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.
