75
Views
13
CrossRef citations to date
0
Altmetric
Original Articles

Fréchet quasidifferential calculus with applications to metric regularity of continuous maps

Pages 469-493 | Received 02 Apr 2004, Published online: 09 Aug 2006

References

References

  • Pallaschke , D , Recht , P and Urbański , R . 1986 . On locally-Lipschitz quasi-differentiable functions in Banach spaces . Optimization , 17 ( 3 ) : 287 – 295 .
  • Borwein , JM and Zhu , QJ . 1999 . A survey on subdifferential calculus with applications . Nonlinear Analysis, TMA , 38 ( 6 ) : 687 – 773 .
  • Ioffe AD 2000 Metric regularity and subdifferential calculus Uspekhi Matematicheskikh Nauk 55 3 103 162 [in Russian]; translated in Russian Mathematical Surveys, 55(3), 501–558
  • Kruger , Ya.A . 2003 . On Fréchet subdifferentials . Journal of Mathematical Sciences (N.Y.) , 116 ( 3 ) : 3325 – 3358 .
  • Phelps RR 1989 Convex Functions, Monotone Operators and Differentiability, 1st Edn Berlin Springer-Verlag
  • Rockafellar RT Wets RJ-B 1998 Variational analysis. In: Fundamental Principles of Mathematical Sciences 317 Berlin Springer-Verlag
  • Castellani M Uderzo A 2002 Dual representations for positively homogeneous functions: an application to constrained extremum problems In: Optimization in Economics, Finance and Industry Milan Datanova pp. 129–143
  • Pallaschke D Urbański R 2002 Pairs of compact convex sets. Fractional arithmetic with convex sets In: Mathematics and its Applications 548 Dordrecht Kluwer Academic Publishers
  • Giannessi , F . 1989 . Semidifferentiable functions and necessary optimality conditions . Journal of Optimization Theory and Applications , 60 ( 2 ) : 191 – 241 .
  • Pappalardo , M . 1992 . On semidifferentiable functions . Optimization , 24 ( 3–4 ) : 207 – 217 .
  • Pappalardo , M and Uderzo , A . 1999 . G-semidifferentiability in Euclidean spaces . Journal of Optimization Theory and Applications , 101 ( 1 ) : 221 – 229 .
  • Yen , ND . 1995 . On G-Semidifferentiable functions in Euclidean spaces . Journal of Optimization Theory and Applications , 85 ( 2 ) : 377 – 392 .
  • Glover , BM , Ishizuka , Y , Jeyakumar , V and Tuan , HD . 1996 . Complete characterization of global optimality for problems involving the pointwise minimum of sublinear functions . SIAM Journal on Optimization , 6 : 367 – 385 .
  • Demyanov VF Rubinov AM 1995 Constructive Nonsmooth Analysis Frankfurt am Main Peter Lang
  • Demyanov , VF and Rubinov , AM . 1980 . On quasidifferentiable functionals . Doklady of USSR Academy of Sciences , 250 ( 1 ) : 21 – 25 .
  • Demyanov VF Rubinov AM 1986 Quasidifferential Calculus New York Optimization Software
  • Clarke , FH . 1976 . A new approach to Lagrange multipliers . Mathematics of Operations Research , 1 : 165 – 174 .
  • Rubinov , A and Dzalilov , Z . 2002 . Abstract convexity of positively homogeneous functions. Generalized convexity, generalized monotonicity, optimality conditions and duality in scalar and vector optimization . Journal of Statistics & Management Systems , 5 ( 1–3 ) : 1 – 20 .
  • Ekeland , I . 1974 . On the variational principle . Journal of Mathematical Analysis and Applications , 47 ( 2 ) : 324 – 353 .
  • Borwein , JM and Preiss , D . 1987 . A smooth variational principle with applications to subdifferentiability and to differentiability of convex functions . Transactions of the American Mathematical Society , 303 ( 2 ) : 517 – 527 .
  • Fabian , M and Mordukhovich , Sh.B . 1998 . Nonsmooth characterizations of Asplund spaces and smooth variational principles . Set-Valued Analysis , 6 ( 4 ) : 381 – 406 .
  • Mordukhovich Sh.B Wang B Necessary suboptimality and optimality conditions via variational principles (Forthcoming on SIAM Journal on Control and Optimization)
  • Asplund , E . 1968 . Fréchet differentiability of convex functions . Acta Mathematica , 121 : 31 – 47 .
  • Fabian , M . 1989 . Subdifferentiability and trustworthiness in the light of a new variational principle of Borwein and Preiss . Acta Universitatis Carolinae , 30 : 51 – 56 .
  • Dmitruk AV Milyutin AA Osmolovskii NP 1980 Lyusternik's theorem and the theory of extrema Uspekhi Matematicheskikh Nauk 35 6 11 46 [in Russian]; translated in Russian Mathematical Surveys, 35(6), 11–51
  • Ioffe AD 1999 Variational methods in local and global nonsmooth analysis NATO Adv. Sci. Inst., Ser. C Math. Phys. Sci. 528 Dordrecht Kluwer Academic Publishers pp. 447–502
  • Jourany , A . 1998 . Necessary conditions for extremality and separation theorems with applications to multiobjective optimization . Optimization , 44 ( 4 ) : 327 – 350 .
  • Dontchev , AL . 1996 . The Graves theorem revisited . Journal of Convex Analysis , 3 ( 1 ) : 45 – 53 .
  • Graves , LM . 1950 . Some mapping theorems . Duke Mathematical Journal , 17 : 111 – 114 .
  • Lyusternik LA 1934 On conditional extrema of functionals Matematicheskii Sbornik Mathematics 41 3 390 401 [in Russian]
  • De Giorgi , E , Marino , A and Tosques , M . 1980 . Problemi di evoluzione in spazi metrici e curve di massima pendenza . Atti della Accademia Nazionale dei Lincei Rendiconti Lincei. Classe di Scienze Fisiche, Matematiche e Naturali , 68 ( 3 ) : 180–187 [in Italian]
  • Demyanov , VF . 2000 . Conditions for an extremum in metrix spaces . Journal of Global Optimization , 17 ( 1–4 ) : 55 – 63 .
  • Ioffe , AD . 1998 . Fuzzy principles and characterization of trustworthiness . Set-Valued Analysis , 6 ( 3 ) : 265 – 276 .
  • Zhu , QJ . 1998 . The equivalence of several basic theorems for subdifferentials . Set-Valued Analysis , 6 ( 2 ) : 171 – 185 .

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.