Scipy optimization with multiple bounds

Question

I'm solving a portfolio optimization problem where I need to allocate weights (capital) in such way that the end portfolio would have the lowest historical volatility possible. Currently I have two constraints on the total sum of weights of each stock and on their squared weights. Bounds for allocation in to each stock are (0.00, 0,02).

Currently my code looks like this:

def portfolio_vol(w):
    #compute porfolio volatility
    portfolio_volatility = np.sqrt(w.T.dot(cov_matrix).dot(w))
    return portfolio_volatility

w is a vector of equal weights calculated as w = 1.0/len(composition)

def find_optimal_allocations():
    bnds = tuple((0.00, 0.02) for x in weights)
    cons = ({'type': 'eq', 'fun': lambda x: 1 - sum(x)}, {'type': 'ineq', 'fun': lambda x: -sum(x**2) + 0.02}) 
    result = spo.minimize(portfolio_vol, weights, method='SLSQP', bounds = bnds, constraints = cons)
    return result.x

I store my weights in a MultIndex pandas Series where stocks are grouped by countries and initially are equal weighted:

Country  Ticker         
AS       OMV AV Equity      0.020000
BE       SOLB BB Equity     0.020000
FI       NESTE FH Equity    0.020000
FR       FP FP Equity       0.020000
GB       LAND LN Equity     0.020000
GE       BAS GR Equity      0.020000
         HEI GR Equity      0.020000
GR       TITK GA Equity     0.020000
IR       CRH ID Equity      0.020000
         RYA ID Equity      0.020000
...

US       AMAT US Equity     0.020000
         AMGN US Equity     0.020000
         APA US Equity      0.020000

I would like to include additional constraint in to find_optimal_allocations() so I could set limits for each country maximum weight: s.groupby(level=0,axis=0).sum()

Country
AS    0.020000
BE    0.020000
FI    0.020000
FR    0.020000
GB    0.020000
GE    0.040000
GR    0.020000
IR    0.040000
US    0.400000

For example withing the these bounds:

[(.05,.10), (.05,.10), (.05,.10), (.05,.10), (.05,.10), (.05,.10), (.05,.10), (.05,.10), (.05,.20)]

How can I include these bounds in to my optimization function so it would account for country weight constraints and individual stock constraints at the same time?

score 0 · Answer 1 · answered Jun 13 '16 at 22:33

0

Modify your optimization function such that there is a high penalty if the weight of any asset goes outside its bound. Using this new function you can try to find the gradient and perform gradient ascent to optimize your return / reward.

answered Jun 13 '16 at 22:33

Brandon Ogle

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Thanks for your reply. I don't understand why I should modify the optimization function. I'm looking to add additional bounds for country weights, similarly how you added bounds for asset classes in this [post](http://stackoverflow.com/questions/18218355/scipy-optimization-with-grouped-bounds?rq=1) As I understand you made the optimization in two steps 1) optimize for individual weights 2) optimize for weights grouped by asset classes. However I do not fully understand the code. Maybe you can explain the logic behind your solution? – Artem Korol Jun 14 '16 at 14:06
That was a non optimal approach. The bottom line is you are hoping to find some local optima in n-dimensional space. For simplicity sake, let's assume you are optimizing only three assets, therefore your goal is to find a peak on a 3D surface. By changing your function you are essentially modifying that surface such that walking outside that bound is equivalent to stepping off a cliff. – Brandon Ogle Jun 14 '16 at 18:41

Scipy optimization with multiple bounds

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