國立虎尾科技大學 |

Optimization in Banach Spaces

紀錄類型:	書目-語言資料,印刷品 : Monograph/item
正題名/作者:	Optimization in Banach Spaces/ by Alexander J. Zaslavski.
作者:	Zaslavski, Alexander J.
面頁冊數:	VIII, 126 p.online resource. :
Contained By:	Springer Nature eBook
標題:	Numerical Analysis. -
電子資源:	https://doi.org/10.1007/978-3-031-12644-4
ISBN:	9783031126444

Optimization in Banach Spaces
Zaslavski, Alexander J.

Optimization in Banach Spaces[electronic resource] /by Alexander J. Zaslavski. - 1st ed. 2022. - VIII, 126 p.online resource. - SpringerBriefs in Optimization,2191-575X. - SpringerBriefs in Optimization,.

Preface -- Introduction -- Convex optimization -- Nonconvex optimization -- Continuous algorithms -- References.

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. Such problems are well studied in a finite-dimensional space and in an infinite-dimensional Hilbert space. When the space is Hilbert there are many algorithms for solving optimization problems including the gradient projection algorithm which is one of the most important tools in the optimization theory, nonlinear analysis and their applications. An optimization problem is described by an objective function and a set of feasible points. For the gradient projection algorithm each iteration consists of two steps. The first step is a calculation of a gradient of the objective function while in the second one we calculate a projection on the feasible set. In each of these two steps there is a computational error. In our recent research we show that the gradient projection algorithm generates a good approximate solution, if all the computational errors are bounded from above by a small positive constant. It should be mentioned that the properties of a Hilbert space play an important role. When we consider an optimization problem in a general Banach space the situation becomes more difficult and less understood. On the other hand such problems arise in the approximation theory. The book is of interest for mathematicians working in optimization. It also can be useful in preparation courses for graduate students. The main feature of the book which appeals specifically to this audience is the study of algorithms for convex and nonconvex minimization problems in a general Banach space. The book is of interest for experts in applications of optimization to the approximation theory. In this book the goal is to obtain a good approximate solution of the constrained optimization problem in a general Banach space under the presence of computational errors. It is shown that the algorithm generates a good approximate solution, if the sequence of computational errors is bounded from above by a small constant. The book consists of four chapters. In the first we discuss several algorithms which are studied in the book and prove a convergence result for an unconstrained problem which is a prototype of our results for the constrained problem. In Chapter 2 we analyze convex optimization problems. Nonconvex optimization problems are studied in Chapter 3. In Chapter 4 we study continuous algorithms for minimization problems under the presence of computational errors. The algorithm generates a good approximate solution, if the sequence of computational errors is bounded from above by a small constant. The book consists of four chapters. In the first we discuss several algorithms which are studied in the book and prove a convergence result for an unconstrained problem which is a prototype of our results for the constrained problem. In Chapter 2 we analyze convex optimization problems. Nonconvex optimization problems are studied in Chapter 3. In Chapter 4 we study continuous algorithms for minimization problems under the presence of computational errors.

ISBN: 9783031126444

Standard No.: 10.1007/978-3-031-12644-4doiSubjects--Topical Terms:

671433
Numerical Analysis.

LC Class. No.: QA402.5-402.6

Dewey Class. No.: 519.6

Optimization in Banach Spaces
LDR:04584nam a22003975i 4500 001 1083766
003 DE-He213
005 20220929102557.0
007 cr nn 008mamaa
008 221228s2022 sz | s |||| 0|eng d
020 $a 9783031126444 $9 978-3-031-12644-4
024 7 $a 10.1007/978-3-031-12644-4 $2 doi
035 $a 978-3-031-12644-4
050 4 $a QA402.5-402.6
072 7 $a PBU $2 bicssc
072 7 $a MAT003000 $2 bisacsh
072 7 $a PBU $2 thema
082 0 4 $a 519.6 $2 23
100 1 $a Zaslavski, Alexander J. $4 aut $4 http://id.loc.gov/vocabulary/relators/aut $3 1022503
245 1 0 $a Optimization in Banach Spaces $h [electronic resource] / $c by Alexander J. Zaslavski.
250 $a 1st ed. 2022.
264 1 $a Cham : $b Springer International Publishing : $b Imprint: Springer, $c 2022.
300 $a VIII, 126 p. $b online resource.
336 $a text $b txt $2 rdacontent
337 $a computer $b c $2 rdamedia
338 $a online resource $b cr $2 rdacarrier
347 $a text file $b PDF $2 rda
490 1 $a SpringerBriefs in Optimization, $x 2191-575X
505 0 $a Preface -- Introduction -- Convex optimization -- Nonconvex optimization -- Continuous algorithms -- References.
520 $a 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. Such problems are well studied in a finite-dimensional space and in an infinite-dimensional Hilbert space. When the space is Hilbert there are many algorithms for solving optimization problems including the gradient projection algorithm which is one of the most important tools in the optimization theory, nonlinear analysis and their applications. An optimization problem is described by an objective function and a set of feasible points. For the gradient projection algorithm each iteration consists of two steps. The first step is a calculation of a gradient of the objective function while in the second one we calculate a projection on the feasible set. In each of these two steps there is a computational error. In our recent research we show that the gradient projection algorithm generates a good approximate solution, if all the computational errors are bounded from above by a small positive constant. It should be mentioned that the properties of a Hilbert space play an important role. When we consider an optimization problem in a general Banach space the situation becomes more difficult and less understood. On the other hand such problems arise in the approximation theory. The book is of interest for mathematicians working in optimization. It also can be useful in preparation courses for graduate students. The main feature of the book which appeals specifically to this audience is the study of algorithms for convex and nonconvex minimization problems in a general Banach space. The book is of interest for experts in applications of optimization to the approximation theory. In this book the goal is to obtain a good approximate solution of the constrained optimization problem in a general Banach space under the presence of computational errors. It is shown that the algorithm generates a good approximate solution, if the sequence of computational errors is bounded from above by a small constant. The book consists of four chapters. In the first we discuss several algorithms which are studied in the book and prove a convergence result for an unconstrained problem which is a prototype of our results for the constrained problem. In Chapter 2 we analyze convex optimization problems. Nonconvex optimization problems are studied in Chapter 3. In Chapter 4 we study continuous algorithms for minimization problems under the presence of computational errors. The algorithm generates a good approximate solution, if the sequence of computational errors is bounded from above by a small constant. The book consists of four chapters. In the first we discuss several algorithms which are studied in the book and prove a convergence result for an unconstrained problem which is a prototype of our results for the constrained problem. In Chapter 2 we analyze convex optimization problems. Nonconvex optimization problems are studied in Chapter 3. In Chapter 4 we study continuous algorithms for minimization problems under the presence of computational errors.
650 2 4 $a Numerical Analysis. $3 671433
650 1 4 $a Optimization. $3 669174
650 0 $a Numerical analysis. $3 527939
650 0 $a Mathematical optimization. $3 527675
710 2 $a SpringerLink (Online service) $3 593884
773 0 $t Springer Nature eBook
776 0 8 $i Printed edition: $z 9783031126437
776 0 8 $i Printed edition: $z 9783031126451
830 0 $a SpringerBriefs in Optimization, $x 2190-8354 $3 1254063
856 4 0 $u https://doi.org/10.1007/978-3-031-12644-4
912 $a ZDB-2-SMA
912 $a ZDB-2-SXMS
950 $a Mathematics and Statistics (SpringerNature-11649)
950 $a Mathematics and Statistics (R0) (SpringerNature-43713)