve_lsq_trust_region,
    solve_trust_region_2d, minimize_quadratic_1d, build_quadratic_1d,
    evaluate_quadratic, right_multiplied_operator, regularized_lsq_operator,
    CL_scaling_vector, compute_grad, compute_jac_scale, check_termination,
    update_tr_radius, scale_for_robust_loss_function, print_header_nonlinear,
    print_iteration_nonlinear)


def trf(fun, jac, x0, f0, J0, lb, ub, ftol, xtol, gtol, max_nfev, x_scale,
        loss_function, tr_solver, tr_options, verbose):
    # For efficiency, it makes sense to run the simplified version of the
    # algorithm when no bounds are imposed. We decided to write the two
    # separate functions. It violates the DRY principle, but the individual
    # functions are kept the most readable.
    if np.all(lb == -np.inf) and np.all(ub == np.inf):
        return trf_no_bounds(
            fun, jac, x0, f0, J0, ftol, xtol, gtol, max_nfev, x_scale,
            loss_function, tr_solver, tr_options, verbose)
    else:
        return trf_bounds(
            fun, jac, x0, f0, J0, lb, ub, ftol, xtol, gtol, max_nfev, x_scale,
            loss_function, tr_solver, tr_options, verbose)


def select_step(x, J_h, diag_h, g_h, p, p_h, d, Delta, lb, ub, theta):
    """Select the best step according to Trust Region Reflective algorithm."""
    if in_bounds(x + p, lb, ub):
        p_value = evaluate_quadratic(J_h, g_h, p_h, diag=diag_h)
        return p, p_h, -p_value

    p_stride, hits = step_size_to_bound(x, p, lb, ub)

    # Compute the reflected direction.
    r_h = np.copy(p_h)
    r_h[hits.astype(bool)] *= -1
    r = d * r_h

    # Restrict trust-region step, such that it hits the bound.
    p *= p_stride
    p_h *= p_stride
    x_on_bound = x + p

    # Reflected direction will cross first either feasible region or trust
    # region boundary.
    _, to_tr = intersect_trust_region(p_h, r_h, Delta)
    to_bound, _ = step_size_to_bound(x_on_bound, r, lb, ub)

    # Find lower and upper bounds on a step size along the reflected
    # direction, considering the strict feasibility requirement. There is no
    # single correct way to do that, the chosen approach seems to work best
    # on test problems.
    r_stride = min(to_bound, to_tr)
    if r_stride > 0:
        r_stride_l = (1 - theta) * p_stride / r_stride
        if r_stride == to_bound:
            r_stride_u = theta * to_bound
        else:
            r_stride_u = to_tr
    else:
        r_stride_l = 0
        r_stride_u = -1

    # Check if reflection step is available.
    if r_stride_l <= r_stride_u:
        a, b, c = build_quadratic_1d(J_h, g_h, r_h, s0=p_h, diag=diag_h)
        r_stride, r_value = minimize_quadratic_1d(
            a, b, r_stride_l, r_stride_u, c=c)
        r_h *= r_stride
        r_h += p_h
        r = r_h * d
    else:
        r_value = np.inf

    # Now correct p_h to make it strictly interior.
    p *= theta
    p_h *= theta
    p_value = evaluate_quadratic(J_h, g_h, p_h, diag=diag_h)

    ag_h = -g_h
    ag = d * ag_h

    to_tr = Delta / norm(ag_h)
    to_bound, _ = step_size_to_bound(x, ag, lb, ub)
    if to_bound < to_tr:
        ag_stride = theta * to_bound
    else:
        ag_stride = to_tr

    a, b = build_quadratic_1d(J_h, g_h, ag_h, diag=diag_h)
    ag_stride, ag_value = minimize_quadratic_1d(a, b, 0, ag_stride)
    ag_h *= ag_stride
    ag *= ag_stride

    if p_value < r_value and p_value < ag_value:
        return p, p_h, -p_value
    elif r_value < p_value and r_value < ag_value:
        return r, r_h, -r_value
    else:
        return ag, ag_h, -ag_value


def trf_bounds(fun, jac, x0, f0, J0, lb, ub, ftol, xtol, gtol, max_nfev,
               x_scale, loss_function, tr_solver, tr_options, verbose):
    x = x0.copy()

    f = f0
    f_true = f.copy()
    nfev = 1

    J = J0
    njev = 1
    m, n = J.shape

    if loss_function is not None:
        rho = loss_function(f)
        cost = 0.5 * np.sum(rho[0])
        J, f = scale_for_robust_loss_function(J, f, rho)
    else:
        cost = 0.5 * np.dot(f, f)

    g = compute_grad(J, f)

    jac_scale = isinstance(x_scale, str) and x_scale == 'jac'
    if jac_scale:
        scale, scale_inv = compute_jac_scale(J)
    else:
        scale, scale_inv = x_scale, 1 / x_scale

    v, dv = CL_scaling_vector(x, g, lb, ub)
    v[dv != 0] *= scale_inv[dv != 0]
    Delta = norm(x0 * scale_inv / v**0.5)
    if Delta == 0:
        Delta = 1.0

    g_norm = norm(g * v, ord=np.inf)

    f_augmented = np.zeros((m + n))
    if tr_solver == 'exact':
        J_augmented = np.empty((m + n, n))
    elif tr_solver == 'lsmr':
        reg_term = 0.0
        regularize = tr_options.pop('regularize', True)

    if max_nfev is None:
        max_nfev = x0.size * 100

    alpha = 0.0  # "Levenberg-Marquardt" parameter

    termination_status = None
    iteration = 0
    step_norm = None
    actual_reduction = None

    if verbose == 2:
        print_header_nonlinear()

    while True:
        v, dv = CL_scaling_vector(x, g, lb, ub)

        g_norm = norm(g * v, ord=np.inf)
        if g_norm < gtol:
            termination_status = 1

        if verbose == 2:
            print_iteration_nonlinear(iteration, nfev, cost, actual_reduction,
                                      step_norm, g_norm)

        if termination_status is not None or nfev == max_nfev:
            break

        # Now compute variables in "hat" space. Here, we also account for
        # scaling introduced by `x_scale` parameter. This part is a bit tricky,
        # you have to write down the formulas and see how the trust-region
        # problem is formulated when the two types of scaling are applied.
        # The idea is that first we apply `x_scale` and then apply Coleman-Li
        # approach in the new variables.

        # v is recomputed in the variables after applying `x_scale`, note that
        # components which were identically 1 not affected.
        v[dv != 0] *= scale_inv[dv != 0]

        # Here, we apply two types of scaling.
        d = v**0.5 * scale

        # C = diag(g * scale) Jv
        diag_h = g * dv * scale

        # After all this has been done, we continue normally.

        # "hat" gradient.
        g_h = d * g

        f_augmented[:m] = f
        if tr_solver == 'exact':
            J_augmented[:m] = J * d
            J_h = J_augmented[:m]  # Memory view.
            J_augmented[m:] = np.diag(diag_h**0.5)
            U, s, V = svd(J_augmented, full_matrices=False)
            V = V.T
            uf = U.T.dot(f_augmented)
        elif tr_solver == 'lsmr':
            J_h = right_multiplied_operator(J, d)

            if regularize:
                a, b = build_quadratic_1d(J_h, g_h, -g_h, diag=diag_h)
                to_tr = Delta / norm(g_h)
                ag_value = minimize_quadratic_1d(a, b, 0, to_tr)[1]
                reg_term = -ag_value / Delta**2

            lsmr_op = regularized_lsq_operator(J_h, (diag_h + reg_term)**0.5)
            gn_h = lsmr(lsmr_op, f_augmented, **tr_options)[0]
            S = np.vstack((g_h, gn_h)).T
            S, _ = qr(S, mode='economic')
            JS = J_h.dot(S)  # LinearOperator does dot too.
            B_S = np.dot(JS.T, JS) + np.dot(S.T * diag_h, S)
            g_S = S.T.dot(g_h)

        # theta controls step back step ratio from the bounds.
        theta = max(0.995, 1 - g_norm)

        actual_reduction = -1
        while actual_reduction <= 0 and nfev < max_nfev:
            if tr_solver == 'exact':
                p_h, alpha, n_iter = solve_lsq_trust_region(
                    n, m, uf, s, V, Delta, initial_alpha=alpha)
            elif tr_solver == 'lsmr':
                p_S, _ = solve_trust_region_2d(B_S, g_S, Delta)
                p_h = S.dot(p_S)

            p = d * p_h  # Trust-region solution in the original space.
            step, step_h, predicted_reduction = select_step(
                x, J_h, diag_h, g_h, p, p_h, d, Delta, lb, ub, theta)

            x_new = make_strictly_feasible(x + step, lb, ub, rstep=0)
            f_new = fun(x_new)
            nfev += 1

            step_h_norm = norm(step_h)

            if not np.all(np.isfinite(f_new)):
                Delta = 0.25 * step_h_norm
                continue

            # Usual trust-region step quality estimation.
            if loss_function is not None:
                cost_new = loss_function(f_new, cost_only=True)
            else:
                cost_new = 0.5 * np.dot(f_new, f_new)
            actual_reduction = cost - cost_new
            Delta_new, ratio = update_tr_radius(
                Delta, actual_reduction, predicted_reduction,
                step_h_norm, step_h_norm > 0.95 * Delta)

            step_norm = norm(step)
            termination_status = check_termination(
                actual_reduction, cost, step_norm, norm(x), ratio, ftol, xtol)
            if termination_status is not None:
                break

            alpha *= Delta / Delta_new
            Delta = Delta_new

        if actual_reduction > 0:
            x = x_new

            f = f_new
            f_true = f.copy()

            cost = cost_new

            J = jac(x, f)
            njev += 1

            if loss_function is not None:
                rho = loss_function(f)
                J, f = scale_for_robust_loss_function(J, f, rho)

            g = compute_grad(J, f)

            if jac_scale:
                scale, scale_inv = compute_jac_scale(J, scale_inv)
        else:
            step_norm = 0
            actual_reduction = 0

        iteration += 1

    if termination_status is None:
        termination_status = 0

    active_mask = find_active_constraints(x, lb, ub, rtol=xtol)
    return OptimizeResult(
        x=x, cost=cost, fun=f_true, jac=J, grad=g, optimality=g_norm,
        active_mask=active_mask, nfev=nfev, njev=njev,
        status=termination_status)


def trf_no_bounds(fun, jac, x0, f0, J0, ftol, xtol, gtol, max_nfev,
                  x_scale, loss_function, tr_solver, tr_options, verbose):
    x = x0.copy()

    f = f0
    f_true = f.copy()
    nfev = 1

    J = J0
    njev = 1
    m, n = J.shape

    if loss_function is not None:
        rho = loss_function(f)
        cost = 0.5 * np.sum(rho[0])
        J, f = scale_for_robust_loss_function(J, f, rho)
    else:
        cost = 0.5 * np.dot(f, f)

    g = compute_grad(J, f)

    jac_scale = isinstance(x_scale, str) and x_scale == 'jac'
    if jac_scale:
        scale, scale_inv = compute_jac_scale(J)
    else:
        scale, scale_inv = x_scale, 1 / x_scale

    Delta = norm(x0 * scale_inv)
    if Delta == 0:
        Delta = 1.0

    if tr_solver == 'lsmr':
        reg_term = 0
        damp = tr_options.pop('damp', 0.0)
        regularize = tr_options.pop('regularize', True)

    if max_nfev is None:
        max_nfev = x0.size * 100

    alpha = 0.0  # "Levenberg-Marquardt" parameter

    termination_status = None
    iteration = 0
    step_norm = None
    actual_reduction = None

    if verbose == 2:
        print_header_nonlinear()

    while True:
        g_norm = norm(g, ord=np.inf)
        if g_norm < gtol:
            termination_status = 1

        if verbose == 2:
            print_iteration_nonlinear(iteration, nfev, cost, actual_reduction,
                                      step_norm, g_norm)

        if termination_status is not None or nfev == max_nfev:
            break

        d = scale
        g_h = d * g

        if tr_solver == 'exact':
            J_h = J * d
            U, s, V = svd(J_h, full_matrices=False)
            V = V.T
            uf = U.T.dot(f)
        elif tr_solver == 'lsmr':
            J_h = right_multiplied_operator(J, d)

            if regularize:
                a, b = build_quadratic_1d(J_h, g_h, -g_h)
                to_tr = Delta / norm(g_h)
                ag_value = minimize_quadratic_1d(a, b, 0, to_tr)[1]
                reg_term = -ag_value / Delta**2

            damp_full = (damp**2 + reg_term)**0.5
            gn_h = lsmr(J_h, f, damp=damp_full, **tr_options)[0]
            S = np.vstack((g_h, gn_h)).T
            S, _ = qr(S, mode='economic')
            JS = J_h.dot(S)
            B_S = np.dot(JS.T, JS)
            g_S = S.T.dot(g_h)

        actual_reduction = -1
        while actual_reduction <= 0 and nfev < max_nfev:
            if tr_solver == 'exact':
                step_h, alpha, n_iter = solve_lsq_trust_region(
                    n, m, uf, s, V, Delta, initial_alpha=alpha)
            elif tr_solver == 'lsmr':
                p_S, _ = solve_trust_region_2d(B_S, g_S, Delta)
                step_h = S.dot(p_S)

            predicted_reduction = -evaluate_quadratic(J_h, g_h, step_h)
            step = d * step_h
            x_new = x + step
            f_new = fun(x_new)
            nfev += 1

            step_h_norm = norm(step_h)

            if not np.all(np.isfinite(f_new)):
                Delta = 0.25 * step_h_norm
                continue

            # Usual trust-region step quality estimation.
            if loss_function is not None:
                cost_new = loss_function(f_new, cost_only=True)
            else:
                cost_new = 0.5 * np.dot(f_new, f_new)
            actual_reduction = cost - cost_new

            Delta_new, ratio = update_tr_radius(
                Delta, actual_reduction, predicted_reduction,
                step_h_norm, step_h_norm > 0.95 * Delta)

            step_norm = norm(step)
            termination_status = check_termination(
                actual_reduction, cost, step_norm, norm(x), ratio, ftol, xtol)
            if termination_status is not None:
                break

            alpha *= Delta / Delta_new
            Delta = Delta_new

        if actual_reduction > 0:
            x = x_new

            f = f_new
            f_true = f.copy()

            cost = cost_new

            J = jac(x, f)
            njev += 1

            if loss_function is not None:
                rho = loss_function(f)
                J, f = scale_for_robust_loss_function(J, f, rho)

            g = compute_grad(J, f)

            if jac_scale:
                scale, scale_inv = compute_jac_scale(J, scale_inv)
        else:
            step_norm = 0
            actual_reduction = 0

        iteration += 1

    if termination_status is None:
        termination_status = 0

    active_mask = np.zeros_like(x)
    return OptimizeResult(
        x=x, cost=cost, fun=f_true, jac=J, grad=g, optimality=g_norm,
        active_mask=active_mask, nfev=nfev, njev=njev,
        status=termination_status)