5 Turbo C Version (TCGM)

1 General Discussion

A conceptual description of methods is given in [Mockus et al., 1997].
Descriptions of real life examples of global optimization are in [Mockus et al., 1997] and in the next part of this book.

1 Purpose

The Turbo C version of the Global Minimizer TCGM is to solve global optimization problems similar to those of the Fortran version GMF. The optimization problem is defined in terms of variables describing the parameters that we can change, an objective function of those variables and some constraints restricting the possible change of variables. Formally, that is a nonlinear programming problem

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

Here $x$ is a vector of $n$ variables. f(x) is an objective function. $a<=x<=b$ defines rectangular constraints. $gi(x) \le 0,\ i=1,...,p$ defines inequality constraints. $gi(x)=0,\ i=p+1,...,m$ defines equality constraints. The objective function is deterministic or stochastic (for some methods). Rectangular constraints are defined for all the methods. The linear and non-linear constraints are defined just for two local methods: the Simplex by Nelder and Mead $Flexi$ [Himmelblau, 1972], and the non-linear programming by Schittkowski $Nlp$ [Schittkowski, 1985].

2 User's Reference

1 Requirements

• Turbo C version 2.0 or above.
• An experience of writing simple C functions.

2 Installation

• Copy the file $tcgm.arj$ from a web-site (see Section 4). Extract the GM files from the $tcgm.arj$ archive
  
  

3 Initialization

• Run the batch file GM.BAT using the command GM, and enter the Turbo C editor.
• Define the objective function as some C function with the name $f$. This function has two parameters:
  • double array of values of each variable,
  • the number of variables.
• Define the general linear or non-linear constraints as a C function named constr. It has four parameters:
  • double array of values of variables,
  • number of variables,
  • double array of values of constraints,
  • number of constraints.
• Compile file FI.C using directive Alt+C and correct possible errors.
• Link file FI.C to GLOBAL MINIMUM system using directive Alt+L.
• Exit Turbo C using directive Alt+X.

Now the GLOBAL MINIMUM is ready to minimize the problem that is defined in the file FI.C. Descriptions of objective functions and constraints have to be in the same file. Running GM.BAT, the file FI.C is displayed on the screen by Turbo C editor. Rectangular constraints are defined using the screen. Two algorithms, Nlp and Flexi, are exceptions. Here the constraints are defined by C function constr in the file FI.C. The four basic steps of initialization are

• run GM.BAT,
• edit file FI.C,
• compile FI.C,
• link file FI.C to system GLOBAL MINIMUM.

4 Minimization

The system will ask the following questions.

• Question: Number of variables.
  Action: Enter the number of variables of the problem.
  
• Question: Number of constraints.
  Action: Enter the number of linear and non-linear constraints of the problem (not counting the number of rectangular constraints that are defined interactively).

Now one see the GLOBAL MINIMUM copyright notice. GLOBAL MINIMUM is implemented as menu, dialog box and windows system. Each object can be either active or passive. There can be only one currently active object. A user can work only with active objects. There is a rectangular caption at the top of each object (menu, dialog box, or window). If this caption is bright, the object is active.

5 Menu System

• Main Menu
  • press any key and enter the MAIN MENU,
  • select the type of optimization or exit the menu system selecting EXIT.
• Global Optimization Menu
  • select the GLOBAL item from the main menu,
  • select any of six available global optimization methods: LPMIN, MIG1, BAYES1, EXTR, UNT, GLOPT.
• Local Optimization Menu
  • select LOCAL item from the main menu,
  • select any of three available local optimization methods: NLP, FLEXI, or LBAYES. The initial point for local optimization is the result of previous global optimization, by default.
• Parameter Box
  • display of the PARAMETER DIALOG BOX follows the selection of method,
  • enter corresponding parameters of methods and press RETURN. Recommendations how to choose parameters are given in the description of methods [Mockus et al., 1997].
• Confirmation Menu
  • CONFIRMATION MENU appears following the selection of parameters,
  • RUN means run the method,
  • EXIT means return to LOCAL or GLOBAL OPTIMIZATION MENU.
• Operation Menu
  • OPERATION MENU appears following the selection of RUN item,
  • RUN means run the method, in case if it was stopped by selecting STOP item,
  • STOP means temporary stopping of the method and activating the OUTPUT DIALOG BOX,
  • press ENTER to return to the OPERATION MENU and to make the OUTPUT DIALOG BOX inactive,
  • EXIT means exit the method and return to GLOBAL or LOCAL in the OPTIMIZATION MENU,
  • ENTER means return to GLOBAL or LOCAL in the OPTIMIZATION MENU, after the method finishes optimization,
  • RECORD means turn RECORD WINDOW on/off,
  • PROJECTION means turn PROJECTION WINDOW on/off.
• Information Display
  • OUTPUT DIALOG BOX shows the results of each iteration (usually best point and goal function value at this point) when the method runs (see Figure 5.1),

    Figure 5.1: Table of TCGM results.

    
  • RECORD WINDOW shows how the best function value depends on the iteration number (see Figure 5.2),

    Figure 5.2: Graph of TCGM results.

    
  • RECORD WINDOW is on when the method is started,
  • if the graph on RECORD WINDOW is horizontal for a long time it can suggest that the minimum is reached or that the method exhausted itself for this problem and it is better to try another method,
  • if one selects RECORD item in OPERATION MENU while running the method then the RECORD WINDOW becomes invisible, if it was visible, and vice versa,
  • PROJECTION WINDOW shows the projection of the observed (calculated) objective function values, the numbers of projection planes correspond to the numbers of variables,
  • if one selects a PROJECTION item in OPERATION MENU while running the method, then one will see projection menu consisting of two items: ON and OFF, meaning on/off the PROJECTION WINDOW,
  • if one selects ON then the projection dialog box appears and we can enter numbers defining the projection planes,
  • select the number, press ENTER and one see how the observed values of the goal function depend on the corresponding values of the selected variable, the values of all other variables will not be seen.
• RESULT DIALOG BOX shows the results of calculations after the method finishes optimization.

6 Navigation

• On Screen
  • screen is divided into three areas: bottom, middle and top, the functionality of these areas is different,
  • bottom area is for general information, such as description of variables, displaying error messages, displaying the state of an active object,
  • middle area is only for displaying and editing variables,
  • the top is a working area; menus, dialog boxes and windows are displayed here.
• On Menu
  • if some menu item is invisible, one opens it using corresponding cursor keys: UP, DOWN, LEFT, RIGHT,
  • the contents of the menu will scroll until the last available item will be reached.
• Dialog Box
  • on the DIALOG BOX one can see such variables as the parameters of method, coordinates of the best and current point, best and current values of goal function and so on, to view a variable one selects the variables name,
  • if the variable is a scalar, one see the corresponding value, next to the variable's name, the description of this variable is displayed at the bottom area of a screen, the exact value of the variable is displayed at the middle area,
  • if the variable is a vector then, after selecting the variables name, one can select the corresponding component of the vector using keys LEFT and RIGHT, the value of the selected component will be in the middle area, the number of component will be in the bottom area, along with the description of variable,
  • to see invisible components of the vector, use the keys LEFT and RIGHT, the vector scrolls, if there are invisible components,
  • press SPACE, to edit a variable, press ENTER, to exit the editor.

7 Moving

• Press the F3 functional key and the red border will appear, move the border with keys UP, DOWN, RIGHT, LEFT and press ENTER to stop.
• The previous image of the object will be deleted and the object will be placed inside the new border.

8 Sizing

• Press the F4 functional key, move the object and press ENTER. The red border appears.
• Size the border using the keys UP, DOWN, RIGHT, LEFT and press ENTER to stop.
• The previous image of the object is deleted and the object is placed into new borders.

No changes happens, if borders are greater than maximal size of the object, or less than its minimal size.