NNLPG

Solves a general nonlinear programming problem using a sequential equality constrained quadratic programming method with user supplied gradients.

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. IACT = 1 to ME for equality constraints and IACT = ME +1 to M for inequality constraints. (Input)

RESULT – If IACT is zero, then RESULT is the computed objective 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 NNLPG_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.

GRAD — User-supplied subroutine to evaluate the gradients at a given point. The usage is CALL GRAD (X, IACT, RESULT), where

X – The point at which the gradient of the objective function or gradient of a constraint is evaluated. (Input)

IACT – Integer indicating whether evaluation of the function gradient is requested or evaluation of a constraint gradient is requested. If IACT is zero, then an objective function gradient evaluation is requested. If IACT is nonzero then the value if IACT indicates the index of the constraint gradient to evaluate. (Input)
IACT = 1 to ME for equality constraints and IACT = ME +1 to M for inequality constraints.

RESULT – If IACT is zero, then RESULT is the computed gradient of the objective function at the point X. If IACT is nonzero, then RESULT is the computed gradient of the requested constraint value at the point X. (Output)

The routine GRAD must be use-associated in a user module that uses NNLPG_INT, or else declared EXTERNAL in the calling program. If GRAD is a separately compiled routine, not in a module, then is 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)

IBTYPE	Action
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 the variables. (Input, if IBTYPE = 0; output, if IBTYPE = 1 or 2; input/output, if IBTYPE = 3) If there is no lower bound on a variable, then the corresponding XLB value should be set to ‑huge(x(1)).

XUB — Vector of length N containing the upper bounds on the variables. (Input, if IBTYPE = 0; output, if IBTYPE = 1 or 2; input/output, if IBTYPE = 3) If there is no upper bound on 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. If XGUESS is located outside or in the boundary region of the bound constraints set, then the algorithm first projects XGUESS into the interior of the bound constraints set before starting the main computations. (Input)
Default: XGUESS = x, where x is the vector with the smallest l2 – norm that satisfies the bounds.

IPRINT — Parameter indicating the desired output level. (Input)

IPRINT	Action
0	No output printed.
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.

Default: IPRINT = 0.

MAXITN — Maximum number of iterations allowed. (Input)
Default: MAXITN = 200.

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

TAU0 — A universal bound describing how much the unscaled penalty-term may deviate from zero. (Input)
NNLPG 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 NNLPG until such a point is found. A small TAU0 diminishes the efficiency of NNLPG, 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

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) ( = epsilon(x(1))2/3 )

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(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)

LGMULT — Vector of length M containing the Lagrange multiplier estimates of the constraints. (Output)

CONSTRES — Vector of length M containing the constraint residuals. (Output)

FORTRAN 90 Interface

Generic: CALL NNLPG (FCN, GRAD, M, ME, IBTYPE, XLB, XUB, X [, …])

Specific: The specific interface names are S_NNLPG and D_NNLPG.

Description

The routine NNLPG 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, NNLPG 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 NNLPG.

A good initial starting point is very problem specific and should be provided by the calling program whenever possible. See optional argument XGUESS.

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. NNLPG 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.

Comments

1. Informational errors

Type	Code	Description
4	1	Constraint evaluation returns an error with current point.
4	2	Objective evaluation returns an error with current point.
4	3	Working set is singular in dual extended QP.
4	4	QP problem is seemingly infeasible.
4	5	A stationary point located or termination criteria too strong.
4	8	Maximum number of iterations exceeded.
4	9	Stationary point not feasible.
4	10	Very slow primal progress.
4	11	The problem is singular.
4	12	Matrix of gradients of binding constraints is singular or very ill-conditioned.
4	13	Small changes in the penalty function.

Examples

Example 1

The problem

is solved.

USE NNLPG_INT

USE WRRRN_INT

IMPLICIT NONE

INTEGER IBTYPE, M, ME

PARAMETER (IBTYPE=0, M=2, ME=1)

REAL(KIND(1E0)) FVALUE, X(2), XGUESS(2), XLB(2), XUB(2)

EXTERNAL FCN, GRAD

XLB = -HUGE(X(1))

XUB = HUGE(X(1))

CALL NNLPG (FCN, GRAD, M, ME, IBTYPE, XLB, XUB, X)

CALL WRRRN ('The solution is', X)

END

SUBROUTINE FCN (X, IACT, RESULT, IERR)

INTEGER IACT

REAL(KIND(1E0)) X(*), RESULT

LOGICAL IERR

SELECT CASE (IACT)

CASE(0)

RESULT = (X(1)-2.0E0)**2 + (X(2)-1.0E0)**2

CASE(1)

RESULT = X(1) - 2.0E0*X(2) + 1.0E0

CASE(2)

RESULT = -(X(1)**2)/4.0E0 - X(2)**2 + 1.0E0

END SELECT

RETURN

END

SUBROUTINE GRAD (X, IACT, RESULT)

INTEGER IACT

REAL(KIND(1E0)) X(*),RESULT(*)

SELECT CASE (IACT)

CASE(0)

RESULT (1) = 2.0E0*(X(1)-2.0E0)

RESULT (2) = 2.0E0*(X(2)-1.0E0)

CASE(1)

RESULT (1) = 1.0E0

RESULT (2) = -2.0E0

CASE(2)

RESULT (1) = -0.5E0*X(1)

RESULT (2) = -2.0E0*X(2)

END SELECT

RETURN

END

Output

The solution is

1 0.8229

2 0.9114

Example 2

The same problem from Example 1 is solved, but here we use central differences to compute the gradient of the first constraint. This example demonstrates how NNLPG can be used in cases when analytic gradients are known for only a portion of the constraints and/or objective function. The subroutine CDGRD is used to compute an approximation to the gradient of the first constraint.

USE NNLPG_INT

USE CDGRD_INT

USE WRRRN_INT

IMPLICIT NONE

INTEGER IBTYPE, M, ME

PARAMETER (IBTYPE=0, M=2, ME=1)

REAL(KIND(1E0)) FVALUE, X(2), XGUESS(2), XLB(2), XUB(2)

EXTERNAL FCN, GRAD

XLB = -HUGE(X(1))

XUB = HUGE(X(1))

CALL NNLPG (FCN, GRAD, M, ME, IBTYPE, XLB, XUB, X)

CALL WRRRN ('The solution is', X)

END

SUBROUTINE FCN (X, IACT, RESULT, IERR)

INTEGER IACT

REAL(KIND(1E0)) X(2), RESULT

LOGICAL IERR

EXTERNAL CONSTR1

SELECT CASE (IACT)

CASE(0)

RESULT = (X(1)-2.0E0)**2 + (X(2)-1.0E0)**2

CASE(1)

CALL CONSTR1(2, X, RESULT)

CASE(2)

RESULT = -(X(1)**2)/4.0E0 - X(2)**2 + 1.0E0

END SELECT

RETURN

END

SUBROUTINE GRAD (X, IACT, RESULT)

USE CDGRD_INT

INTEGER IACT

REAL(KIND(1E0)) X(2),RESULT(2)

EXTERNAL CONSTR1

SELECT CASE (IACT)

CASE(0)

RESULT (1) = 2.0E0*(X(1)-2.0E0)

RESULT (2) = 2.0E0*(X(2)-1.0E0)

CASE(1)

CALL CDGRD(CONSTR1, X, RESULT)

CASE(2)

RESULT (1) = -0.5E0*X(1)

RESULT (2) = -2.0E0*X(2)

END SELECT

RETURN

END

SUBROUTINE CONSTR1 (N, X, RESULT)

INTEGER N

REAL(KIND(1E0)) X(*), RESULT

RESULT = X(1) - 2.0E0*X(2) + 1.0E0

RETURN

END

Output

The solution is

1 0.8229

2 0.9114