Luus–Jaakola

[1] Luus has applied LJ in optimal control,[3] [4] transformer design,[5] metallurgical processes,[6] and chemical engineering.

The LJ heuristic iterates the following steps: Luus notes that ARS (Adaptive Random Search) algorithms proposed to date differ in regard to many aspects.

The worst-case complexity of minimization on the class of unimodal functions grows exponentially in the dimension of the problem, according to the analysis of Yudin and Nemirovsky, however.

The Yudin-Nemirovsky analysis implies that no method can be fast on high-dimensional problems that lack convexity: "The catastrophic growth [in the number of iterations needed to reach an approximate solution of a given accuracy] as [the number of dimensions increases to infinity] shows that it is meaningless to pose the question of constructing universal methods of solving ... problems of any appreciable dimensionality 'generally'.

It is interesting to note that the same [conclusion] holds for ... problems generated by uni-extremal [that is, unimodal] (but not convex) functions.

"[9]When applied to twice continuously differentiable problems, the LJ heuristic's rate of convergence decreases as the number of dimensions increases.

When the current position x is far from the optimum the probability is 1/2 for finding an improvement through uniform random sampling.
As we approach the optimum the probability of finding further improvements through uniform sampling decreases towards zero if the sampling-range d is kept fixed.