INTEROPT is a computer program specifically designed to solve difficult optimization problems. It establishes a convenient, easy-to-understand, dialogue with the user, in which the user is prompted with natural language questions, and asked to describe his/her optimization problem. INTEROPT then writes a computer program based on this dialogue, and runs that program to solve the specified problem.
INTEROPT has several capabilities that are unique in optimization theory:
Although completely general-purpose and easy to use, INTEROPT does suffer from some limitations:
INTEROPT was written with the following philosophy: to develop an optimization program which can solve hard optimization problems as effectively and as fast as possible, while requiring as little software development as possible on the part of the user. Following this philosophy, INTEROPT talks to the user, asking questions like
How many variables are there in this problem?
and the user replies with simple responses. A sample dialogue is given below.
Most of the use of INTEROPT is self-explanatory, and the interactive help feature is generally sufficient. However, in this section, we provide an overview of the operation of interopt, and discuss some terminology which might otherwise be confusing.
INTEROPT operates in basically two modes, function minimization and data fitting. Actually, data fitting is simply a user-friendly front-end to the function minimization capability, where the function to be minimized is the means-squared error between the data provided and the specified function.
y = sin(x) / x for * < x < 3*
Then, we simply specify the function when asked to, and specify the bounds of the parameter x, and INTEROPT will find the value of x which minimizes y.
When fitting data, we are to find the values of certain parameters which minimize the mean-squared-error between the data and the function. Before we continue with this discussion, we need to carefully define a couple of terms:
The terms variable and parameter are used in different ways in INTEROPT, and may be confused. When minimizing a function, as in the example above, we are solving for the value of the variable x which minimizes y. However, consider a problem in function fitting (again in one dimension, just to make the explanation clearer). Suppose we are to find the values of p, *, and u which minimize the fit of
y = p exp ( - ( x - u)^2 / sigma^2 )
to some x, y pairs of data which the user has stored in a file. In this context, x is our variable, and p, u, and sigma are the parameters of the fit. Unlike the function minimization case, in the data fitting case, INTEROPT is solving for the parameters, and is given values for the variables.
The variables or the parameters, in both the cases of function minimization and data fitting, may be vectors. In addition, the outputs may be vectors in the case of data fitting. This is discussed more in the next section.
In the MSE fitting case, the data to be fit must be specified in an ascii data file. The outputs are specified first. In the data fitting example given above, the first two lines of the data file might be
1.0 0.2
0.8 0.3 for a Gaussian with mean u = 0.2. In a fitting problem with vector-valued outputs, for example
y1 = p1 exp ( - ( x - u)^2 / sigma ^2 ) y2 = p2 exp ( - ( x - u)^2 / sigma ^2 )the first two lines of the data file might be
1.0 1.0 0.2 0.9 0.8 0.3In this example, the data correspondence is
y1 y2 x
A function is specified in the syntax of the C programming language.
When asked to type in the function to be minimized, the user should type an equation of the form. y = f ( x ). For example:
y = a*a + b * exp ( 3 * x * x )
When asked to type in the function to be fit, the user should type the equation(s) in the same form, likewise using the syntax of C, but in this case, the user must specify which of the possible outputs is computed. For example,
y[0] = a*a + b1 * exp ( 9 * x * x ); \ y[1] = a*a + b2 * exp ( 3 * x * x )
The use of the array structure for y is required here, since INTEROPT can find the set of parameters which simultaneously fit several outputs. Note that no semicolon is required after the last line.
When specifying a function to INTEROPT, lines may be continued by terminating the line with a \ (backslash).
NOTE: Some shells (csh in particular) treat backslash as a special symbol. In such a case, the backslash must be escaped {type two backslashes}
The expressions entered as functions are arbitrarily complex C expressions, and may contain IF statements and local variables. In this case, enclose the entire expression in curly brackets. For example:
{\
double xtemp;\
xtemp = x - z;\
if (xtemp < 0.0) y[0] = 3.14159;\
else y[0] = ln(xtemp);\
}
Note that no semicolon is required after the last line.
INTEROPT cannot deal with singularities. In the case of positive
singularities, (for example, y = sin(x) / x ),
INTEROPT CAN fail
with the famous Floating point exception error message. It is
necessary for the user to be sure that no such singularities exist
within the range of possible values. The example of sin(x)/x is
particularly bad, since the singluarity actually occurs at the minimum.
Negative singularities cause a different problem: INTEROPT tries very hard to find the minimum of the function. A negative singularity is therefore extremely attractive, and INTEROPT will end up in the singularity every time.
The other condition which can cause problems is the case of transcendental functions such as exp or ln, in which the arguments can only take on particular ranges of values. That is, exponents with arguments larger than 50 or smaller than -50 are not permitted, nor can one have logarithms with negative arguments. These two special cases (ln and exp) are explicitly trapped and handled by INTEROPT. The expression entered by the user is scanned before compilation and converted into another function call which will not allow illegal argument values. If such arguments are passed to the function, an error message is given, a reasonable value is returned, and the program continues to run.
NOTE: Deep within the guts of INTEROPT, the exp function is called, even though it may not be called explicitly by the user. Therefore, the user will occasionally see the argument out of range to exp function error message occur. Just ignore such warnings.
In this section, we provide some description of the files and directory structures used by Interopt:
INTEROPT is set up to run very effectively on UNIX and UNIX look-alike systems. As discussed earlier, speed of operation was a primary consideration, and therefore, the most effective means of software generation was used. INTEROPT actually write programs, and then, via operating system calls, compiles and executes those programs. To install INTEROPT on a UNIX system, simply copy the existing files in the directory structure specified, using ftp, rcp, or whatever other file transfer program you wish, or reading from the distribution TAR tape.
Disk space requirements for INTEROPT are minimal. A fully configured system generally requires around 500 Kbytes.
Within the home directory for INTEROPT (which we refer to henceforth as HOME/), several subdirectories are required.
when you receive your distribution. If it does not exist, INTEROPT will automatically make it. Afterward, under certain conditions, INTEROPT will try to make this directory even though it already exists. If that happens, you will see the message:
interopt.workdir: directory exists
Just ignore this message.
After copying the INTEROPT directories, build INTEROPT by changing directories to HOME/ and typing
make interopt
If you wish INTEROPT to plot its output (in the case of fitting data to curves) and your computer supports INTEROPT11, type
make INTEROPTinteropt
INTEROPT uses a number of files, some automatically generated, and some modified at run time. In the following, only the files which the user may have questions about are documented.
This file is the source for INTEROPT. It is a rather complex sequence of I/0 functions which result in the generation of a number of temporary files. Do NOT hack around with this program!
src/opt.c
This file is modified by INTEROPT, and, when compiled and linked, be- comes the program which actually performs the minimization.
Two programs are of primary interest to the user: INTEROPT itself as interopt, and OPT. interopt is run by the user, and generates OPT automatically as a result of an interactive dialogue with the user.
Another program, THEORY, is likewise automatically generated by INTEROPT, and used if results are to be plotted on an INTEROPTwindows graphics device.
After optimizations have been run, answers may be found in two places:
Answers.dat is the more useful of these output files. Furthermore, it will contain accumulated results in the case of fitting several sets of data to the same function. parameters.h contains only the last result. Parameters are identified in Answers.dat by the logical name given by the user. Answers.dat is reinitialized at the beginning of each INTEROPT run.
INTEROPT logs all commands which are typed to it into interopt.log. This file may be renamed, edited if needed, and used for input (by redirecting standard in) on later runs. Using this file is the preferred means of operation, since interactive dialogues take so long.
compare.com is a command file used when INTEROPTinteropt is run. This script will generate plots of theoretical vs. actual data, using the theoretical parameters derived by the fit. Since graphics operations are very site-specific.
INTEROPT is designed to make optimal use of computer performance by writing, compiling, and running other programs as sub-processes of INTEROPT itself. For that reason, its structure is highly operating-system dependent. To modify INTEROPT for other operating systems, you must have a unix specialist available.
Most of the operating-system dependent software is located in the file nonstandard.c, and modifying this subroutine to comply with the nature of your operating system will be most of the work, however, there may be other dependencies elsewhere within INTEROPT. (Suggestion, buy a UNIX-based machine. It's cheaper than hiring a software guru).
Once INTEROPT is installed in the system-wide INTEROPT directory, you will need to make a special directory in which to run it, and you will need some of the files from the system directory. The easiest way to do this is to create a directory in your area and call it INT. Then, within this directory, establish symbolic links to the system INTEROPT directory. For example, suppose the system INTEROPT directory is /usr/local/interopt. From your home directory, type
mkdir myinteropt cd myinteropt ln -s /usr/local/interopt/src src ln -s /usr/local/interopt/include include ln -s /usr/local/interopt/libinteropt.a libinteropt.a mkdir demos alias interopt /usr/local/interopt/interopt
In this structure, you would put your data files in your demos directory, and would run INTEROPT by typing
interopt
or , if you are redirecting standard input,
interopt < demos/commandfile
Of course, you don't have to use a demos subdirectory, you can put your data files anywhere you wish.
INTEROPT has only 3 control parameters. They are specified on the command line as follows:
Example:
interopt -d .001
or, if redirecting standard input
interopt -d .001 < commandfil
INTEROPT is a computer program specifically designed to solve difficult optimization problems. It establishes a convenient, easy-to-understand, dialogue with the user, in which the user is prompted with natural language questions, and asked to describe his/her optimization problem. INTEROPT then writes a computer program based on this dialogue, and runs that program to solve the specified problem.
INTEROPT has several capabilities that are unique in optimization theory:
Although completely general-purpose and easy to use, INTEROPT does suffer from some limitations:
INTEROPT was written with the following philosophy: to develop an optimization program which can solve hard optimization problems as effectively and as fast as possible, while requiring as little software development as possible on the part of the user. Following this philosophy, INTEROPT talks to the user, asking questions like
How many variables are there in this problem?
and the user replies with simple responses. A sample dialogue is given below.
Most of the use of INTEROPT is self-explanatory, and the interactive help feature is generally sufficient. However, in this section, we provide an overview of the operation of interopt, and discuss some terminology which might otherwise be confusing.
INTEROPT operates in basically two modes, function minimization and data fitting. Actually, data fitting is simply a user-friendly front-end to the function minimization capability, where the function to be minimized is the means-squared error between the data provided and the specified function.
y = sin(x) / x for * < x < 3*
Then, we simply specify the function when asked to, and specify the bounds of the parameter x, and INTEROPT will find the value of x which minimizes y.
When fitting data, we are to find the values of certain parameters which minimize the mean-squared-error between the data and the function. Before we continue with this discussion, we need to carefully define a couple of terms:
The terms variable and parameter are used in different ways in INTEROPT, and may be confused. When minimizing a function, as in the example above, we are solving for the value of the variable x which minimizes y. However, consider a problem in function fitting (again in one dimension, just to make the explanation clearer). Suppose we are to find the values of p, *, and u which minimize the fit of
y = p exp ( - ( x - u)^2 / sigma^2 )
to some x, y pairs of data which the user has stored in a file. In this context, x is our variable, and p, u, and sigma are the parameters of the fit. Unlike the function minimization case, in the data fitting case, INTEROPT is solving for the parameters, and is given values for the variables.
The variables or the parameters, in both the cases of function minimization and data fitting, may be vectors. In addition, the outputs may be vectors in the case of data fitting. This is discussed more in the next section.
In the MSE fitting case, the data to be fit must be specified in an ascii data file. The outputs are specified first. In the data fitting example given above, the first two lines of the data file might be
Output Variable (Index)value
1.0 0.2
0.8 0.3 for a Gaussian with mean u = 0.2. In a fitting problem with vector-valued outputs, for example
y1 = p1 exp ( - ( x - u)^2 / sigma ^2 ) y2 = p2 exp ( - ( x - u)^2 / sigma ^2 )the first two lines of the data file might be
y1 y2 x 1.0 1.0 0.2 0.9 0.8 0.3In this example, the data correspondence is
y1 y2 x
A function is specified in the syntax of the C programming language.
When asked to type in the function to be minimized, the user should type an equation of the form. y = f ( x ). For example:
y = a*a + b * exp ( 3 * x * x )
When asked to type in the function to be fit, the user should type the equation(s) in the same form, likewise using the syntax of C, but in this case, the user must specify which of the possible outputs is computed. For example,
y[0] = a*a + b1 * exp ( 9 * x * x ); \ y[1] = a*a + b2 * exp ( 3 * x * x )
The use of the array structure for y is required here, since INTEROPT can find the set of parameters which simultaneously fit several outputs. Note that no semicolon is required after the last line.
When specifying a function to INTEROPT, lines may be continued by terminating the line with a \ (backslash).
NOTE: Some shells (csh in particular) treat backslash as a special symbol. In such a case, the backslash must be escaped {type two backslashes}
The expressions entered as functions are arbitrarily complex C expressions, and may contain IF statements and local variables. In this case, enclose the entire expression in curly brackets. For example:
{\
double xtemp;\
xtemp = x - z;\
if (xtemp < 0.0) y[0] = 3.14159;\
else y[0] = ln(xtemp);\
}
Note that no semicolon is required after the last line.
INTEROPT cannot deal with singularities. In the case of positive
singularities, (for example, y = sin(x) / x ),
INTEROPT CAN fail
with the famous Floating point exception error message. It is
necessary for the user to be sure that no such singularities exist
within the range of possible values. The example of sin(x)/x is
particularly bad, since the singluarity actually occurs at the minimum.
Negative singularities cause a different problem: INTEROPT tries very hard to find the minimum of the function. A negative singularity is therefore extremely attractive, and INTEROPT will end up in the singularity every time.
The other condition which can cause problems is the case of transcendental functions such as exp or ln, in which the arguments can only take on particular ranges of values. That is, exponents with arguments larger than 50 or smaller than -50 are not permitted, nor can one have logarithms with negative arguments. These two special cases (ln and exp) are explicitly trapped and handled by INTEROPT. The expression entered by the user is scanned before compilation and converted into another function call which will not allow illegal argument values. If such arguments are passed to the function, an error message is given, a reasonable value is returned, and the program continues to run.
NOTE: Deep within the guts of INTEROPT, the exp function is called, even though it may not be called explicitly by the user. Therefore, the user will occasionally see the argument out of range to exp function error message occur. Just ignore such warnings.
In this section, we provide some description of the files and directory structures used by Interopt:
INTEROPT is set up to run very effectively on UNIINTEROPT and UNIINTEROPT look-alike systems. As discussed earlier, speed of operation was a primary consideration, and therefore, the most effective means of software generation was used. INTEROPT actually write programs, and then, via operating system calls, compiles and executes those programs. To install INTEROPT on a UNIINTEROPT system, simply copy the existing files in the directory structure specified, using ftp, rcp, or whatever other file transfer program you wish, or reading from the distribution TAR tape.
Disk space requirements for INTEROPT are minimal. A fully configured system generally requires around 500 Kbytes.
Within the home directory for INTEROPT (which we refer to henceforth as HOME/), several subdirectories are required.
when you receive your distribution. If it does not exist, INTEROPT will automatically make it. Afterward, under certain conditions, INTEROPT will try to make this directory even though it already exists. If that happens, you will see the message:
interopt.workdir: directory exists
Just ignore this message.
After copying the INTEROPT directories, build INTEROPT by changing directories to HOME/ and typing
make interopt
If you wish INTEROPT to plot its output (in the case of fitting data to curves) and your computer supports INTEROPT11, type
make INTEROPTinteropt
INTEROPT uses a number of files, some automatically generated, and some modified at run time. In the following, only the files which the user may have questions about are documented.
This file is the source for INTEROPT. It is a rather complex sequence of I/0 functions which result in the generation of a number of temporary files. Do NOT hack around with this program!
src/opt.c
This file is modified by INTEROPT, and, when compiled and linked, be- comes the program which actually performs the minimization.
Two programs are of primary interest to the user: INTEROPT itself as interopt, and OPT. interopt is run by the user, and generates OPT automatically as a result of an interactive dialogue with the user.
Another program, THEORY, is likewise automatically generated by INTEROPT, and used if results are to be plotted on an INTEROPTwindows graphics device.
After optimizations have been run, answers may be found in two places:
Answers.dat is the more useful of these output files. Furthermore, it will contain accumulated results in the case of fitting several sets of data to the same function. parameters.h contains only the last result. Parameters are identified in Answers.dat by the logical name given by the user. Answers.dat is reinitialized at the beginning of each INTEROPT run.
INTEROPT logs all commands which are typed to it into interopt.log. This file may be renamed, edited if needed, and used for input (by redirecting standard in) on later runs. Using this file is the preferred means of operation, since interactive dialogues take so long.
compare.com is a command file used when INTEROPTinteropt is run. This script will generate plots of theoretical vs. actual data, using the theoretical parameters derived by the fit. Since graphics operations are very site-specific.
INTEROPT is designed to make optimal use of computer performance by writing, compiling, and running other programs as sub-processes of INTEROPT itself. For that reason, its structure is highly operating-system dependent. To modify INTEROPT for other operating systems, you must have a unix specialist available.
Most of the operating-system dependent software is located in the file nonstandard.c, and modifying this subroutine to comply with the nature of your operating system will be most of the work, however, there may be other dependencies elsewhere within INTEROPT. (Suggestion, buy a UNIINTEROPT-based machine. It's cheaper than hiring a software guru).
Once INTEROPT is installed in the system-wide INTEROPT directory, you will need to make a special directory in which to run it, and you will need some of the files from the system directory. The easiest way to do this is to create a directory in your area and call it INT. Then, within this directory, establish symbolic links to the system INTEROPT directory. For example, suppose the system INTEROPT directory is /usr/local/interopt. From your home directory, type
mkdir myinteropt cd myinteropt ln -s /usr/local/interopt/src src ln -s /usr/local/interopt/include include ln -s /usr/local/interopt/libinteropt.a libinteropt.a mkdir demos alias interopt /usr/local/interopt/interopt
In this structure, you would put your data files in your demos directory, and would run INTEROPT by typing
interopt
or , if you are redirecting standard input,
interopt < demos/commandfile
Of course, you don't have to use a demos subdirectory, you can put your data files anywhere you wish.
INTEROPT has only 3 control parameters. They are specified on the command line as follows:
Example:
interopt -d .001
or, if redirecting standard input
interopt -d .001 < commandfil