Nonlinear optimization in finite dimensions. Morse theory, Chebyshev approximation, transversality, flows, parametric aspects
This book presents methods and ideas from differential topology for readers interested in finite-dimensional nonlinear optimization theory. After an introduction into some basic concepts and notions from critical point theory in chapter 1, the chapters 2, 3 and 5 are devoted to Morse theory. The classical results on the homotopy of sublevel sets are proved in chapter 2 for functions \(f:\mathbb R^n\rightarrow\mathbb R\) which satisfy the Palais-Smale condition. These results are extended in chapter 3 to functions defined on ''manifolds with generalized boundary''. These are subsets of \(\mathbb R^n\) which locally look like \(\mathbb R^k\times(\mathbb R_+)^p\). In chapter 5 singular homology is introduced and the Morse relations are proved. This presentation of Morse theory is interrupted in chapter 4 for an application of Morse theory to Chebyshev approximation where one wants to approximate a function \(f\in{\mathcal{C}}^2(\mathbb R^n)\) by an element from a family \(\{F(.,y)\in{\mathcal{C}}^2(\mathbb R^N):y\in M\subset\mathbb R^m\}\) of functions parametrized by a compact manifold \(M\) with generalized boundary. The remaining chapters cover a variety of topics. In chapter 6 the stability of nondegenerate critical points under \({\mathcal{C}}^2\)-perturbations of \(f\) is proved. Chapter 7 contains an introduction to transversality in order to treat Morse-Smale gradient flows in chapter 8. Chapter 9 deals with Newton flows, in particular with gradient Newton flows. If \(f:\mathbb R^n\rightarrow\mathbb R\) is smooth and \(F=\text{grad} f:\mathbb R^n\rightarrow\mathbb R^n\), then the (gradient) Newton flow is the flow associated to the differential equation \(\dot{x}=-DF(x)^{-1}F(x)\) defined on the set \(\{x\in\mathbb R^n:\det DF(x)\not=0\}\). The Euler discretization of this equation yields a Newton-Raphson iteration formula for finding critical points, in particular local minima of \(f\). The last chapter 10 treats parameter dependent problems like one-parameter families of optimization problems. The book is an updated version of the two volumes ``Nonlinear Optimization in \(\mathbb R^n\). I, II'' (Frankfurt/M. 1983/6; Zbl 0527.90064, Zbl 0611.90071) by the same authors. It is essentially self-contained except for the proofs of some technical results (like Sard's theorem, the existence of partitions of unity, the properties of singular homology). The presentation is leisurely; for instance, the fact that the sublevel set \(f^b\) is homotopy equivalent to \(f^a\) with a \(k\)-cell attached (if \(f\) has precisely one critical point of index \(k\) in \(f^b\setminus f^a\)) is proved in four versions, beginning with \(f\in{\mathcal{C}}^4(\mathbb R^n)\) and ending with \(f\in{\mathcal{C}}^2(M)\), \(M\) a manifold with generalized boundary. The book contains many examples and pictures which are very helpful for the novice. It will be useful for graduate students and researchers who want to learn about the differential topology behind nonlinear global optimization.
scientific article; zbMATH DE number 1530909
scientific article