EDP Sciences logo
Subscriber Authentication Point
Search
Menu
All issues Volume 27 (2021) ESAIM: COCV, 27 (2021) 79 References
Open Access
Issue
ESAIM: COCV
Volume 27, 2021
Article Number 79
Number of page(s) 35
DOI https://doi.org/10.1051/cocv/2021074
Published online 22 July 2021
  1. R.A. Adams, Sobolev Spaces. Academic Press, New York (1975). [Google Scholar]
  2. J.-P. Aubin and H. Frankowska, Set-Valued Analysis. Birkhäuser, Boston (1990). [Google Scholar]
  3. L. Bayón, J.M. Grau, M.M. Ruiz and P.M. Suárez, Nonsmooth optimization of hydrothermal problems. J. Comput. Appl. Math. 192 (2006) 11–19. [Google Scholar]
  4. L. Bayón, J.M. Grau, M.M. Ruiz and P.M. Suárez, A constrained and non-smooth hydrothermal problem. Appl. Math. Comput. 209 (2009) 10–18. [Google Scholar]
  5. L. Bayón, J.M. Grau, M.M. Ruiz and P.M. Suárez, A hydrothermal problem with non-smooth Lagrangian. J. Ind. Manag. Optim. 10 (2014) 761–776. [Google Scholar]
  6. S. Bellaassali, Contributions à l’optimisation multicritère, Ph.D. thesis, Université de Bourgogne, Laboratoire Analyse Appliquée et Optimisation, Dijon, France (2003). Available at: https://tel.archives-ouvertes.fr/file/index/docid/46039/filename/tel-00004421.pdf. [Google Scholar]
  7. D.N. Bessis, Yu. S. Ledyaev and R.B. Vinter, Dualization of the Euler and Hamiltonian inclusions. Nonlinear Anal. 43 (2001) 861–882. [Google Scholar]
  8. V.I. Bogachev, Vol. I of Measure Theory. Springer-Verlag, Berlin, Heidelberg (2007). [CrossRef] [Google Scholar]
  9. G. Bonfanti and A. Cellina, The validity of the Euler-Lagrange equation. Discret. Contin. Dyn. Syst. 28 (2010) 511–517. [Google Scholar]
  10. P. Bousquet, The Euler equation in the multiple integrals calculus of variations. SIAM J. Control Optim. 51 (2013) 1047–1062. [Google Scholar]
  11. F. Clarke, Necessary Conditions in Dynamic Optimization. American Mathematical Society, Providence, Rhode Island (2005). [Google Scholar]
  12. F. Clarke, Functional Analysis, Calculus of Variations and Optimal Control. Springer-Verlag, London (2013). [Google Scholar]
  13. F.H. Clarke, The Euler-Lagrange differential inclusion. J. Differ. Equ. 19 (1975) 80–90. [Google Scholar]
  14. F.H. Clarke, The generalized problem of Bolza. SIAM J. Control Optim. 14 (1976) 682–699. [Google Scholar]
  15. F.H. Clarke, Multiple integrals of Lipschitz functions in the calculus of variations. Proc. Am. Math. Soc. 64 (1977) 260–264. [Google Scholar]
  16. F.H. Clarke, The Erdmann condition and Hamiltonian inclusions in optimal control and the calculus of variations. Can. J. Math. 32 (1980) 494–509. [Google Scholar]
  17. F.H. Clarke, Optimization and Nonsmooth Analysis. Wiley–Interscience, New York (1983). [Google Scholar]
  18. F.H. Clarke, A decoupling principle in the calculus of variations. J. Math. Anal. Appl. 172 (1993) 92–105. [Google Scholar]
  19. F.H. Clarke and M.R. de Pinho, The nonsmooth maximum principle. Control Cybern. 38 (2009) 1151–1167. [Google Scholar]
  20. F.H. Clarke, Yu. S. Ledyaev, R.J. Stern and P.R. Wolenski, Nonsmooth Analysis and Control Theory. Springer-Verlag, New York (1998). [Google Scholar]
  21. G. Cupini, M. Guidorzi and C. Marcelli, Necessary conditions and non-existence results for autonomous nonconvex variational problems. J. Differ. Equ. 243 (2007) 329–348. [Google Scholar]
  22. B. Dacorogna, Direct Methods in the Calculus of Variations. Springer-Verlag, New York (2008). [Google Scholar]
  23. V.F. Demyanov, Continuous generalized gradients for nonsmooth functions, in Optimization, Parallel Processing and Applications, edited by A. Kurzhanski, K. Neumann and D. Pallaschke. Springer Berlin, Heidelberg (1988) 24–27. [Google Scholar]
  24. V.F. Demyanov, On codifferentiable functions. Vestn. Leningr. Univ., Math. 2 (1988) 22–26. [Google Scholar]
  25. V.F. Demyanov, Smoothness of nonsmooth functions, in Nonsmooth Optimization and Related Topics, edited by F. Clarke, V. Demyanov and F. Giannesssi. Springer, Boston (1989) 79–88. [Google Scholar]
  26. V.F. Demyanov and L.C.W. Dixon, Quasidifferential Calculus. Springer Berlin, Heidelberg (1986). [Google Scholar]
  27. V.F. Demyanov and A.M. Rubinov, Constructive Nonsmooth Analysis. Peter Lang, Frankfurt am Main (1995). [Google Scholar]
  28. V.F. Demyanov and A.M. Rubinov, Quasidifferentiability and Related Topics. Kluwer Academic Publishers, Dordrecht (2000). [Google Scholar]
  29. V.F. Demyanov, G.E. Stavroulakis, L.N. Polyakova and P.D. Panagiotopoulos, Quasidifferentiability and Nonsmooth Modelling in Mechanics, Engineering and Economics. Kluwer Academic Publishers, Dordrecht (1996). [Google Scholar]
  30. M.V. Dolgopolik, Codifferential calculus in normed spaces. J. Math. Sci. 173 (2011) 441–462. [Google Scholar]
  31. M.V. Dolgopolik, Nonsmooth problems of calculus of variations via codifferentiation. ESAIM: COCV 20 (2014) 1153–1180. [EDP Sciences] [Google Scholar]
  32. M.V. Dolgopolik, Abstract convex approximations of nonsmooth functions. Optim 64 (2015) 1439–1469. [Google Scholar]
  33. M.V. Dolgopolik, A convergence analysis of the method of codifferential descent. Comput. Optim. Appl. 71 (2018) 879–913. [Google Scholar]
  34. M.V. Dolgopolik, Metric regularity of quasidifferentiable mappings and optimality conditions for nonsmooth mathematical programming problems. Set-Valued Var. Anal. 28 (2019) 427–449. [Google Scholar]
  35. M.V. Dolgopolik, A new constraint qualification and sharp optimality conditions for nonsmooth mathematical programming problems in terms of quasidifferentials. SIAM J. Optim. 30 (2020) 2603–2627. [Google Scholar]
  36. N. Dunford and J.T. Schwartz, Linear Operators Part 1: General Theory. John Wiley & Sons, New Jersey (1958). [Google Scholar]
  37. I. Ekeland and R. Temam, Convex Analysis and Variational Problems. SIAM, Philadelphia (1999). [CrossRef] [Google Scholar]
  38. G.B. Folland, Real Analysis. Modern Techniques and Their Applications. Interscience Publishers, New York (1984). [Google Scholar]
  39. Y. Gao, On the minimal quasidifferential in the one-dimensional case. Soochow J. Math. 24 (1998) 211–218. [Google Scholar]
  40. F. Giannessi, A common understanding or a common misunderstanding? Numer. Funct. Anal. Optim. 16 (1995) 1359–1363. [Google Scholar]
  41. J. Grzybowski, D. Pallaschke and R. Urbański, On the reduction of pairs of bounded closed convex sets. Studia Math. 189 (2008) 1–12. [Google Scholar]
  42. J. Grzybowski, D. Pallaschke and R. Urbański, On the amount of minimal pairs of convex sets. Optim. Methods Softw. 25 (2010) 89–96. [Google Scholar]
  43. J. Grzybowski and R. Urbański, Minimal pairs of bounded closed convex sets. Studia Math. 126 (1997) 95–99. [Google Scholar]
  44. J. Grzybowski and R. Urbański, Three criteria of minimality for pairs of compact convex sets. Optim 55 (2006) 569–576. [Google Scholar]
  45. M. Handschug, On equivalent quasidifferentials in the two-dimensional case. Optim 20 (1989) 37–43. [Google Scholar]
  46. A. Ioffe, Euler-Lagrange and Hamiltonian formalisms in dynamic optimization. Trans. Am. Math. Soc. 349 (1997) 2871–2900. [Google Scholar]
  47. A.D. Ioffe, On necessary conditions for a minimum. J. Math. Sci. 217 (2016) 751–772. [Google Scholar]
  48. A.D. Ioffe, On generalized Bolza problems and its application to dynamic optimization. J. Optim. Theory Appl. 182 (2019) 285–309. [Google Scholar]
  49. A.D. Ioffe and R.T. Rockafellar, The Euler and Weierstrass conditions for nonsmooth variational problems. Calc. Var. Partial Differ. Equ. 4 (1996) 59–87. [Google Scholar]
  50. A.D. Ioffe and V.M. Tihomirov, Theory of Extremal Problems. North-Holland Publishing Company, Amsterdam (1979). [Google Scholar]
  51. A. Jourani, Lagrangian and Hamiltonian necessary conditions for the generalized Bolza problem and applications. J. Nonlinear Convex Anal. 10 (2009) 437–454. [Google Scholar]
  52. A. Jourani and L. Thibault, Approximate subdifferential and metric regularity: the finite-dimensional case. Math. Program. 47 (1990) 203–218. [Google Scholar]
  53. L. Kuntz, A characterization of continuously codifferentiable functions and some consequences. Optim 22 (1991) 539–547. [Google Scholar]
  54. G. Leoni, A First Course in Sobolev spaces. American Mathematical Society, Providence, RI (2009). [Google Scholar]
  55. P.D. Loewen, Optimal Control via Nonsmooth Analysis. American Mathematical Society, Providence, Rhode Island (1993). [Google Scholar]
  56. P.D. Loewen and R.T. Rockafellar, The adjoint arc in nonsmooth optimization. Trans. Am. Math. Soc. 325 (1991) 39–72. [Google Scholar]
  57. P.D. Loewen and R.T. Rockafellar, Optimal control of unbounded differential inclusions. SIAM J. Control Optim. 32 (1994) 442–470. [Google Scholar]
  58. P.D. Loewen and R.T. Rockafellar, New necessary conditions for the generalized problem of Bolza. SIAM J. Control Optim. 34 (1996) 1496–1511. [Google Scholar]
  59. P.D. Loewen and R.T. Rockafellar, Bolza problem with general time constraints. SIAM J. Control Optim. 35 (1997) 2050–2069. [Google Scholar]
  60. C. Marcelli, Variational problems with nonconvex, noncoercive, highly discontinuous integrands: characterization and existence of minimizers. SIAM J. Control Optim. 40 (2002) 1473–1490. [Google Scholar]
  61. C. Marcelli, Necessary and sufficient conditions for optimality of nonconvex, noncoercive autonomous variational problems with constraints. Trans. Am. Math. Soc. 360 (2008) 5201–5227. [Google Scholar]
  62. C. Marcelli, E. Outkine and M. Sytchev, Remarks on necessary conditions for minimizers of one-dimensional variational problems. Math. Prepr. Arch. 2001 (2001) 1145–1163. [Google Scholar]
  63. B.S. Mordukhovich, Approximation Methods in Problems of Optimization and Control. Nauka, Moscow (1988). [in Russian]. [Google Scholar]
  64. B.S. Mordukhovich, Discrete approximation and refined Euler-Lagrange conditions for nonconvex differential inclusions. SIAM J. Control Optim. 33 (1995) 882–915. [Google Scholar]
  65. B.S. Mordukhovich, Variational Analysis and Generalized Differentiation I: Basic Theory. Springer-Verlag, Berling, Heidelberg (2006). [Google Scholar]
  66. B.S. Mordukhovich, Variational Analysis and Generalized Differentiation II: Applications. Springer-Verlag, Berling, Heidelberg (2006). [Google Scholar]
  67. I.V. Orlov and A.V. Tsygankova, Multidimensional variational functionals with subsmooth integrands. Eurasian Math. J. 6 (2015) 54–75. [Google Scholar]
  68. D. Pallaschke and R. Urbański, Some criteria for the minimality of pairs of compact convex sets. ZOR — Methods Models Oper. Res. 37 (1993) 129–150. [Google Scholar]
  69. D. Pallaschke and R. Urbański, Quasidifferentiable calculus and minimal pairs of compact convex sets. Schedae Informaticae 21 (2012) 107–125. [Google Scholar]
  70. E.S. Polovinkin, Differential inclusions with unbounded right-hand side and necessary optimality conditions. Proc. Stekov Inst. Math. 291 (2015) 237–252. [Google Scholar]
  71. E.S. Polovinkin, Necessary optimality conditions for the Mayer problem with unbounded differential inclusion. IFAC-PapersOnline 51 (2018) 521–524. [Google Scholar]
  72. E.S. Polovinkin, Pontryagin’s direct method for optimization problems with differential inclusions. Proc. Stekov Inst. Math. 304 (2019) 241–256. [Google Scholar]
  73. B.N. Pshenichnyi, Necessary Conditions for an Extremum. Marcel Dekker, New York (1971). [Google Scholar]
  74. R.T. Rockafellar, Conjugate convex functions in optimal control and the calculus of variations. J. Math. Anal. Appl. 32 (1970) 174–222. [Google Scholar]
  75. R.T. Rockafellar, Generalized Hamiltonian equations for convex problems of Lagrange. Pac. J. Math. 33 (1970) 411–427. [Google Scholar]
  76. R.T. Rockafellar, Existence and duality theorems for convex problems of Bolza. Trans. Am. Math. Soc. 159 (1971) 1–40. [Google Scholar]
  77. R.T. Rockafellar, Dualization of subgradient conditions for optimality. Nonlinear Anal. 20 (1993) 627–646. [Google Scholar]
  78. S. Scholtes, Minimal pairs of convex bodies in two dimensions Mathematika 39 (1992) 267–273. [CrossRef] [MathSciNet] [Google Scholar]
  79. M.H.N. Skandari, A.V. Kamyad and S. Effati, Generalized Euler-Lagrange equation for nonsmooth calculus of variations. Nonlinear Dyn. 75 (2014) 85–100. [Google Scholar]
  80. R. Vinter and H. Zheng, The extended Euler-Lagrange condition for nonconvex variational problems. SIAM J. Control Optim. 35 (1997) 56–77. [Google Scholar]
  81. R.B. Vinter, Optimal Control. Birkhäuser, Boston (2000). [Google Scholar]
  82. A. Zaffaroni, Codifferentiable mappings with applications to vector optimality. Pilska Studia Mathematica Bulgarica 12 (1998) 255–266. [Google Scholar]
  83. A. Zaffaroni, Continuous approximations, codifferentiable functions and minimization methods, in Quasidifferentiability and related Topics, edited by V.F. Demyanov and A.M. Rubinov. Kluwer Academic Publishers, Dordrecht (2000) 361–391. [Google Scholar]

Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.

Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.

Initial download of the metrics may take a while.