Global Optimization with Non-Convex Constraints pp 445-549 | Cite as

# Peano-Type Space-Filling Curves as Means for Multivariate Problems

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## Abstract

A large number of decision problems in the world of applications may be formulated as searching for a , where the , 1 ≤ , then, by introducing the transformation , , and the extra constraint it is possible to keep up the initial presentation (8.1.2) for the domain of search (which is assumed to be the standard one) not altering the relations of Lipschitzian properties in dimensions.

*constrained global optimum*(minimum, for certainty)(8.1.1)

*domain of search*(8.1.2)

*R*^{ N }is the*N*-dimensional Euclidean space and the*objective function φ*(*y*) (henceforth denoted*g*_{ m }_{+1}(*y*)) and the left-hand sides*g*_{ i }(*y*) 1 ≤*i*≤*m*, of the constraints are*Lipschitzian*with respective constants*L*_{ i }, 1 ≤*i*≤*m*+ 1, i.e., for any two points*y*′*y*″ ∈*D*it is true that$$
\left| {{g_i}\left( {y'} \right) - {g_i}\left( {y''} \right)} \right| \leqslant {L_i}\left\| {y' - y''} \right\| = {L_i}{\left\{ {\sum\limits_{j = 1}^N {{{\left( {{{y'}_j} - {{y''}_j}} \right)}^2}} } \right\}^{1/2}},
$$

(8.1.3)

*i*≤*m*+ 1. Note that (8.1.1) is the obvious generalization of the one-dimensional constrained problem (6.1.1) for the case of*N*dimensions, i.e., for the case when the sought decision is described by some*N*-dimensional vector*y** ∈*D*. If the domain of search is defined by the hyperparallelepiped(8.1.4)

(8.1.5)

(8.1.6)

(8.1.7)

## Keywords

Global Minimizer Limit Point Inverse Image Index Scheme Trial Point
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© Springer Science+Business Media Dordrecht 2000