By Altannar Chinchuluun, Panos M. Pardalos (auth.), Aimo Törn, Julius Žilinskas (eds.)
The study of Antanas Žilinskas has concerned about constructing versions for international optimization, imposing and investigating the corresponding algorithms, and utilising these algorithms to functional difficulties. This quantity, devoted to Professor Žilinskas at the party of his sixtieth birthday, comprises new survey papers within which top researchers from the sphere current a number of types and algorithms for fixing worldwide optimization difficulties.
Audience
This booklet is meant for scientists and graduate scholars in desktop technology and utilized arithmetic who're attracted to optimization algorithms and numerical analysis.
Read Online or Download Models and Algorithms for Global Optimization: Essays Dedicated to Antanas Žilinskas on the Occasion of His 60th Birthday PDF
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Extra info for Models and Algorithms for Global Optimization: Essays Dedicated to Antanas Žilinskas on the Occasion of His 60th Birthday
Sample text
XiL ] was proven to be the only shift- and scale-invariant technique and thus, the only one optimal under an arbitrary shift-invariant and scale-invariant optimality criterion [KSM97] (see also [Rum98]); by using shift-invariance, we explain why the probability proportional to exp(-y . , [KESOl]). 1 Proof of Proposition 1 For cu = 1, the condition (3) takes the form where we denoted C(s) the variables: sf6(1, s). To simplify this equation, let us separate 36 Christodoulos A. Floudas and Vladik Kreinovich let us move all terms containing xL to the left-hand side - by dividing both sides by (g(x + s) - g(xL + s)), and let us move all terms containing xu to the right-hand side - by dividing both sides by (h(xU)- h(x)).
By definition, the value A depends on x, s, and xL. , A(x, s) = A(s). Thus, (13) takes the form ef l/A(s). where we denoted a(s) The function g(x) is smooth, hence the function a(s) is smooth too - as the ratio of two smooth functions. Differentiating both sides of (16) with respect to s and taking s = 0, we get def where a = at(0). , let us move all the term depending on x to the right-hand side and all the terms depending on xL to the left-hand side. As a result, we arrive at the following: Optimal Techniques for Solving Global Optimization Problems 37 The right-hand side is a function of x only, but since it is equal to the lefthand side - which does not depend on x at all - it is simply a constant.
So, if we use scale invariance to select a convex underestimator, we end up with a new parameter y which only attains integervalued values and is, thus, less flexible than the continuous-valued parameters coming from scale-invariance. 10 Auxiliary Shift-Invariance Results Instead of an expression (2), we can consider an even more general expression Whet can we conclude from shift-invariance in this more general case? Definition 3. A pair of smooth functions (a(%,xL),b(x, xu)) from real numbers to real numbers is shift-invariant i f f o r every s and a , there exists G ( a ,s ) such that for every xL, x, and xu, we have G(a, s ) .