solnp: Nonlinear optimization using augmented Lagrange method (original version)

Description

Nonlinear optimization using augmented Lagrange method (original version)

Usage

solnp(
  pars,
  fun,
  eqfun = NULL,
  eqB = NULL,
  ineqfun = NULL,
  ineqLB = NULL,
  ineqUB = NULL,
  LB = NULL,
  UB = NULL,
  control = list(),
  ...
)

Value

An list with the following slot:

pars: Optimal Parameters.
convergence: Indicates whether the solver has converged (0) or not (1 or 2).
values: Vector of function values during optimization with last one the value at the optimal.
lagrange: The vector of Lagrange multipliers.
hessian: The Hessian of the augmented problem at the optimal solution.
ineqx0: The estimated optimal inequality vector of slack variables used for transforming the inequality into an equality constraint.
nfuneval: The number of function evaluations.
elapsed: Time taken to compute solution.

Arguments

pars: an numeric vector of decision variables (length n).
fun: the objective function (must return a scalar).
eqfun: an optional function for calculating equality constraints.
eqB: a vector of the equality bounds (if eq_fn provided).
ineqfun: an optional function for calculating inequality constraints.
ineqLB: the lower bounds for the inequality (must be finite)
ineqUB: the upper bounds for the inequalitiy (must be finite)
LB: lower bounds on decision variables
UB: upper bounds on decision variables
control: a list of solver control parameters (see details).
...: additional arguments passed to the supplied functions (common to all functions supplied).

Author

Alexios Galanos

Details

The optimization problem solved by csolnp is formulated as: $$ \begin{aligned} \min_{x \in \mathbb{R}^n} \quad & f(x) \\ \text{s.t.} \quad & g(x) = b \\ & h_l \le h(x) \le h_u\\ & x_l \le x \le x_u\\ \end{aligned} $$

where $f(x)$ is the objective function, $g(x)$ is the vector of equality constraints with target value $b$, $h(x)$ is the vector of inequality constraints bounded by $h_l$ and $h_u$, with parameter bounds $x_l$ and $x_u$. Internally, inequality constraints are converted into equality constraints using slack variables and solved using an augmented Lagrangian approach. The control is a list with the following options:

rho: This is used as a penalty weighting scaler for infeasibility in the augmented objective function. The higher its value the more the weighting to bring the solution into the feasible region (default 1). However, very high values might lead to numerical ill conditioning or significantly slow down convergence.
outer.iter: Maximum number of major (outer) iterations (default 400).
inner.iter: Maximum number of minor (inner) iterations (default 800).
delta: Relative step size in forward difference evaluation (default 1.0e-7).
tol: Relative tolerance on feasibility and optimality (default 1e-8).
trace: The value of the objective function and the parameters is printed at every major iteration (default 1).

Examples

Run this code

{
# From the original paper by Y.Ye
# see the unit tests for more....
# POWELL Problem
fn1 = function(x)
{
    exp(x[1] * x[2] * x[3] * x[4] * x[5])
}
eqn1 = function(x){
    z1 = x[1] * x[1] + x[2] * x[2] + x[3] * x[3] + x[4] * x[4] + x[5] * x[5]
    z2 = x[2] * x[3] - 5 * x[4] * x[5]
    z3 = x[1] * x[1] * x[1] + x[2] * x[2] * x[2]
    return(c(z1, z2, z3))
}
x0 = c(-2, 2, 2, -1, -1)
}
powell = solnp(x0, fun = fn1, eqfun = eqn1, eqB = c(10, 0, -1))

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