By Thomas Weise
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Additional info for Global Optimization Algorithms - Theory and Application
Global optimization algorithms are optimization algorithms that employs measures that prevent convergence to local optima and increase the probability of ﬁnding a global optimum. 38 would hold. In reality, it can be considered questionable whether such an algorithm can actually be built. 16: An example optimization problem. Let us now give a simple example for problem landscapes and how they are inﬂuenced by the optimization algorithm applied to them. 16 illustrates one objective function, 50 1 Introduction deﬁned over a ﬁnite subset X of the two-dimensional real plane, which we are going to optimize.
30 on page 481. 2 What is an optimum? 17 (Prevalence). An element x1 prevails over an element x2 (x1 ≻ x2 ) if the application-dependent prevalence comparator function cmpF (x1 , x2 ) ∈ R returns a value less than 0. 17) It is easy to see that we can deﬁne Pareto domination relations and Method of Inequalities-based comparisons, as well as the weighted sum combination of objective values based on this notation. 3 on page 111), it covers many of the most sophisticated multi-objective techniques that are proposed, for instance, in [717, 1134, 2012].
It is denoted by ∇f or grad(f ). Optimization algorithms depend on some form of gradient in objective or ﬁtness space in order to ﬁnd good individuals. In most cases, the problem space X is not a vector space over the real numbers R, so we cannot directly diﬀerentiate the objective functions with Nabla operator41 ∇F . Generally, samples of the search space are used to approximate the gradient. If we compare to elements x1 and x2 of problem space and ﬁnd x1 ≻ x2 , we can assume that there is some sort of gradient facing downwards from x2 to x1 .
Global Optimization Algorithms - Theory and Application by Thomas Weise