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128 Eligius M.T. Hendrix1.1 EffectivenessFocusing on effectiveness there are several targets a user of the algorithm may have:1. To discover all global minimum points. This of course can only be realised when thenumber of global minimum points is finite.2. To detect at least one global optimal point.3. To find a solution with a function value as low as possible.4. To produce a uniform covering: This idea as introduced by [3], can be relevant for populationbased algorithms.The first and second targets are typical satisfaction targets; was the search successful or not?What are good measures of success? In the older literature, often convergence was used i.e.x k → x ∗ , where x ∗ is one of the minimum points. Alternatively one observes f(x k ) → f(x ∗ ).In tests and analyses, to make results comparable, one should be explicit in the definitions ofsuccess. We need not only to specify ɛ and/or δ such that‖x k − x ∗ ‖ < ɛ and/or f(x k ) < f(x ∗ ) + δ (2)but we should also specify whether success means that there is an index K such that (2) is truefor all k > K. Alternatively, a record min k f(x k ) may have reached level f(x ∗ ) + δ and this canbe considered a success. Whether the algorithm is effective also depends on the stochastic natureof it. When we are dealing with stochastic algorithms, the effectiveness can be expressedas the probability that a success has been reached. In analysis, this probability can be derivedwhen having sufficient assumptions on the behaviour of the algorithm and in numerical experimentsit can be estimated by looking over repeated runs how many times the algorithmleads to convergence. We will give some examples of such analysis.1.2 EfficiencyGlobally efficiency is defined as the effort the algorithm needs to be successful. A usual indicatorfor algorithms is the (expected) number of function evaluations necessary to reach theoptimum. This indicator depends on many factors such as the shape of the test function andthe termination criteria used. The indicator more or less suggests that the calculation effortof function evaluations dominates the other calculation effort of the algorithm. Several otherindicators appear in literature that can be considered to be derived from the main indicator.In nonlinear programming (e.g. [5] and [2]) the concept of convergence speed is common.It is a limiting concept on the convergence of the series x k . Let x 0 , x 1 , . . . , x k , . . . converge topoint x ∗ . The largest number α for which‖x k+1 − x ∗ ‖limk→∞ ‖x k − x ∗ ‖ α = β < ∞ (3)gives the order of convergence, whereas β is called the convergence factor. In this terminology,among others the following concepts appearlinear convergence means α = 1 and β < 1quadratic convergence means α = 2 and 0 < β < 1superlinear convergence: 1 < α < 2 and β < 1, i.e. β = 0 for α = 1.Mainly in deterministic GO algorithms, information on the past evaluations is stored in thecomputer memory. This requires efficient data handling for looking up necessary information