The book is devoted to the study of approximate solutions of
optimization problems in the presence of computational errors. 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. The research presented in the book is the
continuation and the further development of the author's (c) 2016 book
Numerical Optimization with Computational Errors, Springer 2016. Both
books study the algorithms taking into account computational errors
which are always present in practice. 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.
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. This
fact, which was not taken into account in the previous book, is indeed
important in practice. For example, the subgradient projection algorithm
consists of two steps. 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. 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.
