Solves a general nonlinear programming problem using a sequential equality constrained quadratic programming method.
Required Arguments
FCN
User-supplied subroutine to evaluate the objective function and constraints at a
given point. The internal usage is CALL FCN (X, IACT, RESULT, IERR),
where
X The point at which the objective function or constraint is evaluated. (Input)
IACT Integer indicating whether evaluation of the objective function is requested or evaluation of a constraint is requested. If IACT is zero, then an objective function evaluation is requested. If IACT is nonzero then the value if IACT indicates the index of the constraint to evaluate. (Input)
RESULT If IACT is zero, then RESULT is the computed function value at the point X. If IACT is nonzero, then RESULT is the computed constraint value at the point X. (Output)
IERR Logical variable. On input IERR is set to .FALSE. If an error or other undesirable condition occurs during evaluation, then IERR should be set to .TRUE. Setting IERR to .TRUE. will result in the step size being reduced and the step being tried again. (If IERR is set to .TRUE. for XGUESS, then an error is issued.)
The routine FCN must be use-associated in a user module that uses NNLPF_INT, or else declared EXTERNAL in the calling program. If FCN is a separately compiled routine, not in a module, then it must be declared EXTERNAL.
M Total number of constraints. (Input)
ME Number of equality constraints. (Input)
IBTYPE Scalar indicating the types of bounds on variables. (Input)
0 User will supply all the bounds.
1 All variables are nonnegative.
2 All variables are nonpositive.
3 User supplies only the bounds on 1st variable; all other variables will have the same bounds.
XLB Vector of
length N containing the lower
bounds on variables. (Input, if IBTYPE = 0; output, if
IBTYPE = 1 or 2;
input/output, if IBTYPE = 3)
If
there is no lower bound for a variable, then the corresponding XLB value should be
set to -Huge(X(1)).
XUB Vector of
length N containing the upper
bounds on variables. (Input, if IBTYPE = 0; output, if
IBTYPE = 1 or 2;
input/output, if IBTYPE = 3).
If
there is no upper bound for a variable, then the corresponding XUB value should be
set to Huge(X(1)).
X Vector of length N containing the computed solution. (Output)
Optional Arguments
N Number of
variables. (Input)
Default: N = size(X).
XGUESS Vector
of length N containing an
initial guess of the solution. (Input)
Default: XGUESS = x, (with the
smallest value of 
) that
satisfies the bounds.
XSCALE Vector
of length N setting the internal
scaling of the variables. The initial value given and the objective
function and gradient evaluations however are always in the original unscaled
variables. The first internal variable is obtained by dividing values X(I) by XSCALE(I).
(Input)
In the absence of other information, set all entries to
1.0.
Default: XSCALE(:) = 1.0.
IPRINT Parameter indicating the desired output level. (Input)
1 One line of intermediate results is printed in each iteration.
2 Lines of intermediate results summarizing the most important data for each step are printed.
3 Lines of detailed intermediate results showing all primal and dual variables, the relevant values from the working set, progress in the backtracking and etc are printed
4 Lines of detailed intermediate results showing all primal and dual variables, the relevant values from the working set, progress in the backtracking, the gradients in the working set, the quasi-Newton updated and etc are printed.
MAXITN Maximum
number of iterations allowed. (Input)
Default: MAXITN = 200.
EPSDIF Relative
precision in gradients. (Input)
Default: EPSDIF = epsilon(x(1))
TAU0 A
universal bound describing how much the unscaled penalty-term may deviate from
zero. (Input)
NNLPF assumes that
within the region described by
all functions may be evaluated
safely. The initial guess, however, may violate these requirements. In that case
an initial feasibility improvement phase is run by NNLPF until such a
point is found. A small TAU0 diminishes the efficiency of NNLPF, because the
iterates then will follow the boundary of the feasible set closely. Conversely,
a large TAU0 may degrade the
reliability of the code.
Default TAU0 = 1.E0
DEL0 In the initial phase of minimization a constraint is considered binding if
Good values are between .01 and
1.0. If DEL0 is chosen too small then
identification of the correct set of binding constraints may be delayed.
Contrary, if DEL0 is too large, then the method
will often escape to the full regularized SQP method, using individual slack
variables for any active constraint, which is quite costly. For well-scaled
problems DEL0=1.0 is reasonable.
(Input)
Default: DEL0 = .5*TAU0
EPSFCN Relative
precision of the function evaluation routine. (Input)
Default: EPSFCN = epsilon(x(1))
IDTYPE Type of
numerical differentiation to be used. (Input)
Default: IDTYPE = 1
IDTYPE
Action
1 Use a forward difference quotient with discretization stepsize 0.1(EPSFCN2 componentwise relative.
2 Use the symmetric difference quotient with discretization stepsize 0.1(EPSFCN1/3) componentwise relative
3 Use the sixth order approximation computing a Richardson extrapolation of three symmetric difference quotient values. This uses a discretization stepsize 0.01(EPSFCN1/7 )
TAUBND Amount
by which bounds may be violated during numerical differentiation.
Bounds are violated by TAUBND (at most) only
if a variable is on a bound and finite differences are taken for gradient
evaluations. (Input)
Default: TAUBND = 1.E0
SMALLW Scalar
containing the error allowed in the multipliers. For example, a negative
multiplier of an inequality constraint is accepted (as zero) if its absolute
value is less than SMALLW.
(Input)
Default: SMALLW =
exp(2*log(epsilon(x(1)/3)))
DELMIN Scalar
which defines allowable constraint violations of the final accepted
result. Constraints are satisfied if |gi(x)|
DELMIN , and gj(x)
(-DELMIN ) respectively. (Input)
Default: DELMIN
= min(DEL0/10, max(EPSDIF, min(DEL0/10,
max(1.E-6*DEL0, SMALLW))))
SCFMAX Scalar
containing the bound for the internal automatic scaling of the objective
function. (Intput)
Default: SCFMAX = 1.0E4
FVALUE Scalar containing the value of the objective function at the computed solution. (Output)
FORTRAN 90 Interface
Generic: CALL NNLPF (FCN, M, ME, IBTYPE, XLB, XUB, X [, ])
Specific: The specific interface names are S_NNLPF and D_NNLPF .
Description
The routine NNLPF provides an interface to a licensed version of subroutine DONLP2, a FORTRAN code developed by Peter Spellucci (1998). It uses a sequential equality constrained quadratic programming method with an active set technique, and an alternative usage of a fully regularized mixed constrained subproblem in case of nonregular constraints (i.e. linear dependent gradients in the working sets). It uses a slightly modified version of the Pantoja-Mayne update for the Hessian of the Lagrangian, variable dual scaling and an improved Armjijo-type stepsize algorithm. Bounds on the variables are treated in a gradient-projection like fashion. Details may be found in the following two papers:
P. Spellucci: An SQP method for general nonlinear programs using only equality constrained subproblems. Math. Prog. 82, (1998), 413-448.
P. Spellucci: A new technique for inconsistent problems in the SQP method. Math. Meth. of Oper. Res. 47, (1998), 355-500. (published by Physica Verlag, Heidelberg, Germany).
The problem is stated as follows:
Although default values are provided for optional input arguments, it may be necessary to adjust these values for some problems. Through the use of optional arguments, NNLPF allows for several parameters of the algorithm to be adjusted to account for specific characteristics of problems. The DONLP2 Users Guide provides detailed descriptions of these parameters as well as strategies for maximizing the perfomance of the algorithm. The DONLP2 Users Guide is available in the help subdirectory of the main IMSL product installation directory. In addition, the following are a number of guidelines to consider when using NNLPF.
A good initial starting point is very problem specific and should be provided by the calling program whenever possible. See optional argument XGUESS.
Gradient approximation methods can have an effect on the success of NNLPF. Selecting a higher order appoximation method may be necessary for some problems. See optional argument IDTYPE.
If a
two sided constraint 
is
transformed into two constraints 
and 
, then choose 
, or at least try to provide an estimate for that value.
This will increase the efficiency of the algorithm. See optional
argument DEL0.
The parameter IERR provided in the interface to the user supplied function FCN can be very useful in cases when evaluation is requested at a point that is not possible or reasonable. For example, if evaluation at the requested point would result in a floating point exception, then setting IERR to .TRUE. and returning without performing the evaluation will avoid the exception. NNLPF will then reduce the stepsize and try the step again. Note, if IERR is set to .TRUE. for the initial guess, then an error is issued.
Example
PARAMETER (IBTYPE=0, M=2, ME=1)
REAL(KIND(1E0)) FVALUE, X(2), XGUESS(2), XLB(2), XUB(2)
CALL NNLPF (FCN, M, ME, IBTYPE, XLB, XUB, X)
CALL WRRRN ('The solution is', X)
SUBROUTINE FCN (X, IACT, RESULT, IERR)
RESULT = (X(1)-2.0E0)**2 + (X(2)-1.0E0)**2
RESULT = X(1) - 2.0E0*X(2) + 1.0E0
RESULT = -(X(1)**2)/4.0E0 - X(2)**2 + 1.0E0
Output
The solution is
1
0.8229
2 0.9114
|
Visual Numerics, Inc. PHONE: 713.784.3131 FAX:713.781.9260 |