On global optimization using interval arithmetic
Consider an objective function \(f:\mathbb{R}^ n \to\mathbb{R}\) and a box \(B\) in \(\mathbb{R}^ n\). The author aims at the global minimizers of \(f\) in \(B\). An interval analysis based algorithm for the stated problem has been described by \textit{E. R. Hansen} [J. Optimization Theory Appl. 29, 331-344 (1979; Zbl 0388.65023) and Numer. Math. 34, 247-270 (1980; Zbl 0442.65052)]. Hansen's method requires an interval extension \(F\) of the objective function \(f\). The author investigates the case where such an interval extension \(F\) is not available or too expensive. He approximates the objective function \(f\) by a model function \(m\) which piecewise has the form \(m(x)=\text{sinh}(x^ T Ax+b^ T x+c)\) with \(A\in\mathbb{R}^{n\times n}\), \(b\in\mathbb{R}^ n\), \(c\in\mathbb{R}\). Then he applies Hansen's method to the model function \(m\). The quantities \(A\), \(b\), \(c\) are determined using only function values of \(f\). The main disadvantage of this approach is that the global minimizers of \(m\) are only approximations of the global minimizers of \(f\). Nevertheless, numerical tests with objective functions from magnetic optics yield results of practical relevance.
scientific article; zbMATH DE number 86184