Shor's R algorithm for unconstrained minimization of smooth and non smooth functions.

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`fn` |
A function to be minimized. fn(x) takes input as a vector of parameters over which minimization is to take place. fn() returns a scaler. |

`gr` |
A function to return the gradient for fn(x). |

`nvar` |
Number of parameters that fn() takes. |

`nstart` |
Number of initial guesses. Default is 10. |

`x0` |
Matrix, with dimension (nvar x nstart), each column represent an initial guess. |

`upper` |
upper bound for the initial value |

`lower` |
lower bound for the initial value |

`maxit` |
maximum number of iterations. |

`fvalquit` |
quit if f drops below this value. |

`beta` |
the key parameter, between 0 and 1, or nan to minimize the error in the secant equation each iteration (default 1/2) beta = 1: steepest descent; beta -> 0: conjugate gradient (beta = 0 will give divide by 0) note: beta = 1 - gamma, where gamma is notation in IMAJNA paper |

`normtol` |
termination tolerance on d: smallest vector in convex hull of up to ngrad gradients (default: 1e-6) |

`xnormquit` |
quit if norm(x) drops below this. |

`evaldist` |
the gradients used in the termination test qualify only if they are evaluated at points approximately within distance options.evaldist of x |

`ngrad` |
number of gradients willing to save and use in solving QP to check optimality tolerance on smallest vector in their convex hull; see also next two options |

`rescale` |
1 if rescale B to have inf norm 1 every iteration |

`strongwolfe` |
if this is 1, strong line search is used, otherwise, weak line search is used. Default is 0. |

`useprevstep` |
if 1, line search is initialized with previous steps. 1 seemed to perform better, but hard to justify this rationally. |

`wolfe1` |
wolfe line search parameter 1. |

`wolfe2` |
wolfe line search parameter 2. |

`quitLSfail` |
1 if want to quit when line search fails. 0 otherwise. |

`prtlevel` |
if 1, prints in code messages. |

Returns a list containing the following fields:

`x` |
a matrix with k'th column containing final value of x obtained from k'th column of x0. |

`f` |
a vector of final obtained minimum values of fn() at the initial points. |

`g` |
each column is the gradient at the corresponding column of x |

`B` |
list of the final shor matrices |

`frec` |
record of function evaluations |

`betarec` |
record of beta |

`xrec` |
record of iterates |

`svrec` |
record of singular value's of the Hessian |

Copyright (c) 2010 Michael Overton for Matlab code and documentation, with permission converted to R by Abhirup Mallik (and Hans W Borchers).

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