这本书致力于研究存在计算误差的最优化问题的近似解。
The book is devoted to the study of approximate solutions of optimization problems in the presence of computational errors.
包含了有关Hilbert空间中算法收敛行为的一些结果，这些结果被认为是解决优化问题的重要工具。
It contains a number of results on the convergence behavior of algorithms in a Hilbert space, which are known as important tools for solving optimization problems.
本书中介绍的研究内容是作者2016年《计算误差的数值优化》一书的延续和进一步发展，Springer 2016。
The research presented in the book is the continuation and the further development of the author’s © 2016 book Numerical Optimization with Computational Errors, Springer 2016.
这两本书研究的算法都考虑到计算误差，这是经常出现在实践中。
Both books study the algorithms taking into account computational errors which are always present in practice.
主要目标是，对于已知的计算误差，找出可以得到什么样的近似解，以及为此需要多少次迭代。
The main goal is, for a known computational error, to find out what an approximate solution can be obtained and how many iterates one needs for this.
这本新书与2016年的书的主要区别在于，在本书中，讨论考虑到了这样一个事实：对于每一个算法，其迭代都由几个步骤组成，而不同步骤的计算误差通常是不同的。
The main difference between this new book and the 2016 book is that in this present book the discussion takes into consideration the fact that for every algorithm, its iteration consists of several steps and that computational errors for different steps are generally, different.
上一本书没有考虑到这一事实，但在实践中确实很重要。
This fact, which was not taken into account in the previous book, is indeed important in practice.
例如，次梯度投影算法包括两个步骤。
For example, the subgradient projection algorithm consists of two steps.
第一步计算目标函数的次梯度，第二步计算可行集上的投影。
The first step is a calculation of a subgradient of the objective function while in the second one we calculate a projection on the feasible set.
在这两个步骤中，每一步都有一个计算误差，这两个计算误差一般是不同的。
In each of these two steps there is a computational error and these two computational errors are different in general.
It may happen that the feasible set is simple and the objective function is complicated.
As a result, the computational error, made when one calculates the projection, is essentially smaller than the computational error of the calculation of the subgradient.
Clearly, an opposite case is possible too.
Another feature of this book is a study of a number of important algorithms which appeared recently in the literature and which are not discussed in the previous book.
This monograph contains 12 chapters.
Chapter 1 is an introduction.
In Chapter 2 we study the subgradient projection algorithm for minimization of convex and nonsmooth functions.
We generalize the results of [NOCE] and establish results which has no prototype in [NOCE].
In Chapter 3 we analyze the mirror descent algorithm for minimization of convex and nonsmooth functions, under the presence of computational errors.
For this algorithm each iteration consists of two steps.
The first step is a calculation of a subgradient of the objective function while in the second one we solve an auxiliary minimization problem on the set of feasible points.
In each of these two steps there is a computational error.
We generalize the results of [NOCE] and establish results which has no prototype in [NOCE].
In Chapter 4 we analyze the projected gradient algorithm with a smooth objective function under the presence of computational errors.
In Chapter 5 we consider an algorithm, which is an extension of the projection gradient algorithm used for solving linear inverse problems arising in signal/image processing.
In Chapter 6 we study continuous subgradient method and continuous subgradient projection algorithm for minimization of convex nonsmooth functions and for computing the saddle points of convex-concave functions, under the presence of computational errors.
All the results of this chapter has no prototype in [NOCE].
In Chapters 7-12 we analyze several algorithms under the presence of computational errors which were not considered in [NOCE].
Again, each step of an iteration has a computational errors and we take into account that these errors are, in general, different.
An optimization problems with a composite objective function is studied in Chapter 7.
A zero-sum game with two-players is considered in Chapter 8.
A predicted decrease approximation-based method is used in Chapter 9 for constrained convex optimization.
Chapter 10 is devoted to minimization of quasiconvex functions.
Minimization of sharp weakly convex functions is discussed in Chapter 11.
Chapter 12 is devoted to a generalized projected subgradient method for minimization of a convex function over a set which is not necessarily convex.
The book is of interest for researchers and engineers 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 the influence of computational errors for several important optimization algorithms.
The book is of interest for experts in applications of optimization to engineering and economics.
更多精彩文章请关注公众号：
