Scipy Minimize 限制变量总和？

Question

Problem

Take a sum of cash allocated to multiple currencies and re-allocate the cash based on a defined yield schedule. If a cash balance is negative, it will use the credit rate schedule (debit if positive). Additionally, a cash balance can't "go negative" or go further negative than it already is (I have not yet built this constraint yet).

When I build try to optimize (as per below), the display options show no iteration taking place and return the same results. What I would expect to have happen is for all of the cash to move into the highest rate (in this case GBP), thus maximizing the total_return.

Data

+----------+---------+
| Currency | Balance |
+----------+---------+
| GBP      | 799181  |
+----------+---------+
| JPY      | -411087 |
+----------+---------+
| EUR      | 141234  |
+----------+---------+
| USD      | 2650988 |
+----------+---------+

+---------------+--------+--------+--------+--------+
| Currency      | GBP    | JPY    | EUR    | USD    |
+---------------+--------+--------+--------+--------+
| Rate (Credit) | .00196 | .05320 | -.0003 | 0.0    |
+---------------+--------+--------+--------+--------+
| Rate (Debit)  | .01    | .00865 | -.0076 | .00028 |
+---------------+--------+--------+--------+--------+

Current Scipy Minimize Setup

import numpy as np
from scipy.optimize import shgo

def objective(x):             
     applied_rate = np.where((x<0), rates_sched[0], rates_sched[1])
     total_return = np.dot(applied_rate, x)
     return -total_return

def constraint1(x):
     return sum(x) - sum(cash_start) # constrain cash to starting point

cash_start = np.array([799181, -411087, 141234,2650988])

rates_sched = np.array([[.00196, .05320, -.0003, 0.0],
                        [0.01, .00865, -.0076, .00028]])


bnds = []

for value in cash_start:
    if value <0:
        bnd = (value, sum(cash_start))
    else:
        bnd = (0, sum(cash_start))
    bnds.append(bnd)

con1 = {'type': 'eq', 'fun': constraint1}
solution = shgo(objective, bounds = bnds, constraints = con1)

Update

I switched the scipy.optimize algorithm to shgo and I am getting closer. Current issue is that the resulting array (x) isn't respecting the constraint.

Answer 1

In the documentation of shgo I read:

Only the COBYLA and SLSQP local minimize methods currently support constraint arguments. If the constraints sequence used in the local optimization problem is not defined in minimizer_kwargs and a constrained method is used then the global constraints will be used. (Defining a constraints sequence in minimizer_kwargs means that constraints will not be added so if equality constraints and so forth need to be added then the inequality functions in constraints need to be added to minimizer_kwargs too).

Maybe that is why these lines fixes the problem for you:

solution2 = shgo(objective, bounds = bnds, iters=5, constraints = [con1],
                   minimizer_kwargs={'method':'SLSQP'}, 
                   sampling_method='sobol'
                )

But I am still not sure why this code works...

Note that performing simple local minimization:

from scipy.optimize import minimize

...

solution = minimize(objective, x0=cash_start, method='SLSQP', 
    bounds = bnds, constraints = con1)

gives a bad result.