Subroutine Example 2: Hock-Schittkowski Problem Number 14

This example is intended to demonstrate how to set up and solve a constrained problem using the subroutine interface. In particulare, we will show you how to solve Hock-Schittkowski Problem Number 14, which is a two-dimensional problem with one linear and one nonlinear constraint.

minimize

$(x_1 - 2.0)^2 + (x_2 - 1.0)^2$

subject to

$x_1 - 2.0 x_2 = -1.0,$

$0.25 x_1^2 - x_2^2 + 1.0 \ge 0.0,$

For this example, we will assume that that analytic first derivatives are available (but not second derivatives). We will also assume that the subroutines that initialize, evaluate the function, and evaluate the nonlinear constraint are in the same file. We step through the specifics below.

User-Provided Subroutines

This section contains examples of the user-defined functions that are required. The first performs the initialization of the problem. The second performs the evaluation of the function.

First, include the necessary header files. In this case, we need iostream so we can print error messages and the OPT++ header file, NLP.h, for some definitions. Also, because we are going to great a dynamically loadable library, we need to surround all of our code by an extern "C" statement.

#include <iostream> #include "NLP.h" extern "C" {

The subroutine that initializes the problem should perform any one-time tasks that are needed for the problem. One part of that is checking for error conditions in the setup. In this case, the dimension, ndim, can only take on a value of 2. Using "exit" is not the ideal way to deal with error conditions, but it serves well as an example. In addition to checking the error condition, we also do some manipulation of the initial values.

void init_hs14(int ndim, ColumnVector& x) { if (ndim != 2) { cerr << "Number of variables for Hock Problem 14 should be 2." << " The number of variables given is " << ndim << endl; exit (1); } //end if x(1) = 2+(x(1) -1)* 1.1771243447; x(2) = 2+(x(2) -1)* 1.0885621722; } //end init_hs14

The second required subroutine will evaluate the function. In this problem, we are trying to find the minimum value of Hock-Schittkowski Problem 14, so it is necessary to write the code that computes the value of that function given some set of optimization parameters. In addition, since we are assuming that first derivatives are available, we must also provide the code to compute the gradient. Mathematically, the function is given by:

$(x_1 - 2.0)^2 + (x_2 - 1.0)^2$

The following code will compute the value of f(x).

First, some manipulation of the optimization parameters, x, is done.

void hs14(int mode, int n, const ColumnVector& x, double& fx, ColumnVector& g, int& result) { double f1, f2, x1, x2; x1 = x(1); x2 = x(2); f1 = x1 - 2.0; f2 = x2 - 1.0;

Then the function or gradient is computed. Notice how the mode and result variables are used to determine/report the type of evaluation done.

if (mode & NLPFunction) { fx = f1*f1 + f2*f2; result = NLPFunction; } //end f(x) if (mode & NLPGradient) { g(1) = 2*(x1 - 2.0); g(2) = 2*(x2 - 1.0); result = NLPGradient; } //end g(x) } //end hs14

In addition to the function, we must also provide the code to evaluate the nonlinear constraint. We will also include code for the first derivative of the constraint, and we will put this code in the same library as the function evaluation. The code looks very similar to that for the function, except now the expression is the following:

$0.25 x_1^2 - x_2^2 + 1.0 \ge 0.0,$

Also, notice the differences in the argument list. This is due to the fact that there could be multiple constraints computed in this function. In the current GUI/XML set-up, however, we are restricted to only one constraint per function. This will change in future releases.

First, some manipulation of the optimization parameters, x, is done.

void ineq_hs14(int mode, int n, const ColumnVector& x, ColumnVector& fx, Matrix& g, int& result) { double f1, f2, x1, x2; x1 = x(1); x2 = x(2); f1 = x1*x1; f2 = x2*x2;

Then the function or gradient is computed. Notice how the mode and result variables are used to determine/report the type of evaluation done.

if (mode & NLPFunction) { fx = -.25*f1 - f2 + 1.0; result = NLPFunction; } //end f(x) if (mode & NLPGradient) { g(1,1) = -0.5*x1; g(2,1) = -2.0*x2; result = NLPGradient; } //end g(x) } //end ineq_hs14 } //end extern "C"

Now that we have all of the code necessary to initialize and evaluate Hock-Schittkowski Problem 14, give it a try!

Building and Running the Example

If you want to try running this example, the following steps should do the trick.

cd into your favorite directory.
Write the code described above. You can organize it however you like, but we recommend putting both subroutines in the same file (e.g., testexample.C).
Copy the Makefile from the tests/xml directory into the directory where your code resides. (WARNING: Since the Makefile contains platform-dependent information, it should be one that was configured for the platform on which you are doing this example.)
Edit the Makefile by replacing the files listed in the existing SOURCES line with your file. For example,
```
           SOURCES = testexample.C
        
```
Type "make". This step will create the library, testexample.so.

Set the LD_LIBRARY_PATH environment variable to the directory where your library resides. For us, the directory was /home/pdhough/TooMuchFun, so:

	   for csh or tcsh, <br>
	   setenv LD_LIBRARY_PATH /home/pdhough/TooMuchFun

	   for bash, <br>
	   set LD_LIBRARY_PATH=/home/pdhough/TooMuchFun
	   export LD_LIBRARY_PATH

Create the XML input file for OPT++. The initialization subroutine is init_hs14, the function subroutine is hs14, and the library is testexample.so. The number or variables is 2, and there are first derivatives but not second. If you want to duplicate our results, we used an initial guess of 0.0 for the first variable and 0.0 for the second. There is one linear constraint for which you will need to enter the variables, coefficients, and right-hand side. There is one nonlinear constraint with a subroutine interface and first derivatives available. We used the NIPS algorithm with the default values for all of the parameters. We recommend putting the XML file in the same directory as your code so that you have all of your problem information in one place.
If you have set up OPT++ and the various environment variables as described in the documentations, you should now be able to run the problem by issuing the following command:
```
           ./testexample
        
```
If you like, you can compare your output to our results. There may be slight differences, but if you used the same input that we did, the results should look pretty much the same.

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Last revised July 25, 2006

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, last revised August 30, 2006.