This focused monograph presents a study of subgradient algorithms for
constrained minimization problems in a Hilbert space. The book is of
interest for experts in applications of optimization to engineering and
economics. The goal is to obtain a good approximate solution of the
problem in the presence of computational errors. 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 different, in general. The book is especially useful for the
reader because it contains solutions to a number of difficult and
interesting problems in the numerical optimization. The subgradient
projection algorithm is one of the most important tools in optimization
theory and its applications. An optimization problem is described by an
objective function and a set of feasible points. For this algorithm each
iteration consists of two steps. The first step requires a calculation
of a subgradient of the objective function; the second requires a
calculation of a projection on the feasible set. The computational
errors in each of these two steps are different. This book shows that
the algorithm discussed, generates a good approximate solution, if all
the computational errors are bounded from above by a small positive
constant. Moreover, if computational errors for the two steps of the
algorithm are known, one discovers an approximate solution and how many
iterations one needs for this. In addition to their mathematical
interest, the generalizations considered in this book have a significant
practical meaning.
