Evolution and Optimum Seeking

88 Random Strategies
Random directions that are not oriented with respect to the structure of the objective
function and the allowed region also imply a higher cost because they do not take optimal
single steps. They can, however, be applied in every case.
Many deterministic optimization methods, especially those which are guided by the
gradient of the objective function, haveconvergence di culties at points where the partial
derivatives are discontinuous. On the contour diagram of a two parameter objective
function, of which the maximum is sought, such positions correspond to sharp ridges
leading to the summit (e.g., Zwart, 1970). Anarrowvalley{the geometric picture in
the case of minimization{leads to the same problem if the nite step lengths are greater
than its width. Then all attempts fail to make improvements in the coordinate directions
or, from trial steps in these directions, fail to predict a locally best direction in which
to continue (gradient direction). The same phenomenon can also occur when the partial
derivatives are speci ed analytically, because of the rounding errors involved in computing
with a nite number of signi cant gures. To avoid premature termination of a search in
such cases, Norkin (1961) has suggested the following procedure. When the optimization
according to the conventional scheme has ended, a step is taken away from the supposed
optimum in an arbitrary coordinate direction. The extremum is sought again, excluding
this one variable, and the search is only nally ended when deviations in all directions
have led back to the same point. This rule should also prevent stagnation at saddle points.
Even the simplex method of linear programming makes random decisions if the search
for the extremum threatens to be endless because the problem is degenerate. Then following
Dantzig's suggestion (1966) the iteration scheme should be interrupted in favor of a
random exchange step. A problem is only degenerate, however, because the general rules
do not cover the special case (see also Chap. 6, Sect. 6.2). A further example of resorting
to chance when a dead end has been reached is Brent's modi cation of the strategy with
conjugate directions (Brent, 1973). Powell's algorithm (Powell, 1964) when applied to
problems in many dimensions tends to generate linearly dependent directions and then
to proceed within a subspace of IR n . For this reason Brent now and then interrupts the
line searches with steps in randomly chosen directions (see also Chap. 3, Sect. 3.2.2.1).
One very frequently comes across proposals to let chance take control when the problem
is to nd global minimaofmultimodal objective functions. Such problems frequently
crop up in process design (Motskus, 1967 Mockus, 1971) but can also be the result of recasting
discrete problems into continuous form (Katkovnik and Shimelevich, 1972). Practically
all sequential search procedures can only lead to a local optimum{as a rule, the one
nearest to the starting point. There are a few proposals for ensuring global convergence of
sequential optimization methods (e.g., Motskus and Feldbaum, 1963 Chichinadze, 1967,
1969 Goldstein and Price, 1971 Ueing, 1971, 1972 Branin and Hoo, 1972 McCormick,
1972 Sutti, Trabattoni, and Brughiera, 1972 Treccani, Trabattoni, and Szego, 1972
Brent, 1973 Hesse, 1973 Opacic, 1973 Ritter and Tui as mentioned by Zwart, 1973).
They are often in the form of additional, heuristic rules. Gran (1973), for example, considers
gradient methods that are supposed to achieve globalconvergence by the addition
of a random process to the deterministic changes. Hill (1964 see also Hill and Gibson,
1965) suggests subdividing the interval to be explored and gathering su cient information
in each section to carry out a cubic interpolation. The best of the results for the