La migración de colombianos a Ecuador, un tema ‘bajo reflectores’

This thesis has brought together reviews of the current position in the fields of interactive computer graphics and non-linear optimization, and, through the development of and experiments with a comprehensive system, has shown that there are advantages to be gained by using inter­ active graphics to solve optimization problems.

The system developed on an IBM 370/135 linked to a Tektronix 4014 graphics display device has been used in several experiments.

The behaviour of existing routines on standard test problems in one and two dimensions has been illustrated graphically, and the system has been used as a means of. interactively controlling conventional method for N-dimensional problems.

Experiments have also been conducted into the direct interactive visual optimization of low order N-dimensional functions and an algorithm has been derived for quadratic forms.

Finally, the system has been used to solve a constrained optimiza­ tion problem arising in molecular vibrational analysis.

In each application described there has been clear evidence that interactive computer graphics provides an extra degree of insight into the behaviour of both existing and new techniques. Graphics is seen to enhance both analysis and presentation and often points the way ahead when other approaches falter. Consequently, i t is believed that graphics has a substantial contribution to make in teaching, in method research and in practical problem solving.

With the equipment used here, the price paid for those advantages

gained has generally been extended c.p.u. time, largely through the

use of an increased number of function evaluations. However, in terms of hardware, a faster central processor and a refreshed-type graphics display unit would undoubtedly enable more complex or larger problems to be tackled. On the software side, a faster con-

f

touring package would be of value, although the fa c ility currently offered for user control of step size does enable economies to be made. Having said this, all of the problems described in this thesis have been capable of being dealt with in standard session times on the graphics system currently installed.

Turning now to further possible developments, there would seem to be ample scope for attempting solution of more problems involving constraints. Time has prevented the investigation of more than one such problem in this work, but the results were encouraging. Con­ strained problems are often more d ifficu lt to solve than unconstrained ones using conventional procedures and yet, with the aid of a capable graphics system, they could pose little extra difficulty using

approaches along the lines of those proposed here.

Similarly, problems known to have several local minima would appear to offer interesting possibilities.

There is interest in graphics applications within optimization at other centres. A very recent paper by L G Birta (46) describes a package which limits its attention to the control of conventional non-linear optimization procedures, in a manner similar to that fa c ility incorporated into the system and described earlier. The Numerical Optimization Centre at Hatfield Polytechnic is also active in this specific area and has investigated one-dimensional procedures using graphics.

The practical problem dealt with in this thesis was particu­ larly suitable for the interactive graphical approach, and i t seems likely that other such problems may exist within the fieldsof pure

and applied science or operational research. It would be worthwhile_!

to conduct investigation into the potential in other fields, for

there are relatively few reported applications of interactive computer graphics to problem solving in general.

It is hoped that the work described in this thesis might inspire further, more ambitious applications of interactive computer graphics to the solution of non-linear optimization problems, whatever their source, and that the ideas expressed here might lead to the further use of interactive graphics as a means of deriving, testing and implementing methods for non-linear optimization.

APPENDIX

• During the period of this work the author has attended the

following conferences/meetings:

(i) "New methods for non-linear constraints in optimization

calculation" by Professor M J D Powell - Sheffield University, May 1977.

( ii) Symposium on "Mathematical methods in computer graphics and design" - University of Leicester, September 1978.

( i ii ) 6IN0 Users Group meeting ~ University of Salford, July 1979. The author has also regularly attended seminars held by the Institute of Mathematics and Its Applications and the Operational Research Society at Sheffield University and Sheffield City Polytechnic.

The guided reading lis t associated with this submission is as follows: O) Fundamentals of optimization;

(a) "Methods of optimization" - G R Walsh (Wiley 1975).

(b) "Non-linear optimization techniques" - M J Box, D Davies,

W H Swann, (Oliver & Boyd 1969).

(c) "Non-linear optimization" - L C W Dixon (EUP 1972).

(2) State of art;

(d) "Towards global optimization", L C W Dixon, G P Szego

(N Holland 1975).

(e) "A preliminary investigation of function optimization by a

combination of methods" -D A Philips (Comp. J. 17, No. 1, 1974).

(f) "An algorithm for drawing the curve f(X,Y) =0" - D C Sutcliffe

(Comp. J. 19, No. 3, 1976).

(3) Computer graphics;

(g) "Principles of interactive computer graphics" - W N Newman and

R F Sproull (McGraw-Hill 1973).

(h) "Computer graphics - whence and hence" - J Potts (CAG Vol. 1, 1975).

(i) "Applications of interactive computing in scientific environment" - T J Martin and A Sykes (On-line Conference 72).

The reading list also includes the journals of Computer Aided Design, Computer and Graphics, Computers and Education, and several others in Operational Research, Mathematics and Computing have been consulted.

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