A Stochastic Approach to Global Optimization

Cover A Stochastic Approach to Global Optimization
A Stochastic Approach to Global Optimization
A H G Rinnooy Kan
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Step h. If i " N, then stop. Else, set i := i+1 and go to Step 3.
Note that a sample point x can only be linked to a point with smaller function value that is within distance x^ (provided that a local search has not been applied unnecessarily, and the starting point is added to the resulting minimum for that reason only). Moreover (under the same provision), if X is assigned to a cluster which ]. S initiated by a local minimum x*, then there exists an rfj-descent sequence connecting x and x* .
...The sample point x can be assigned to several clusters, if there exist rj^-descent sequences from X to each of the corresponding local minima.
Unfortunately, even if v^ tends to 0, then the fact that there e;. Ists an rj^— descent sequence from x to a local minimum x*, does not necessarily imply that X € R ^. If P is applied to x, then it is still possible ttiat it will follow another descent path, and find another (possibly undetected) local minimum. However, as we will see later, this cannot happen if x is located in the interior of a component which includes some local minimum as its only r vtiionary point.


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