Example 1: Unconstrained Quasi-Newton Without Derivatives

This example is intended to demonstrate how to set up and solve a very simple problem. We will show you how to solve (unconstrained) Rosenbrock's function in two dimensions, i.e.,

minimize

$100(x_2 - x_{1}^2)^2 + (1 - x_1)^2$

For demonstration purposes, we will assume that there are no analytic derivatives available so we will use finite-difference approximations to the gradient and a quasi-Newton algorithm with a BFGS approximation to the Hessian. Recall that it is necessary to write C++ code for the main routine that sets up the problem and the algorithm and for the subroutines that initialize and evaluate the function. We step through the specifics below.

Main Routine

First include the necessary header files. Start with any C++/C header files that are needed. In this case, a bit of I/O. The other two header files are OPT++ header files. NLF contains objects, data, and methods required for setting up the function/problem. OptQNewton contains the objects, data, and methods required for using an unconstrained quasi-Newton optimization method. The last statement using NEWMAT::ColumnVector introduces the data member ColumnVector from the matrix library namespace NEWMAT. The use of namespaces prevents potential conflicts with third party libraries that may also have a data member named ColumnVector.

#include <fstream> #include "NLF.h" #include "OptQNewton.h" using NEWMAT::ColumnVector;

The following two lines serve as the declarations of the pointers to the subroutines that initialize the problem and evaluate the objective function, respectively.

void init_rosen(int ndim, ColumnVector& x); void rosen(int ndim, const ColumnVector& x, double& fx, int& result);

The next few lines complete the setup of the problem, which include setting the dimension of the problem and creating the nonliner function object. To create the nonlinear function object use the dimension of the problem and the pointers to the subroutines declared above. The FDNLF1 object has built-in finite-difference approximations to the gradient.

int main() { int ndim = 2; FDNLF1 nlp(ndim, rosen, init_rosen);

Now, let's build a quasi-Newton algorithm object using the nonlinear problem that has just been created. The quasi-Newton algorithm will use BFGS updates to approximate the Hessian. In addition, set any of the algorithmic parameters to desired values. All parameters have default values, so it is not necessary to set them unless you have specific values you wish to use. In this example, we set the globalization strategy, the maximum number of function evaluations allowed, the function tolerance (used as a stopping criterion), and the name of the output file.

OptQNewton objfcn(&nlp); objfcn.setSearchStrategy(TrustRegion); objfcn.setMaxFeval(200); objfcn.setFcnTol(1.e-4); // The "0" in the second argument says to create a new file. A "1" // would signify appending to an existing file. if (!objfcn.setOutputFile("example1.out", 0)) cerr << "main: output file open failed" << endl;

Now call the algorithm's optimize method to solve the problem.

objfcn.optimize();

Finally, print out some summary information and clean up before exiting. The summary information is handy, but not necessary. The cleanup flushes the I/O buffers.

objfcn.printStatus("Solution from quasi-newton"); objfcn.cleanup(); }

Now that the main routine is in place, we step through the code required for the initialization and evaluation of the function.

User-Defined Functions

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 the OPT++ header file, NLP, for some definitions. Next, we define the scope of the methods using namespace. The first statment introduces the data member ColumnVector from the namespace NEWMAT. The second statement allows us to refer to all the methods in the OPTPP namespace. These two statements are crucial for a sucessful compilation.

#include "NLP.h" using NEWMAT::ColumnVector; using namespace::OPTPP;

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.

void init_rosen (int ndim, ColumnVector& x) { if (ndim != 2) exit (1);

The initialization is also an ideal place to set the initial values of the optimization parameters, x. This can be hard coded, as done here, or it can be done in some other manner (e.g., reading them in from a file, the code for which should appear here).

// ColumnVectors are indexed from 1, and they use parentheses around // the index. x(1) = -1.2; x(2) = 1.0; }

The last piece of code is a subroutine that will evaluate the function. In this problem, we are trying to find the minimum value of Rosenbrock's function, so it is necessary to write the code that computes the value of that function given some set of optimization parameters. Mathematically, Rosenbrock's function is:

$f(x) = 100(x_2 - x_{1}^2)^2 + (1 - x_1)^2$

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

First, some error checking and manipulation of the optimization parameters, x, are done.

void rosen(int ndim, const ColumnVector& x, double& fx, int& result) { double f1, f2, x1, x2; if (ndim != 2) exit (1); x1 = x(1); x2 = x(2); f1 = (x2 - x1 * x1); f2 = 1. - x1;

Then the function value, fx, is computed, and the variable, result, is set to indicate that a function evaluation was performed.

fx = 100.* f1*f1 + f2*f2; result = NLPFunction; }

On a more general note, this subroutine could serve as a wrapper to a C or Fortran subroutine. Similarly, it could make a system call to a completely independent executable. As long as the values of fx and result are set when all is said and done, it does not matter how the function value is computed.

Now that we have all of the code necessary to set up and solve Rosenbrock's function, give it a try!

Building and Running the Example

Building your executable should be fairly straightforward. Below is the recommended set of steps to follow.

Determine which defines you need. If the C++ compiler you are using supports the ANSI standard style of C header files, you will need
```
		-DHAVE_STD
           
```
If the C++ compiler you are using supports namespaces, you will need
```
		-DHAVE_NAMESPACES
           
```
If you are using the parallel version of OPT++, you will need
```
		-DWITH_MPI
           
```
Determine the location of the header files. If you did a "make install", they will be located in the "include" subdirectory of the directory in which OPT++ is installed. If that directory is not one your compiler normally checks, you will need
```
		-IOPT++_install_directory/include
           
```
If you did not do a "make install", the header files will almost certainly be in a directory not checked by your compiler. Thus, you will need
```
		-IOPT++_top_directory/include -IOPT++_top_directory/newmat11
           
```
Determine the location of the libraries. If you did a "make install", they will be located in the "lib" subdirectory of the directory in which OPT++ is installed. If that directory is not one your compiler normally checks, you will need
```
		-LOPT++_install_directory/lib
           
```
If you did not do a "make install", the libraries will almost certainly be in a directory not checked by your compiler. Thus, you will need
```
		-LOPT++_top_directory/lib/.libs
           
```
If you configured OPT++ for the default behavior of using the BLAS and/or you configure OPT++ to use NPSOL, you will need the appropriate Fortran libraries for linking. The easiest way to get these is to look in the Makefile for the value of FLIBS.
If all is right in the world, the following format for your compilation command should work:
```
		$CXX <defines> <includes> example1.C tstfcn.C <lib \
		directory> -lopt -lnewmat -l$BLAS_LIB $FLIBS 
           
```
$CXX is the C++ compiler you are using. <defines> and <includes> are the flags determined in steps 1-2. example1.C is your main routine, and tstfcn.C contains your function evaluations. (Note: If you have put them both in one file, you need only list that single file here.) <lib_directory was determined in step 3. -lopt and -lnewmat are the two OPT++ libraries. $BLAS_LIB is the BLAS library you are using, and $FLIBS is the list of Fortran libraries determined in step 4.

You should now be able to run the executable (type "./example1"). You can compare the results, found in example1.out, to our results. There may be slight differences due to operating system, compiler, etc., but the results should very nearly match.

Next Example: Example 2: Nonlinear Interior-Point Method With General Constraints | Back to Setting up and Solving an Optimization Problem

Last revised April 27, 2007

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, last revised August 30, 2006.