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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 ) .