Optimization in Banach Spaces

The book is devoted to the study of constrained minimization  problems on closed and convex sets in Banach spaces with a Frechet differentiable objective function. It discusses algorithms for convex and nonconvex minimization problems in a general Banach space, and shows that the gradient projection algorithm generates a good approximate solution if all computational errors are bounded from above by a small positive constant.

Format: Paperback / softback
Length: 126 pages
Publication date: 30 September 2022
Publisher: Springer International Publishing AG

The book delves into the study of constrained minimization problems on closed and convex sets in Banach spaces, with a Frechet differentiable objective function. These problems have received significant attention in both finite-dimensional and infinite-dimensional Hilbert spaces. In the Hilbert space setting, numerous algorithms exist for solving optimization challenges, including the gradient projection algorithm, which holds a prominent position in optimization theory, nonlinear analysis, and their applications. An optimization problem is characterized by an objective function and a set of feasible points. The gradient projection algorithm involves two main steps. In the first step, a gradient of the objective function is calculated, while in the second step, a projection is computed on the feasible set. However, both steps introduce computational errors. In recent research, it has been demonstrated that the gradient projection algorithm can produce a satisfactory approximate solution when all computational errors are bounded above by a small positive constant. It is worth noting that the properties of a Hilbert space play a crucial role in this context. When considering an optimization problem in a general Banach space, the situation becomes more complex and less well understood. On the other hand, such problems arise in the approximation theory.

The book holds appeal for mathematicians engaged in optimization, as well as serving as a valuable resource for graduate students in preparation courses. The book's primary distinguishing feature, which particularly resonates with this audience, is its exploration of algorithms for convex and nonconvex minimization problems in a general Banach space.

Furthermore, the book holds interest for experts in the applications of optimization to the approximation theory.

Weight: 221g
Dimension: 235 x 155 (mm)
ISBN-13: 9783031126437
Edition number: 1st ed. 2022