I am using cvxpy to do a simple portfolio optimization.
I implemented the following dummy code
from cvxpy import *
import numpy as np
np.random.seed(1)
n = 10
Sigma = np.random.randn(n, n)
Sigma = Sigma.T.dot(Sigma)
orig_weight = [0.15,0.25,0.15,0.05,0.20,0,0.1,0,0.1,0]
w = Variable(n)
mu = np.abs(np.random.randn(n, 1))
ret = mu.T*w
lambda_ = Parameter(sign='positive')
lambda_ = 5
risk = quad_form(w, Sigma)
constraints = [sum_entries(w) == 1, w >= 0, sum_entries(abs(w-orig_weight)) <= 0.750]
prob = Problem(Maximize(ret - lambda_ * risk), constraints)
prob.solve()
print 'Solver Status : ',prob.status
print('Weights opt :', w.value)
I am constraining on being fully invested, long only and to have a turnover of <= 75%. However I would like to use turnover as a "soft" constraint in the sense that the solver will use as little as possible but as much as necessary, currently the solver will almost fully max out turnover.
I basically want something like this which is convex and doesn't violate the DCP rules
sum_entries(abs(w-orig_weight)) >= 0.05
I would assume this should set a minimum threshold (5% here) and then use as much turnover until it finds a feasible solution.
I tried rewriting my objective function to
prob = Problem(Maximize(lambda_ * ret - risk - penalty * max(sum_entries(abs(w-orig_weight))+0.9,0)) , constraints)
where penalty is e.g. 2 and my constraint object still looks like
constraints = [sum_entries(w) == 1, w >= 0, sum_entries(abs(w-orig_weight)) <= 0.9]
I have never used soft-constraints and any explanation would be highly appreciated.
EDIT: Intermediate solution
from cvxpy import *
import numpy as np
np.random.seed(1)
n = 10
Sigma = np.random.randn(n, n)
Sigma = Sigma.T.dot(Sigma)
w = Variable(n)
mu = np.abs(np.random.randn(n, 1))
ret = mu.T*w
risk = quad_form(w, Sigma)
orig_weight = [0.15,0.2,0.2,0.2,0.2,0.05,0.0,0.0,0.0,0.0]
min_weight = [0.35,0.0,0.0,0.0,0.0,0,0.0,0,0.0,0.0]
max_weight = [0.35,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0]
lambda_ret = Parameter(sign='positive')
lambda_ret = 5
lambda_risk = Parameter(sign='positive')
lambda_risk = 1
penalty = Parameter(sign='positive')
penalty = 100
penalized = True
if penalized == True:
print '-------------- RELAXED ------------------'
constraints = [sum_entries(w) == 1, w >= 0, w >= min_weight, w <= max_weight]
prob = Problem(Maximize(lambda_ * ret - lambda_ * risk - penalty * max_entries(sum_entries(abs(w-orig_weight)))-0.01), constraints)
else:
print '-------------- HARD ------------------'
constraints = [sum_entries(w) == 1, w >= 0, w >= min_weight, w <= max_weight, sum_entries(abs(w-orig_weight)) <= 0.40]
prob = Problem(Maximize(lambda_ret * ret - lambda_risk * risk ),constraints)
prob.solve()
print 'Solver Status : ',prob.status
print('Weights opt :', w.value)
all_in = []
for i in range(n):
all_in.append(np.abs(w.value[i][0] - orig_weight[i]))
print 'Turnover : ', sum(all_in)
The above code will force a specific increase in weight for item[0], here +20%, in order to maintain the sum() =1 constraint that has to be offset by a -20% decrease, therefore I know it will need a minimum of 40% turnover to do that, if one runs the code with penalized = False the <= 0.4 have to be hardcoded, anything smaller than that will fail. The penalized = True case will find the minimum required turnover of 40% and solve the optimization. What I haven't figured out yet is how I can set a minimum threshold in the relaxed case, i.e. do at least 45% (or more if required).
I found some explanation around the problem here, in chapter 4.6 page 37.
