I am trying to set the factor betas to zero, but scipy minimize 'cobyla' fails due to constraint violation. Anyone?

I get the following error when I set factor betas = zero.

Did not converge to a solution satisfying the constraints. See maxcv for magnitude of violation.

7
Total Returns
--
Alpha
--
Beta
--
Sharpe
--
Sortino
--
Max Drawdown
--
Benchmark Returns
--
Volatility
--
 Returns 1 Month 3 Month 6 Month 12 Month
 Alpha 1 Month 3 Month 6 Month 12 Month
 Beta 1 Month 3 Month 6 Month 12 Month
 Sharpe 1 Month 3 Month 6 Month 12 Month
 Sortino 1 Month 3 Month 6 Month 12 Month
 Volatility 1 Month 3 Month 6 Month 12 Month
 Max Drawdown 1 Month 3 Month 6 Month 12 Month
import math
import numpy as np
import pandas as pd
import talib
from sklearn.decomposition import PCA
import statsmodels.api as smapi
from sklearn.covariance import OAS
from scipy.optimize import minimize
from sklearn.linear_model import LassoCV

def getweights(params, cov, signal):
cons = []
(m,n) = np.shape(params)

for i in range(0, n):
cons.append({'type': 'ineq', 'fun': lambda x, i=i: np.dot(x.T, params[:, i])})
cons.append({'type': 'ineq', 'fun': lambda x, i=i: -np.dot(x.T, params[:, i])})

cons.append({'type': 'ineq', 'fun': lambda x: np.dot(x, signal) - 0.01})

for i in range(0, m):
cons.append({'type': 'ineq', 'fun': lambda x, i=i: 1 - abs(x[i]) })
cons.append({'type': 'ineq', 'fun': lambda x, i=i: abs(x[i]) - 0.1 })

x0 = [1.] * m
res = minimize(lambda x: -np.dot(x.T, signal) / np.dot(np.dot(x.T, cov), x), x0,
constraints = cons, method='cobyla',options={'maxiter':2000})
print res.message
return res.x

def initialize(context):
context.SPY = sid(19655)
context.done = False

def handle_data(context, data):
pass

prices = data.history(context.univ, "price", 120, "1d")
prices = prices.dropna(axis=1)

returns = prices.pct_change().dropna().values
returns = returns * 1000.
cov = OAS().fit(returns).covariance_
e, v = np.linalg.eig(cov)
idx = e.argsort()
comp = v[:, idx[-25:]]

if comp[0, 0] < 0:
comp *= -1

sources = np.dot(returns, comp)
betas = np.zeros((np.shape(returns)[1], np.shape(sources)[1]))

for i in range(0, np.shape(returns)[1]):
betas[i, :] = model.params[1:]

signal = np.sum(returns[-15:, :], axis=0) / np.std(returns, axis=0)
W = getweights(betas, cov, signal)
den = np.sum(np.abs(W))
if den == 0:
den = 1
wsum = 0
for i, sid in enumerate(prices):
val = W[i] / den * context.portfolio.portfolio_value * 2.
order_target_value(sid, val)
wsum += val
order_target_value(context.SPY, -wsum)

if context.done:
return
context.done = True
fundamental_df = get_fundamentals(
query(fundamentals.valuation_ratios.ev_to_ebitda)
.filter(fundamentals.valuation.market_cap != None)
.filter(fundamentals.asset_classification.morningstar_sector_code == 309)
.filter(fundamentals.company_reference.primary_exchange_id != "OTCPK") # no pink sheets
.filter(fundamentals.share_class_reference.security_type == 'ST00000001') # common stock only
.filter(~fundamentals.share_class_reference.symbol.contains('_WI')) # drop when-issued
.filter(fundamentals.share_class_reference.is_primary_share == True) # remove ancillary classes
.filter(fundamentals.valuation_ratios.ev_to_ebitda > 0)
.order_by(fundamentals.valuation.market_cap.desc()).limit(25)).T
context.univ = fundamental_df[0:25].index


There was a runtime error.
10 responses

Pravin -

I changed to the optimizer to SLSQP and it seems to work (except there are a few "Positive directional derivative for linesearch" notices in the logs, whatever that means).

Grant

3
Total Returns
--
Alpha
--
Beta
--
Sharpe
--
Sortino
--
Max Drawdown
--
Benchmark Returns
--
Volatility
--
 Returns 1 Month 3 Month 6 Month 12 Month
 Alpha 1 Month 3 Month 6 Month 12 Month
 Beta 1 Month 3 Month 6 Month 12 Month
 Sharpe 1 Month 3 Month 6 Month 12 Month
 Sortino 1 Month 3 Month 6 Month 12 Month
 Volatility 1 Month 3 Month 6 Month 12 Month
 Max Drawdown 1 Month 3 Month 6 Month 12 Month
import math
import numpy as np
import pandas as pd
import talib
from sklearn.decomposition import PCA
import statsmodels.api as smapi
from sklearn.covariance import OAS
from scipy.optimize import minimize
from sklearn.linear_model import LassoCV

def getweights(params, cov, signal):
cons = []
(m,n) = np.shape(params)

for i in range(0, n):
cons.append({'type': 'ineq', 'fun': lambda x, i=i: np.dot(x.T, params[:, i])})
cons.append({'type': 'ineq', 'fun': lambda x, i=i: -np.dot(x.T, params[:, i])})

cons.append({'type': 'ineq', 'fun': lambda x: np.dot(x, signal) - 0.01})

for i in range(0, m):
cons.append({'type': 'ineq', 'fun': lambda x, i=i: 1 - abs(x[i]) })
cons.append({'type': 'ineq', 'fun': lambda x, i=i: abs(x[i]) - 0.1 })

x0 = [1.] * m
res = minimize(lambda x: -np.dot(x.T, signal) / np.dot(np.dot(x.T, cov), x), x0,
constraints = cons, method='SLSQP',options={'maxiter':2000})
print res.message
return res.x

def initialize(context):
context.SPY = sid(19655)
context.done = False

def handle_data(context, data):
pass

prices = data.history(context.univ, "price", 120, "1d")
prices = prices.dropna(axis=1)

returns = prices.pct_change().dropna().values
returns = returns * 1000.
cov = OAS().fit(returns).covariance_
e, v = np.linalg.eig(cov)
idx = e.argsort()
comp = v[:, idx[-25:]]

if comp[0, 0] < 0:
comp *= -1

sources = np.dot(returns, comp)
betas = np.zeros((np.shape(returns)[1], np.shape(sources)[1]))

for i in range(0, np.shape(returns)[1]):
betas[i, :] = model.params[1:]

signal = np.sum(returns[-15:, :], axis=0) / np.std(returns, axis=0)
W = getweights(betas, cov, signal)
den = np.sum(np.abs(W))
if den == 0:
den = 1
wsum = 0
for i, sid in enumerate(prices):
val = W[i] / den * context.portfolio.portfolio_value * 2.
order_target_value(sid, val)
wsum += val
order_target_value(context.SPY, -wsum)

if context.done:
return
context.done = True
fundamental_df = get_fundamentals(
query(fundamentals.valuation_ratios.ev_to_ebitda)
.filter(fundamentals.valuation.market_cap != None)
.filter(fundamentals.asset_classification.morningstar_sector_code == 309)
.filter(fundamentals.company_reference.primary_exchange_id != "OTCPK") # no pink sheets
.filter(fundamentals.share_class_reference.security_type == 'ST00000001') # common stock only
.filter(~fundamentals.share_class_reference.symbol.contains('_WI')) # drop when-issued
.filter(fundamentals.share_class_reference.is_primary_share == True) # remove ancillary classes
.filter(fundamentals.valuation_ratios.ev_to_ebitda > 0)
.order_by(fundamentals.valuation.market_cap.desc()).limit(25)).T
context.univ = fundamental_df[0:25].index


There was a runtime error.
    for i in range(0, n):
cons.append({'type': 'ineq', 'fun': lambda x, i=i: np.dot(x.T, params[:, i])})
cons.append({'type': 'ineq', 'fun': lambda x, i=i: -np.dot(x.T, params[:, i])})


This code is strange. Are you sure that's what you mean? To judge from the documentation, that's going to result in constraints ∀i, np.dot(x.T, params[:, i]) >= 0 & -np.dot(x.T, params[:, i]) >= 0. I.e., np.dot(x.T, params[:, i]) == 0, i.e., x is orthogonal to every constraint plane, i.e., x == 0.

You probably want some constants in those lambdas, like you have in the subsequent set of constraints, which translate to ∀i, 0.01 <= abs(x[i]) <= 1.

Thanks Grant and Alex.

@Grant, I think COBYLA is used for quadratic optimization. I am not so sure if SLSQP can be used for that.

@Alex, I am trying to set the portfolio exposure to all params to zero and hence the constraint.

Pravin

I've been working on some optimization stuff with Jonathan and we came to the same conclusion as you did in your other post. The CVXOPT user experience is trash, especially to those not familiar with linear algebra or converting equations into standard matrix form. What's more scipy.minimize often does fail to find the correct minima, plus it is inefficient.

Here is a long only optimization ran on the Mean Absolute Deviation technique I posted about last year. Now this is a non-quadratic optimization and can be expressed as a linear program. You can see in the below graph how both scipy.minimize (green) and CVXOPT (blue) perform with optimization time on the Y axis and number of assets on the X axis. Needless to say the CVXOPT lp outperforms.

If you can, use CVXOPT. I have a working knowledge of linear algebra and the only way I was able to use CVXOPT was with their modeling api. I don't know if you've already tried using it but looking at your above implementation it seems it would not be hard to translate over. The trickier part I felt was setting correct bounds and constraints.

Anyhow hopefully this provides you and anyone else who looks at this thread with some decent information.

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Thanks James. I have the same feelings about cvxopt and scipy minimize and am glad that you share the same opinion.I could not use cvxopt for this because their qp uses xPx + xq whereas I need xq / xPx which is nearly impossible to formulate using cvxopt qp.

@ James -

CVXPY (http://www.cvxpy.org/en/latest/) looks friendly. See also http://www.jmlr.org/papers/volume17/15-408/15-408.pdf. I recall that Pravin had requested Quantopian look into it.

I came across CXPY too, I agree it would be good to get onto the platform.

Did not converge to a solution satisfying the constraints. See maxcv for magnitude of violation.

This is happening because your constraints are unsatisfiable. The determinant of params, a 25x25 matrix, is very close to 1. You are effectively requiring that params times x is 0, i.e. x is 0. On the other hand, your abs constraints require that x be nonzero.

Brilliant. Thanks Alex. I will change it so that it is satisfies constraints. Many thanks again.