Evolution and Optimum Seeking

Numerical Comparison of Strategies 179
S. A. Lill (1971) Note on Algorithm 46
Z. Kovacs (1971) Note on Algorithm 46
Some of the parameters a ecting the accuracy were altered, either because the small
values de ned by the author could not be realized on the available computer or
because the closest possible approach to the objective could not have been achieved
with them.
Simplex method of Nelder and Mead:
R. O'Neill (1971) Algorithm AS 47, function minimization using a
simplex procedure
A complete program for the Rosenbrock strategy:
M. Machura, A. Mulawa (1973) Algorithm 450, Rosenbrock function minimization
This was not applied because it could only treat the unconstrained case.
The same applies to the code for the complex method of M. J. Box:
J. A. Richardson, J. L. Kuester
(1973)
Algorithm 454, the complex method for constrained
optimization
The part of the strategy that, when the starting point is not feasible seeks a basis
in the feasible region, is not considered here.
Whenever the procedures named were published in ALGOL they have been translated
into FORTRAN. All the other optimization strategies not mentioned here have also been
programmed in FORTRAN, with close reference to the original publications. If one wanted
to repeat the test series today, amuch larger number of codes could be made use of from
the book of More and Wright (1993).
6.3.3 Results of the Tests
6.3.3.1 First Test: Convergence Rates for a Quadratic Objective Function
In the rst part of the numerical strategy comparison the theoretical predictions of convergence
rates and Q-properties will be tested, or, where these are not available, experimental
data will be supplied instead. For this purpose two quadratic objective functions are used
(Appendix A, Sect. A.1). In the rst (Problem 1.1) the matrix of coe cients is diagonal
with unit diagonal elements, i.e., a scalar matrix. This simplest of all quadratic problems
is characterized by concentric contour lines or surfaces that can be represented or
imagined as circles in the two parameter case, spheres in the three parameter case, and
surfaces of hyperspheres in the general case. The same pattern of contours but with arbitrary
monotonic variation in the objective function occurs in the sphere model for which
the average rates of progress of the evolution strategies could be determined theoretically
(Rechenberg, 1973 and Chap. 5 of this book).
The second objective function (Problem 1.2) has a matrix of coe cients with all nonzero
elements. It represents a full quadratic problem (except for the missing linear term)