sumt | R Documentation |

Solve constrained optimization problems via the Sequential Unconstrained Minimization Technique (SUMT).

sumt(x0, L, P, grad_L = NULL, grad_P = NULL, method = NULL, eps = NULL, q = NULL, verbose = NULL, control = list())

`x0` |
a list of starting values, or a single starting value. |

`L` |
a function to minimize. |

`P` |
a non-negative penalty function such that |

`grad_L` |
a function giving the gradient of |

`grad_P` |
a function giving the gradient of |

`method` |
a character string, or |

`eps` |
the absolute convergence tolerance. The algorithm stops if
the (maximum) distance between successive Defaults to |

`q` |
a double greater than one controlling the growth of the
Defaults to 10. |

`verbose` |
a logical indicating whether to provide some output on minimization progress. Defaults to |

`control` |
a list of control parameters to be passed to the
minimization routine in case |

The Sequential Unconstrained Minimization Technique is a heuristic for
constrained optimization. To minimize a function *L* subject to
constraints, one employs a non-negative function *P* penalizing
violations of the constraints, such that *P(x)* is zero iff *x*
satisfies the constraints. One iteratively minimizes *L(x) +
ρ_k P(x)*, where the *ρ* values are increased according to
the rule *ρ_{k+1} = q ρ_k* for some constant *q > 1*,
until convergence is obtained in the sense that the Euclidean distance
between successive solutions *x_k* and *x_{k+1}* is small
enough. Note that the “solution” *x* obtained does not
necessarily satisfy the constraints, i.e., has zero *P(x)*. Note
also that there is no guarantee that global (approximately)
constrained optima are found. Standard practice would recommend to
use the best solution found in “sufficiently many” replications
of the algorithm.

The unconstrained minimizations are carried out by either
`optim`

or `nlm`

, using analytic
gradients if both `grad_L`

and `grad_P`

are given, and
numeric ones otherwise.

If more than one starting value is given, the solution with the minimal augmented criterion function value is returned.

A list inheriting from class `"sumt"`

, with components `x`

,
`L`

, `P`

, and `rho`

giving the solution obtained, the
value of the criterion and penalty function at `x`

, and the final
*ρ* value used in the augmented criterion function.

A. V. Fiacco and G. P. McCormick (1968).
*Nonlinear programming: Sequential unconstrained minimization
techniques*.
New York: John Wiley & Sons.

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