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1994 | 110 | 3 | 271-282
Tytuł artykułu

The cancellation law for inf-convolution of convex functions

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EN
Abstrakty
EN
Conditions under which the inf-convolution of f and g $f □ g(x):= inf_{y+z=x}(f(y)+g(z))$ has the cancellation property (i.e. f □ h ≡ g □ h implies f ≡ g) are treated in a convex analysis framework. In particular, we show that the set of strictly convex lower semicontinuous functions $f: X → ℝ ∪ {+∞}$ on a reflexive Banach space such that $ lim_{∥x∥ → ∞} f(x)/∥x∥ = ∞$ constitutes a semigroup, with inf-convolution as multiplication, which can be embedded in the group of its quotients.
Twórcy
  • Technical University of Łódź, Żwirki 36, 90-924 Łódź, Poland
Bibliografia
  • [1] H. Attouch, Varriational Convergence for Functions and Operators, Pitman Adv. Publ. Program, Boston, 1984.
  • [2] J.-P. Aubin, Optima and Equilibria. An Introduction to Nonlinear Analysis, Springer, Berlin, 1993.
  • [3] J.-P. Aubin and H. Frankowska, Set-Valued Analysis, Birkhäuser, Boston, 1990.
  • [4] F. H. Clarke, Optimization and Nonsmooth Analysis, Wiley Interscience, New York, 1983.
  • [5] R. Correa, A. Jofré and L. Thibault, Characterization of lower semicontinuous convex functions, Proc. Amer. Math. Soc. 116 (1992), 67-72.
  • [6] K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, 1985.
  • [7] R. B. Holmes, Geometric Functional Analysis and its Applications, Springer, New York, 1975.
  • [8] A. D. Ioffe and V. M. Tihomirov, Theory of Extremal Problems, North-Holland, Amsterdam, 1979.
  • [9] R. R. Phelps, Convex Functions, Monotone Operators and Differentiability, Springer, Berlin, 1989.
  • [10] C. Pontini, Solving in the affirmative a conjecture about a limit of gradients, J. Optim. Theory Appl. 70 (1991), 623-629.
  • [11] T. Rockafellar, Directionally Lipschitzian functions and subdifferential calculus, Proc. London Math. Soc. 39 (1979), 331-355.
  • [12] T. Rockafellar, Generalized directional derivatives and subgradients of nonconvex functions, Canad. J. Math. 32 (1980), 257-280.
  • [13] T. Rockafellar, On a special class of convex functions, J. Optim. Theory Appl. 70 (1991), 619-621.
  • [14] T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, 1970.
  • [15] L. Thibault and D. Zagrodny, Integration of subdifferentials of lower semicontinuous functions on Banach spaces, J. Math. Anal. Appl., to appear.
  • [16] D. Zagrodny, Approximate mean value theorem for upper subderivatives, Nonlinear Anal. 12 (1988), 1413-1428.
  • [17] D. Zagrodny, An example of bad convex function, J. Optim. Theory Appl. 70 (1991), 631-637.
Typ dokumentu
Bibliografia
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