By Even S.

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1007/978-1-4614-9093-7__2, © Remigijus Paulaviˇcius, Julius Žilinskas 2014 21 22 2 Lipschitz Optimization with Different Bounds over Simplices There are at least four approaches to obtain an estimate of the Lipschitz constant L [125]: 1. It can be given a priori [3, 59, 89, 112]. This case is very important from the theoretical viewpoint but is not frequently encountered in practice. 2. Its adaptive global estimate over the whole domain can be used [59,71,109,134]. 3. Local Lipschitz constants can be estimated adaptively [70, 79, 116, 117, 124, 129, 134]; 4.

1. 3x2 C2/ and the feasible region D D Œ0; 1 Œ0; 1 is covered by two right-angled isosceles triangles I1 D Œv1 ; v2 ; v3 and I2 D Œv1 ; v3 ; v4 (see Fig. 5). Let us consider the first simplex I1 D Œv1 ; v2 ; v3 . 5 Tight Lipschitz Bound over Simplices with the 1-norm 39 Fig. 0:4416; 0:1886/: ˆ 0 Ä x1 Ä 0:5 ˆ : 0 Ä x2 Ä 1:0 Here we give four most significant digits. I1 / (see Fig. 32) 40 2 Lipschitz Optimization with Different Bounds over Simplices Fig. I1 /. 2. The subdivision of simplices I1 and I2 produces four simplices I3 ; : : : ; I6 (see Fig.

However, in multidimensional case the same points may be encountered when subdividing different simplices. One possibility is to maintain a list of already evaluated points and before evaluating a point to check it has already been evaluated. Vertex verification reduces the number of function evaluations essentially, and also increases computational time; therefore it is more suitable in the case of computationally expensive functions. 5 too. 7 Parallel Branch-and-Bound with Simplicial Partitions 49 for Lipschitz optimization, described by Hansen and Jaumard [49].