Quasidifferential Calculus, 1986, 301 pp., ISBN 0-911575-35-9, $92.00
First-time publication--not a translation from the Russian.
Spurred by the needs of nondifferentiable optimization, the authors present
a systematic exposition of Quasidifferential Calculus extending results
of traditional classical calculus: formulas for calculating quasidifferentials,
chain rules, inverse and implicit function theorems, necessary and sufficient
conditions for extrema, quasidifferentiable mappings. Includes application
to nonsmooth optimization, as well as extensive bibliographic notes.
TABLE OF CONTENTS
Preface ..... ix
CHAPTER 1. PRELIMINARIES ..... 1
CHAPTER 2. ELEMENTS OF THE THEORY OF MULTIVALUED MAPPINGS ..... 6
CHAPTER 3. DIRECTIONAL DERIVATIVES ..... 19
CHAPTER 4. CONVEX FUNCTIONS ..... 37
CHAPTER 5. SUBLINEAR FUNCTIONS ..... 46
CHAPTER 6. THE SPACE OF CONVEX SETS ..... 65
CHAPTER 7. UPPER SEMICONTINUOUSLY DIRECTIONALLY DIFFERENTIABLE FUNCTIONS ..... 78
CHAPTER 8. THE CLARKE DERIVATIVES ..... 91
CHAPTER 9. UPPER CONVEX AND LOWER CONCAVE APPROXIMATIONS ..... 104
CHAPTER 10. QUASIDIFFERENTIABLE FUNCTIONS ..... 112
CHAPTER 11. EXAMPLES ..... 122
CHAPTER 12. QUASIDIFFERENTIABILITY OF A COMPOSITION ..... 128
CHAPTER 13. A RELATION BETWEEN THE CLARKE SUBDIFFERENTIAL AND THE QUASIDIFFERENTIAL ..... 143
CHAPTER 14. IMPLICIT AND INVERSE FUNCTION THEOREMS ..... 155
CHAPTER 15. CONES OF ADMISSIBLE DIRECTIONS AND A NONDEGENERACY CONDITION ..... 176
CHAPTER 16. NECESSARY AND SUFFICIENT CONDITIONS FOR AN EXTREMUM ..... 188
CHAPTER 17. RATE AND DIRECTIONS OF STEEPEST DESCENT AND ASCENT ..... 203
CHAPTER 18. SADDLE POINTS OF QUASIDIFFERENTIABLE FUNCTIONS ..... 229
CHAPTER 19. STAR-SHAPED SETS AND THEIR APPLICATIONS IN NONSMOOTH OPTIMIZATION ..... 236
CHAPTER 20. APPROXIMATE QUASIDIFFERENTIABILITY ..... 254
Bibliographic Notes ..... 265
References ..... 269
Index ..... 283
Transliteration Table (Russian-English) ..... 289