使用SLSQP的Python的scipy.optimize.minimize失败，显示“用于线搜索的正方向导数”

Question

I have a least squares minimization problem subject to inequality constraints which I am trying to solve using scipy.optimize.minimize. It seems that there are two options for inequality constraints: COBYLA and SLSQP.

I first tried SLSQP since it allow for explicit partial derivatives of the function to be minimized. Depending on the scaling of the problem, it fails with error:

Positive directional derivative for linesearch    (Exit mode 8)

whenever interval or more general inequality constraints are imposed.

This has been observed previously eg, here . Manual scaling of the function to be minimized (along with the associated partial derivatives) seems to get rid of the problem, but I cannot achieve the same effect by changing ftol in the options.

Overall, this whole thing is causing me to have doubts about the routine working in a robust manner. Here's a simplified example:

import numpy as np
import scipy.optimize as sp_optimize

def cost(x, A, y):

    e = y - A.dot(x)
    rss = np.sum(e ** 2)

    return rss

def cost_deriv(x, A, y):

    e = y - A.dot(x)
    deriv0 = -2 * e.dot(A[:,0])
    deriv1 = -2 * e.dot(A[:,1])

    deriv = np.array([deriv0, deriv1])

    return deriv


A = np.ones((10,2)); A[:,0] = np.linspace(-5,5, 10)
x_true = np.array([2, 2/20])
y = A.dot(x_true)
x_guess = x_true / 2

prm_bounds = ((0, 3), (0,1))

cons_SLSQP = ({'type': 'ineq', 'fun' : lambda x: np.array([x[0] - x[1]]),
               'jac' : lambda x: np.array([1.0, -1.0])})

# works correctly
min_res_SLSQP = sp_optimize.minimize(cost, x_guess, args=(A, y), jac=cost_deriv, bounds=prm_bounds, method='SLSQP', constraints=cons_SLSQP, options={'disp': True})
print(min_res_SLSQP)

# fails
A = 100 * A
y = A.dot(x_true)
min_res_SLSQP = sp_optimize.minimize(cost, x_guess, args=(A, y), jac=cost_deriv, bounds=prm_bounds, method='SLSQP', constraints=cons_SLSQP, options={'disp': True})
print(min_res_SLSQP)

# works if bounds and inequality constraints removed
min_res_SLSQP = sp_optimize.minimize(cost, x_guess, args=(A, y), jac=cost_deriv,
method='SLSQP', options={'disp': True})
print(min_res_SLSQP)

How should ftol be set to avoid failure? More generally, can a similar problem arise with COBYLA? Is COBYLA a better choice for this type of inequality constrained least squares optimization problem?

Using a square root in the cost function was found to improve performance. However, for a non-linear re-paramterization of the problem (simpler but closer to what I need to do in practice), it fails again. Here are the details:

import numpy as np
import scipy.optimize as sp_optimize


def cost(x, y, g):

    e = ((y - x[1]) / x[0]) - g

    rss = np.sqrt(np.sum(e ** 2))

    return rss


def cost_deriv(x, y, g):

    e = ((y- x[1]) / x[0]) - g

    factor = 0.5 / np.sqrt(e.dot(e))
    deriv0 = -2 * factor * e.dot(y - x[1]) / (x[0]**2)
    deriv1 = -2 * factor * np.sum(e) / x[0]

    deriv = np.array([deriv0, deriv1])

    return deriv


x_true = np.array([1/300, .1])
N = 20
t = 20 * np.arange(N)
g = 100 * np.cos(2 * np.pi * 1e-3 * (t - t[-1] / 2))
y = g * x_true[0] + x_true[1]

x_guess = x_true / 2
prm_bounds = ((1e-4, 1e-2), (0, .4))

# check derivatives
delta = 1e-9
C0 = cost(x_guess, y, g)
C1 = cost(x_guess + np.array([delta, 0]), y, g)
approx_deriv0 = (C1 - C0) / delta
C1 = cost(x_guess + np.array([0, delta]), y, g)
approx_deriv1 = (C1 - C0) / delta
approx_deriv = np.array([approx_deriv0, approx_deriv1])
deriv = cost_deriv(x_guess, y, g)

# fails
min_res_SLSQP = sp_optimize.minimize(cost, x_guess, args=(y, g), jac=cost_deriv,
bounds=prm_bounds, method='SLSQP', options={'disp': True})
print(min_res_SLSQP)

Answer 1

Instead of minimizing np.sum(e ** 2) , minimize sqrt(np.sum(e ** 2)) , or better (in terms of calculation): np.linalg.norm(e) !

This modification:

• does not change your solution in regards to x
• will need post-processing if the original objective is needed (probably not)
• is much more robust

With this change, all cases work, even using numerical-differentiation (i was too lazy to modify the gradient, which needs to reflect this!).

Example output (number of func-evals gives away num-diff):

Optimization terminated successfully.    (Exit mode 0)
            Current function value: 3.815547437029837e-06
            Iterations: 16
            Function evaluations: 88
            Gradient evaluations: 16
     fun: 3.815547437029837e-06
     jac: array([-6.09663382, -2.48862544])
 message: 'Optimization terminated successfully.'
    nfev: 88
     nit: 16
    njev: 16
  status: 0
 success: True
       x: array([ 2.00000037,  0.10000018])
Optimization terminated successfully.    (Exit mode 0)
            Current function value: 0.0002354577991007501
            Iterations: 23
            Function evaluations: 114
            Gradient evaluations: 23
     fun: 0.0002354577991007501
     jac: array([ 435.97259208,  288.7483819 ])
 message: 'Optimization terminated successfully.'
    nfev: 114
     nit: 23
    njev: 23
  status: 0
 success: True
       x: array([ 1.99999977,  0.10000014])
Optimization terminated successfully.    (Exit mode 0)
            Current function value: 0.0003392807206384532
            Iterations: 21
            Function evaluations: 112
            Gradient evaluations: 21
     fun: 0.0003392807206384532
     jac: array([ 996.57340243,   51.19298764])
 message: 'Optimization terminated successfully.'
    nfev: 112
     nit: 21
    njev: 21
  status: 0
 success: True
       x: array([ 2.00000008,  0.10000104])

While there are probably some problems with SLSQP, it's still one of the most tested and robust codes given that broad application-spectrum!

I would also expect SLSQP to be much better here compared to COBYLA, as the latter is based heavily on linearizations. (but just take it as a guess; it's easy to try given the minimize-interface!)

Alternative

In general, an Interior-point based solver for Convex Quadratic Programming will be the best approach here. But for this, you need to leave scipy. (or maybe an SOCP-solver would be better... i'm not sure).

cvxpy brings a nice-modelling system and a good open-source solver ( ECOS ; although technically a conic-solver -> more general and less robust; but should beat SLSQP).

Using cvxpy and ECOS, this looks like:

import numpy as np
import cvxpy as cvx

""" Problem data """
A = np.ones((10,2)); A[:,0] = np.linspace(-5,5, 10)
x_true = np.array([2, 2/20])
y = A.dot(x_true)
x_guess = x_true / 2

prm_bounds = ((0, 3), (0,1))

# problematic case
A = 100 * A
y = A.dot(x_true)

""" Solve """
x = cvx.Variable(len(x_true))
constraints = [x[0] >= x[1]]
for ind, (lb, ub) in enumerate(prm_bounds):  # ineffecient -> matrix-based expr better!
    constraints.append(x[ind] >= lb)
    constraints.append(x[ind] <= ub)

objective = cvx.Minimize(cvx.norm(A*x - y))
problem = cvx.Problem(objective, constraints)
problem.solve(solver=cvx.ECOS, verbose=False)
print(problem.status)
print(problem.value)
print(x.value.T)

# optimal
# -6.67593652593801e-10
# [[ 2.   0.1]]