# QUANTCONNECT.COM - Democratizing Finance, Empowering Individuals.
# Lean Algorithmic Trading Engine v2.0. Copyright 2014 QuantConnect Corporation.
#
# Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at http://www.apache.org/licenses/LICENSE-2.0
#
# Unless required by applicable law or agreed to in writing, software
# distributed under the License is distributed on an "AS IS" BASIS,
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from AlgorithmImports import *
from scipy.optimize import minimize

### <summary>
### Provides an implementation of a portfolio optimizer that calculate the optimal weights
### with the weight range from -1 to 1 and minimize the portfolio variance with a target return of 2%
### </summary>
### <remarks>The budged constrain is scaled down/up to ensure that the sum of the absolute value of the weights is 1.</remarks>
class MinimumVariancePortfolioOptimizer:
    '''Provides an implementation of a portfolio optimizer that calculate the optimal weights
    with the weight range from -1 to 1 and minimize the portfolio variance with a target return of 2%'''
    def __init__(self,
                 minimum_weight = -1,
                 maximum_weight = 1,
                 target_return = 0.02):
        '''Initialize the MinimumVariancePortfolioOptimizer
        Args:
            minimum_weight(float): The lower bounds on portfolio weights
            maximum_weight(float): The upper bounds on portfolio weights
            target_return(float): The target portfolio return'''
        self.minimum_weight = minimum_weight
        self.maximum_weight = maximum_weight
        self.target_return = target_return

    def optimize(self, historical_returns, expected_returns = None, covariance = None):
        '''
        Perform portfolio optimization for a provided matrix of historical returns and an array of expected returns
        args:
            historical_returns: Matrix of annualized historical returns where each column represents a security and each row returns for the given date/time (size: K x N).
            expected_returns: Array of double with the portfolio annualized expected returns (size: K x 1).
            covariance: Multi-dimensional array of double with the portfolio covariance of annualized returns (size: K x K).
        Returns:
            Array of double with the portfolio weights (size: K x 1)
        '''
        if covariance is None:
            covariance = historical_returns.cov()
        if expected_returns is None:
            expected_returns = historical_returns.mean()

        size = historical_returns.columns.size   # K x 1
        x0 = np.array(size * [1. / size])

        constraints = [
            {'type': 'eq', 'fun': lambda weights: self.get_budget_constraint(weights)},
            {'type': 'eq', 'fun': lambda weights: self.get_target_constraint(weights, expected_returns)}]

        # https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.minimize.html
        opt = minimize(lambda weights: self.portfolio_variance(weights, covariance),     # Objective function
                       x0,                                                        # Initial guess
                       bounds = self.get_boundary_conditions(size),               # Bounds for variables
                       constraints = constraints,                                 # Constraints definition
                       method='SLSQP')     # Optimization method:  Sequential Least Squares Programming (SLSQP)

        if not opt['success']: return x0

        # Scale the solution to ensure that the sum of the absolute weights is 1
        sum_of_absolute_weights = np.sum(np.abs(opt['x']))
        return opt['x'] / sum_of_absolute_weights

    def portfolio_variance(self, weights, covariance):
        '''Computes the portfolio variance
        Args:
            weighs: Portfolio weights
            covariance: Covariance matrix of historical returns'''
        variance = np.dot(weights.T, np.dot(covariance, weights))
        if variance == 0 and np.any(weights):
            # variance can't be zero, with non zero weights
            raise ValueError(f'MinimumVariancePortfolioOptimizer.portfolio_variance: Volatility cannot be zero. Weights: {weights}')
        return variance

    def get_boundary_conditions(self, size):
        '''Creates the boundary condition for the portfolio weights'''
        return tuple((self.minimum_weight, self.maximum_weight) for x in range(size))

    def get_budget_constraint(self, weights):
        '''Defines a budget constraint: the sum of the weights equals unity'''
        return np.sum(weights) - 1

    def get_target_constraint(self, weights, expected_returns):
        '''Ensure that the portfolio return target a given return'''
        return np.dot(np.matrix(expected_returns), np.matrix(weights).T).item() - self.target_return