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
 
    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())