The book presents an application-oriented survey of the most important parts of the nonlinear optimization theory on normed or even Banach spaces. The theory is applied to various mathematical problems so that of instance the Euler-Lagrange equation in the calculus of variations, the generalized Kolmogorov condition, the alternation theorem in approximation theory or the Pontryagin maximum principle in optimal control are derived from the general results obtained.
The book is divided into seven chapters and four appendices. The first two chapters describe basic notions and results, which are used in the further chapters (e.g., problem formulation, existence theorems, convexity results, properties of the set of optimal solutions). Chapter 3 is devoted to various approaches to generalized derivatives (directional derivatives, Gâteaux and Fréchet derivatives, subdifferentials of convex functions, quasidifferentials, Clarke derivatives). Properties of tangent cones and related theoretical results to optimization are presented in Chapter 4. In Chapter 5, the author derives the generalized Lagrange multiplier rule and its corollaries under various regularity assumptions. The results are applied to optimal control problems. Chapter 6 is devoted to the investigation of duality theory for nonlinear optimization problems. The introduction of the concept of the so-called “convex-likeness” of \(f\), \(g\), where \(f\) is a given objective functional and \(g\) is a given constraint mapping of the primal minimization problem makes possible to treat certain nonconvex optimization problems with this duality theory. The obtained results are applied to approximation problems in the concluding part of the chapter. In Chapter 7, a direct treatment of special optimal control problems is suggested (linear-quadratic and time-minimal problems). The four appendices explain some mathematical tools which are necessary for understanding the material of the book (weak convergence, reflexivity of Banach spaces, Hahn-Banach theorem, partially ordered linear spaces). The bibliography gives a survey of books in the area of nonlinear optimization and related areas.