Class PortfolioRiskExactSigma

  • All Implemented Interfaces:
    Function<Vector,​Double>, RealScalarFunction

    public class PortfolioRiskExactSigma
    extends SOCPRiskConstraint
    Constructs the constraint coefficient arrays of the portfolio risk term in the compact form. The constraints are generated during the transformation of the objective function.

    The portfolio risk in the objective function is transformed into the following constraints: \[ (x+w^{0})^{\top}\Sigma(x+w^{0})\leq t_1. \] By letting \(y=x+w^{0}\), it can be written as: \[ y^{\top}\Sigma\;y\leq t_1 \] When the exact covariance matrix \(\Sigma\) is used, then the portfolio risk constraint is equivalent to: \[ y^{\top}\Sigma\;y\leq t_1 \Longleftrightarrow y^{\top}\Sigma\;y+(\frac{t_{1}-1}{2})^{2}\leq(\frac{t_{1}+1}{2})^{2}\Longleftrightarrow ||\left(\begin{array}{c}\Sigma^{\frac{1}{2}}y\\\frac{t_{1}-1}{2}\end{array}\right)||_{2}\leq \frac{t_{1}+1}{2}. \] And the standard SOCP form of the portfolio risk constraint in this case are: \[ ||\left(\begin{array}{c}\Sigma^{\frac{1}{2}}y\\\frac{t_{1}-1}{2}\end{array}\right)||_{2}\leq \frac{t_{1}+1}{2}\Longleftrightarrow ||A_{1}^{\top}z+C_{1}||_{2}\leq b^{\top}_{1}z+d_{1}\\ A_{1}^{\top}=\left(\begin{array}{cc}\Sigma^{\frac{1}{2}} & 0_{n\times 1}\\0_{1\times n} & 1/2\end{array}\right)\nonumber,\; C_{1}=\left(\begin{array}{c}0_{n\times 1}\\-1/2\end{array}\right),\; b_{1}=\left(\begin{array}{c}0_{n\times 1}\\1/2\end{array}\right)\; d_{1}=\frac{1}{2},\; z=\left(\begin{array}{c}y\\t_{1}\end{array}\right). \]
    • Constructor Detail

      • PortfolioRiskExactSigma

        public PortfolioRiskExactSigma​(Matrix Sigma,
                                       PortfolioRiskExactSigma.MatrixRoot root)
        Transforms the portfolio risk term, \(y^{\top}\Sigma\;y\leq t_1\), into the standard SOCP form when the exact covariance matrix is used.
        Parameters:
        Sigma - the covariance matrix
        root - the method to compute the root of a matrix
      • PortfolioRiskExactSigma

        public PortfolioRiskExactSigma​(Matrix Sigma,
                                       Matrix sigmaRoot)
        Transforms the portfolio risk term, \(y^{\top}\Sigma\;y\leq t_1\), into the standard SOCP form when the exact covariance matrix is used.
        Parameters:
        Sigma - the covariance matrix
        sigmaRoot - the root of a matrix
      • PortfolioRiskExactSigma

        public PortfolioRiskExactSigma​(Matrix Sigma)
        Transforms the portfolio risk term, \(y^{\top}\Sigma\;y\leq t_1\), into the standard SOCP form when the exact covariance matrix is used.
        Parameters:
        Sigma - the covariance matrix
    • Method Detail

      • areAllConstraintsSatisfied

        public boolean areAllConstraintsSatisfied​(Vector y)
        Checks whether all SOCP constraints represented by this portfolio constraint are satisfied. The constraint generated by objective function to find the optimal solution. It cannot be "violated".
        Specified by:
        areAllConstraintsSatisfied in class SOCPPortfolioConstraint
        Parameters:
        y - a portfolio solution or allocation; the asset weights
        Returns:
        true
      • evaluate

        public Double evaluate​(Vector y)
        Description copied from interface: Function
        Evaluate the function f at x, where x is from the domain.
        Parameters:
        y - x
        Returns:
        f(x)
      • dimensionOfDomain

        public int dimensionOfDomain()
        Description copied from interface: Function
        Get the number of variables the function has. For example, for a univariate function, the domain dimension is 1; for a bivariate function, the domain dimension is 2.
        Returns:
        the number of variables
      • dimensionOfRange

        public int dimensionOfRange()
        Description copied from interface: Function
        Get the dimension of the range space of the function. For example, for a Rn->Rm function, the dimension of the range is m.
        Returns:
        the dimension of the range