markowitz_soco.py

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"""
Approach: use second-order cone optimization.
 
Copyright (c) by Joao Pedro PEDROSO, Masahiro MURAMATSU and Mikio KUBO, 2012
"""
from pyscipopt import Model, quicksum, multidict
 
def markowitz(I,sigma,r,alpha):
    """markowitz -- simple markowitz model for portfolio optimization.
    Parameters:
        - I: set of items
        - sigma[i]: standard deviation of item i
        - r[i]: revenue of item i
        - alpha: acceptance threshold
    Returns a model, ready to be solved.
    """
    model = Model("markowitz")
 
    x = {}
    for i in I:
        x[i] = model.addVar(vtype="C", name="x(%s)"%i)  # quantity of i to buy
 
    model.addCons(quicksum(r[i]*x[i] for i in I) >= alpha)
    model.addCons(quicksum(x[i] for i in I) == 1)
 
    # set nonlinear objective: SCIP only allow for linear objectives hence the following
    obj = model.addVar(vtype="C", name="objective", lb = None, ub = None)  # auxiliary variable to represent objective
    model.addCons(quicksum(sigma[i]**2 * x[i] * x[i] for i in I) <= obj)
    model.setObjective(obj, "minimize")
 
    model.data = x
    return model
 
 
 
if __name__ == "__main__":
    # portfolio
    import math
    I,sigma,r = multidict(
        {1:[0.07,1.01],
         2:[0.09,1.05],
         3:[0.1,1.08],
         4:[0.2,1.10],
         5:[0.3,1.20]}
        )
    alpha = 1.05
 
    model = markowitz(I,sigma,r,alpha)
    model.optimize()
 
    x = model.data
    EPS = 1.e-6
    print("%5s\t%8s" % ("i","x[i]"))
    for i in I:
        print("%5s\t%8g" % (i,model.getVal(x[i])))
    print("sum:",sum(model.getVal(x[i]) for i in I))
    print
    print("Optimal value:", model.getObjVal())