constrainedNlp¶

Solves a general nonlinear programming problem using a sequential equality constrained quadratic programming method.

Synopsis¶

constrainedNlp (fcn, m, meq, ibtype, xlb, xub)

Required Arguments¶

void fcn (n, x[], iact, result, ierr) (Input)

User supplied function to evaluate the objective function and constraints at a given point.

int n (Input): Number of variables.
float x[] (Input): The point at which the objective function or a constraint is evaluated.
int iact (Input): Integer indicating whether evaluation of the 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 of iact indicates the index of the constraint to evaluate. iact =1 to meq for equality constraints and iact = meq +1 to m for inequality constraints.
float result[] (Output): If iact is zero, then result is the computed objective function at the point x. If iact is nonzero, then result is the requested constraint value at the point x.
int ierr (Output): An integer. On input ierr is set to 0. If an error or other undesirable condition occurs during evaluation, then ierr should be set to 1. Setting ierr to 1 will result in the step size being reduced and the step being tried again. (If ierr is set to 1 for xguess, then an error is issued.)

int m (Input)

Total number of constraints.

int meq (Input)

Number of equality constraints.

int ibtype (Input)

Scalar indicating the types of bounds on variables.

`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 first variable, all other variables will have the same bounds.

float xlb[] (Input, Output, or Input/Output)

Array with n components 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 machine(8).

float xub[] (Input, Output, or Input/Output)

Array with n components 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 machine(7).

Return Value¶

The solution x of the nonlinear programming problem. If no solution can be computed, then None is returned.

Optional Arguments¶

gradient, void grad(n, x[], iact, result[]) (Input)

User-supplied function to evaluate the gradients at a given point where

Arguments

int n (Input): Number of variables.
float x[] (Input): The point at which the gradient of the objective function or gradient of a constraint is evaluated
int iact (Input): 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 of iact indicates the index of the constraint gradient to evaluate. iact =1 to meq for equality constraints and iact =meq+1 to m for inequality constraints.
float result[] (Output): 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.

t_print, int (Input)

Parameter indicating the desired output level. (Input)

`t_print`	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: t_print = 0.

xguess, float[] (Input)

Array of length n containing an initial guess of the solution.

Default: xguess = X, (with the smallest value of \(\|x\|_2\)) that satisfies the bounds.

itmax, int (Input)

Maximum number of iterations allowed.

Default: itmax = 200.

tau0, float (Input)

A universal bound describing how much the unscaled penalty-term may deviate from zero. constrainedNlp assumes that within the region described by

\[\sum_{i=1}^{M_e} \left|g_i(x)\right| - \sum_{i=M_e+1}^{M} \min \left(0, g_i\left(x\right)\right) \leq \mathit{tau}0\]

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 constrainedNlp until such a point is found. A small tau0 diminishes the efficiency of constrainedNlp, 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 = 1.0.

del0, float (Input)

In the initial phase of minimization a constraint is considered binding if

\[\frac{g_i(x)}{\max \left(1, \left\|\nabla g_i(x)\right\|\right)} \leq \mathit{del}0 \phantom{...} i = M_{\mathrm{e}} + 1, \ldots, M\]

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.

Default: del0 = .5* tau0

smallw, float (Input)

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.

Default: smallw = exp(2*log(eps/3)) where eps is the machine precision.

delmin, float (Input)

Scalar which defines allowable constraint violations of the final accepted result. Constraints are satisfied if \(|g_i(x)|\) ≤ delmin for equality constraints, and \(g_i(x)\) ≥ (‑delmin) for equality constraints.

Default: delmin = min(.1*del0, max(epsdif, max(1.e-6*del0, smallw))

scfmax, float (Input)

Scalar containing the bound for the internal automatic scaling of the objective function. (Input)

Default: scfmax = 1.0e4

obj (Output)

Scalar containing the value of the objective function at the computed solution.

lagrangeMultipliers (Output)

An array containing the Lagrange multiplier estimates of the constraints.

constraintResiduals (Output)

An array containing the constraints residuals.

Note: The following optional arguments are valid only if gradient is not supplied.

difftype, int (Input)

Type of numerical differentiation to be used.

Default: difftype = 1

`difftype`	Action
1	Use a forward difference quotient with discretization stepsize \(0.1(\text{epsfcn})^{1/2}\) componentwise relative.
2	Use the symmetric difference quotient with discretization stepsize \(0.1(\text{epsfcn})^{1/3}\) componentwise relative.
3	Use the sixth order approximation computing a Richardson extrapolation of three symmetric difference quotient values. This uses a discretization stepsize \(0.01(\text{epsfcn})^{1/7}\).

xscale, float[] (Input)

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]. In the absence of other information, set all entries to 1.0.

Default: xscale[] = 1.0.

epsdif (Input)

Relative precision in gradients.

Default: epsdif = ɛ where ɛ is the machine precision.

epsfcn, float (Input)

Relative precision of the function evaluation routine.

Default: epsfcn = ɛ where ɛ is the machine precision

taubnd, float (Input)

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.

Default: taubnd = 1.0

Description¶

The function constrainedNlp provides an interface to a licensed version of subroutine DONLP2, a 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:

\[\begin{split}\begin{array}{ll} \min\limits_{x \in R^n} f(x) \\ \text{subject to } & g_j(x) = 0, \text{ for } j = 1, \ldots, m_e \\ & g_j(x) \geq 0, \text{ for } j = m_e + 1, \ldots, m \\ & x_1 \leq x \leq x_u \end{array}\end{split}\]

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, constrainedNlp 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 constrainedNlp.

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 constrainedNlp. Selecting a higher order approximation method may be necessary for some problems. See optional argument difftype.
If a two sided constraint \(l_i\leq g_i(x)\leq u_i\) is transformed into two constraints \(g _{2i}(x)\geq 0\) and \(g_{2i+1}(x)\geq 0\), then choose \(del0<1/2(u_i-l_i)/\max\{1,\|∇g_i(x)\|\}\), 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 1 and returning without performing the evaluation will avoid the exception. constrainedNlp will then reduce the stepsize and try the step again. Note, if ierr is set to 1 for the initial guess, then an error is issued.

On some platforms, constrainedNlp can evaluate the user-supplied functions fcn and grad in parallel. This is done only if the function ompOptions is called to flag user-defined functions as thread-safe. A function is thread-safe if there are no dependencies between calls. Such dependencies are usually the result of writing to global or static variables.

Example¶

The problem

\[\begin{split}\begin{array}{l} \min f(x) = \left(x_1 - 2\right)^2 + \left(x_2 - 1\right)^2 \\ \text{subject to } \begin{array}[t]{l} & g_1(x) = x_1 - 2x_2 + 1 = 0 \\ & g_2(x) = -x_1^2/4 - x_2^2 + 1 \geq 0 \end{array} \\ \end{array}\end{split}\]

is solved.

from numpy import *
from pyimsl.math.constrainedNlp import constrainedNlp
from pyimsl.math.writeMatrix import writeMatrix


# Himmelblau problem 1
def fcn(n, x, iact, result, ierr):
    tmp1 = x[0] - 2.0e0
    tmp2 = x[1] - 1.0e0
    if iact == 0:
        result[0] = tmp1 * tmp1 + tmp2 * tmp2
    elif iact == 1:
        result[0] = x[0] - 2.0e0 * x[1] + 1.0e0
    elif iact == 2:
        result[0] = -(x[0] * x[0]) / 4.0e0 - x[1] * x[1] + 1.0e0
    ierr = 0


m = 2
me = 1
n = 2
ibtype = 0

inf = 1e300000
xlb = [-inf, -inf]
xub = [inf, inf]

x = constrainedNlp(fcn, m, me, ibtype, xlb, xub)
writeMatrix("The solution is ", x)

Output¶

 
    The solution is 
          1            2
     0.8229       0.9114

Fatal Errors¶

IMSL_STOP_USER_FCN

Request from user supplied function to stop algorithm.

User flag = “#”.