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220 Ya. D. Sergeyev and D. E. KvasovFigure 1.Estimation of the lower bound of f(x) over an interval D i = [a i, b i].The goal of this work is to present a new algorithm oriented to solving difficult multidimensionalmultiextremal “black-box” problems (1)–(3). In the algorithm, partition of theadmissible region into a set of smaller hyperintervals is performed by a new efficient diagonalpartition strategy (see [8, 11]). This strategy allows one to accelerate significantly the searchprocedure in terms of function evaluations with respect to the traditional diagonal partitionstrategies (see, e.g., [9]), especially in high dimensions. A new technique balancing usage ofthe local and global information has been also incorporated in the new method.Following the diagonal approach (see [9]), the objective function is evaluated at two verticesof each hyperinterval (see Fig. 1). The procedure of estimating the Lipschitz constant evolvesthe ideas of the center-sampling method DIRECT from [6] to the case of diagonal algorithms.Particularly, in order to calculate the lower bounds of f(x) over hyperintervals, possible estimatesof the Lipschitz constant varying from zero to infinity are considered at each iterationof the proposed diagonal algorithm. An auxiliary function is considered over the main diagonalof each hyperinterval D i = [a i , b i ]. This function is constructed as maximum of two linearfunctions P 1 (x, ˜L) and P 2 (x, ˜L) passing with the slopes ± ˜L through the vertices a i and b i (seeFig. 1). An estimate of the lower bound of f(x) over the main diagonal of D i is calculated atthe intersection of these two lines and is given by the following formula (see [7, 9])R i = R i (˜L) = 1 2 (f(a i) + f(b i ) − ˜L‖b i − a i ‖), 0 < L ≤ ˜L < ∞. (4)For any ˜L ≥ L, the value R i is the lower bound of f(x) over the diagonal [a i , b i ], but notover the whole hyperinterval D i . It is shown, that inequality˜L ≥ √ 2Lguarantees that the value R i from (4) is a valid estimate of the lower bound of f(x) over thewhole hyperinterval D i , i.e.,R i (˜L) ≤ f(x), x ∈ D i .Numerical results performed to compare the new algorithm with two algorithms belongingto the same class of methods for solving problem (1)–(3) – the original DIRECT algorithmfrom [6] and its locally-biased modification DIRECTl from [1, 2] – are presented. In the following,we briefly describe results of wide numerical experiments performed.