6 C++ Version (GMC)

1 Purpose of GMC

The Global Minimizer's C++ version GMC is for continuous global optimization [Mockus et al., 1997]. Users define their optimization problems describing

parameters that can be changed (defined as variables ),
function of those variables (defined as objective function ),
constraints restricting a possible change of variables (defined as inequalities $gi(x) \le 0,\ i=1,...,p$ , or equalities $gi(x)=0,\ i=p+1,...,m$ ).

Formally

$\begin{eqnarray} \ min f(x), \\ a \le x \le b \\ gi(x) \le 0,i=1,...,p, \\ gi(x)=0,i=p+1,...,m. \end{eqnarray}$

The objective function

may be deterministic or (for some methods) with "noise." As usual, the optimization region is defined by rectangular constraints (4.2). For some methods, Nlp and Flexi, this region is defined by linear and non-linear constraints (6.3) and (6.4).

2 User's Reference

1 Requirements

C++ compiler, for example, GNU g++ compiler or any ANSI C++ compliant compiler,
X-Window system,
UNIX (Linux, Solaris, AIX),
experience in writing simple C, or C++ functions.

2 Installation

Differences between various UNIX versions are reflected in the file 'Makefile'. An example for individual Linux users is in Figure 6.1. An AIX version for individual users is in [Mockus et al., 1997]. A classroom (collective user) version for Solaris is there, too.

**Figure 6.1:** Example of 'Makefile' for Linux.
$\begin{figure}\begin{codebox}{4.8in} \begin{verbatim}...$

Using some other UNIX version, the Makefile has to be changed accordingly. Installing GMC one should

copy the archive file from the web-site (see section 4) by the unix command
$\begin{codesamp} > mcopy a: gmc.tgz \end{codesamp}$
extract GM files from the archive 'gmc.tgz' by the command
$\begin{codesamp} > tar -zxf gmc.tgz \end{codesamp}$

3 Initialization

Define the objective as some function . The function has two parameters: an array of values of each variable and the number of variables. The function returns the value of objective, given the values of variables.
Define "non-interactive" linear or non-linear constraints as a function named It has four parameters: an array of values of variables, the number of variables, an array of values of constraints, and the number of constraints. Values of constraints are returned by the function .
A user can also define rectangular constraints interactively, during the optimization
Describe objectives and constraints in a file named (see Figure 6.2)
- the objective function is implemented by the routine
  $\begin{codesamp} double fi (const double *xx, int dim) \end{codesamp}$
- the constraints are defined by the routine
  $\begin{codesamp} void constr (const double *x, int n, double *g, int ng) \end{codesamp}$
Start optimize the problem described in the file .
set X-window
compile
$\begin{codesamp} > rm fi.o > make \end{codesamp}$
run
$\begin{codesamp} > ./test \end{codesamp}$

**Figure 6.2:** Example of the file 'fi.C.'
$\begin{figure}\begin{codebox}{4.8in} \begin{verbatim}/* File: fi.C */ ...$

The opening menu of GMC is in Figure 6.3

**Figure 6.3:** An example of GMC opening menu and parameter box.
$\begin{figure}\centerline{ \epsfig{file=gmcparams.eps,width=12.0cm} }\end{figure}$

The Main Menu:
Global, Local, Operations, Quit, Parameters, Results, Output.
Global Optimization Menu:
Bayes1, Mig1, Unt, Exkor, Glopt, Lpmin.
Local Optimization Menu:
Nlp, Flexi, Lbayes.
By default the initial point for local optimization is the result of previous global optimization.
Operations Menu:
Run, Stop, Exit.
Quit Command.
Parameter Box.
Enter the parameters of methods and point O.K.
An example of the parameter box is in Figure 6.3.
Results Box:
Shows the best objective Y and the best variables X(I). If no changes are needed, point O.K.
Output Menu:
- Convergence, Projection, and Numeric Windows.
- Convergence window shows how the objective depends on the iteration number (see the Figure 6.4.
  
  Figure 6.4: Convergence window.
  $\begin{figure}\centerline{ \epsfig{file=gmcconv.eps,width=12.0cm} }\end{figure}$
- Projection windows show how the objective depends on the different variables. One defines the projection by entering the number of variable. Figures 6.5 and 6.6 show two projections.
  
  Figure 6.5: Projection 1 (shows how depends on the first variable).
  $\begin{figure}\centerline{ \epsfig{file=gmcproj1.eps,width=12.0cm} }\end{figure}$
  
  Figure 6.6: Projection 2 (shows how depends on the second variable).
  $\begin{figure}\centerline{ \epsfig{file=gmcproj2.eps,width=12.0cm} }\end{figure}$
- Numeric window shows the current values of objective and variables (see Figure 6.7).
  
  Figure 6.7: Current numeric results.
  $\begin{figure}\centerline{ \epsfig{file=gmcnum.eps,width=12.0cm} }\end{figure}$
Using Menu:
- To see "invisible" components of a vector, point to 'down' or 'up' arrows. The vector scrolls, if there are invisible components.
- To edit a variable, touch BACKSPACE, then edit the variable and point to O.K.
- Enter the Parameter Box immediately after selecting a method.

**Figure 6.4:** Convergence window.
$\begin{figure}\centerline{ \epsfig{file=gmcconv.eps,width=12.0cm} }\end{figure}$

**Figure 6.5:** Projection 1 (shows how depends on the first variable).
$\begin{figure}\centerline{ \epsfig{file=gmcproj1.eps,width=12.0cm} }\end{figure}$

**Figure 6.6:** Projection 2 (shows how depends on the second variable).
$\begin{figure}\centerline{ \epsfig{file=gmcproj2.eps,width=12.0cm} }\end{figure}$

**Figure 6.7:** Current numeric results.
$\begin{figure}\centerline{ \epsfig{file=gmcnum.eps,width=12.0cm} }\end{figure}$

jonas mockus 2004-03-03