solnp {Rsolnp} | R Documentation |
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(),
...
)
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). |
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).
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.
Author(s)
Alexios Galanos
Examples
{
# 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))