gurobi_robust_genetics_sqp

gurobi_robust_genetics_sqp(sigma, mubar, omega, sires, dams, lam, kappa, dimension, upper_bound, lower_bound, time_limit, max_iterations, robust_gap_tol, model_output, debug)

Solve the robust optimal contribution selection problem using Gurobi. Consider a breeding cohort whose relationships are modelled some matrix $\Sigma$, and whose expected breeding values have mean $\boldsymbol{\bar\mu}$ and variance $\Omega$. This function uses sequential quadratic programming the contribution $\mathbf{w}$ which maximises expected breeding value subject to considering the worst case for group co-ancestry under some constraints.

Parameters

• sigma : ndarray or spmatrix
  Relationship matrix $\Sigma$ for the candidates in the cohort. This can either be a dense NumPy array or a sparse SciPy matrix, but either way must be symmetric and positive definite. If an estimated model is being used then any correction should be done before use here.
• mubar : ndarray
  Vector $\boldsymbol{\bar\mu}$ of expected values of the expected returns for candidates in the cohort for selection.
• sigma : ndarray or spmatrix
  Covariance matrix $\Omega$ for expected returns for candidates in the cohort. This can either be a dense NumPy array or a sparse SciPy matrix, but either way must be symmetric and positive definite. If an estimated model is being used then any correction should be done before use here.
• sires : Any
  An object representing an set of indices for all sires (male candidates) in the cohort. Type is not restricted, so could be a NumPy array, a set, a list, an iterator like range, etc.
• dams : Any
  An object representing an set of indices for all dams (female candidates) in the cohort. Type is not restricted, so could be a NumPy array, a set, a list, an iterator like range, etc.
• lam : float
  Controls the balance between risk and return, with lower values giving riskier selections and higher more conservative ones. This can take any real value, but $\lambda<0$ effectively corresponds to rewarding inbreeding so should not be considered.
• kappa : float
  Controls how resilient the solution must be to variation in expected breeding values. Can take any positive real values, with higher values representing a higher tolerance for uncertainty. Taking $\kappa = 0$ corresponds to assuming is known exactly, while $\kappa = \infty$ corresponds to unlimited tolerance for uncertainty.
• dimension : int
  Number of candidates in the cohort, i.e. the dimension of the problem. While strictly speaking this is evident from the other parameters, it's asked for as a convenience to prevent repeated calculation.
• upper_bound : ndarray or float, optional
  Upper bound on how much each candidate can contribute. This can be an array of differing bounds for each candidate, or a float which applies to all candidates equally. This value should be such that $0\leq l_i \leq u_i \leq 1$ and the default is, $\mathbf{u} = \mathbf{1}$, enforced by the contributions $\mathbf{w}$ being a vector proportions rather than absolute values.
• lower_bound : ndarray or float, optional
  Lower bound on how much each candidate can contribute. This can be an array of differing bounds for each candidate, or a float which applies to all candidates equally. This value should be such that $0\leq l_i \leq u_i \leq 1$ and the default is, $\mathbf{l} = \mathbf{0}$, enforced by the contributions $\mathbf{w}$ being a vector proportions rather than absolute values.
• time_limit : float, optional
  Maximum amount of time in seconds to give Gurobi to solve the problem. Optional, with the default being no time limit.
• max_iterations : int, optional
  Maximum number of iterations that can be taken in solving the problem, i.e. the maximum number of constraints to use to approximate the constraint. Optional, with a default value of 1000.
• robust_gap_tol : float, optional
  Tolerance to use checking whether an approximating constraint is active, and whether the SQP overall has converged. Default value is $10^{-7}$.
• model_output : str, optional
  Name to use if saving the model file to the working directory. If given, the string is used as the file name, '{model_output}.mps'. This is optional, and by default the file isn't saved.
• debug : bool, optional
  Flag which controls whether Gurobi prints its output to terminal. Default value is False.

Returns

• ndarray
  Portfolio vector $\mathbf{w}$ which Gurobi has determined is a solution.
• float
  Auxiliary variable $z$ corresponding to uncertainty associated with the portfolio vector which Gurobi has determined is a solution.
• float
  Value of the objective function for returned solution.

Examples

If a problem had been loaded into Python, then usage of this function might look like

>>> w, z, obj = robustocs.gurobi_robust_genetics_sqp(
...     sigma, mubar, omega, sires, dams, lam=0.5, kappa=1, dimension=4)
>>> w
array([0.38223361, 0.38223361, 0.11776639, 0.11776639])
>>> z
0.3787064407360815
>>> obj
0.7768406227995354

Note that due to differing methods this need not give exactly the same solution as gurobi_robust_genetics_conic or highs_robust_genetics_sqp, though they should be similar to some tolerance.