The theory of convex optimization has been constantly developing over the past 30 years. Most recently, many researchers have been studying more complicated classes of problems that still can be studied by means of convex analysis, so-called "anticonvex" and "convex-anticonvex" optimizaton problems. This manuscript contains an exhaustive presentation of the duality for these classes of problems and some of its generalization in the framework of abstract convexity. This manuscript will be of great interest for experts in this and related fields.

This monograph attempts to present the results known today on bases in Banach spaces and some unsolved problems concerning them. Although this important part of the theory of Banach spaces has been studied for more than forty years by numerous mathematicians, the existing books on functional analysis (e. g. M. M. Day [43], A. Wilansky [263], R. E. Edwards [54]) contain only a few results on bases. A survey of the theory of bases in Banach spaces, up to 1963, has been presented in the expository papers [241], [242] and [243], which contain no proofs; although in the meantime the theory has rapidly deve1oped, much of the present monograph is based on those expository papers. Independently, a useful bibliography of papers on bases, up to 1963, was compiled by B. L. Sanders [219J. Due to the vastness of the field, the monograph is divided into two volumes, ofwhich this is the first (see the tab1e of contents). Some results and problems re1ated to those treated herein have been de1iberately planned to be inc1uded in Volume 11, where they will appear in their natural framework (see [242], [243]).

Since the appearance, in 1970, of Vol. I of the present monograph 1370], the theory of bases in Banach spaces has developed substantially. Therefore, the present volume contains only Ch. III of the monograph, instead of Ch. Ill, IV and V, as was planned initially (cp. the table of contents of Vol. I). Since this volume is a continuation of Vol. I of the same monograph, we shall refer to the results of Vol. I directly as results of Ch. I or Ch. II (without specifying Vol. I). On the other hand, sometimes we shall also mention that certain results will be considered in Vol. III (Ch. IV, V). In spite of the many new advances made in this field, the statement in the Preface to Vol. I, that "the existing books on functional analysis contain only a few results on bases", remains still valid, with the exception of the recent book [248 a] of J. Lindenstrauss and L. Tzafriri. Since we have learned about [248 a] only in 1978, in this volume there are only references to previous works, instead of [248 a]; however, this will cause no inconvenience, since the intersec tion of the present volume with [248 a] is very small. Let us also mention the appearance, since 1970, of some survey papers on bases in Banach spaces (V. D. Milman [287], [288], C. W. McArthur [275]; M. I. Kadec [204], 3 and others).

The theory of convex optimization has been constantly developing over the past 30 years. Most recently, many researchers have been studying more complicated classes of problems that still can be studied by means of convex analysis, so-called "anticonvex" and "convex-anticonvex" optimizaton problems. This manuscript contains an exhaustive presentation of the duality for these classes of problems and some of its generalization in the framework of abstract convexity. This manuscript will be of great interest for experts in this and related fields.