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.
