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Positively Homogeneous Functions Revisited

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Abstract

The paper deals with positively homogeneous functions defined on a finite-dimensional space. Our attention is mainly focused on those subspaces of positively homogeneous functions that are important in nonsmooth analysis and optimization: the subspace of continuous positively homogeneous functions, of Lipschitz continuous positively homogeneous functions, of difference sublinear functions, and of piecewise linear functions. We reproduce some known results and present a number of new ones, in particular, those that concern Lipschitz continuous positively homogeneous functions.

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References

  1. Demyanov, V.F., Malozemov, V.N.: Introduction to Minimax. Wiley, New York (1974)

    Google Scholar 

  2. Demyanov, V.F.: Minimax: Directional Differentiability (in Russian). Leningrad University Press, Leningrad (1974)

    Google Scholar 

  3. Demyanov, V.F., Vasiliev, L.V.: Nondifferentiable Optimization. Springer, New York (1986)

    Google Scholar 

  4. Demyanov, V.F. (ed.): Nonsmooth Problems in the Theory of Optimization and Control (in Russian). Leningrad University Press, Leningrad (1982)

  5. Demyanov, V.F., Dixon, L.C.W. (eds.): Quasidifferential calculus. North-Holland, Amsterdam (1986)

  6. Demyanov, V.F., Rubinov, A.M.: Quasidifferential Calculus. Optimization Software, New York (1986)

    Book  MATH  Google Scholar 

  7. Demyanov, V.F., Rubinov, A.M.: Foundations of Nonsmooth Analysis and Quasidifferential Calculus (in Russian). Nauka, Moscow (1990)

    Google Scholar 

  8. Demyanov, V.F., Rubinov, A.M.: Constructive Nonsmooth Analysis. Verlag Peter Lang, Frankfurt am Main (1995)

    MATH  Google Scholar 

  9. Demyanov, V.F., Rubinov, A.M. (eds.): Quasidifferentiability and Related Topics. Kluwer Academic Publishers, Dordrecht (2000)

  10. Demyanov, V.F.: Exhausters of a positively homogeneous function. Optimization 45, 13–29 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  11. Bourbaki, N.: General Topology: Chapter 5–10. Springer, Berlin (1989)

    Book  MATH  Google Scholar 

  12. Rockafellar, R.T.: Convex analysis. Princeton University Press, Princeton (1970)

    Book  MATH  Google Scholar 

  13. Hiriart-Urruty, J.B., Lemaréchal, C.: Convex Analysis and Minimization Algorithms, Vol. I. Fundamentals. Springer, Berlin (1993)

    MATH  Google Scholar 

  14. Hiriart-Urruty, J.B., Lemaréchal, C.: Convex Analysis and Minimization Algorithms, Vol. II. Advanced Theory and Bundle Methods. Springer, Berlin (1993)

    MATH  Google Scholar 

  15. Kutateladze, S.S., Rubinov, A.M.: Minkowski duality and its applications. Russian Mathem. Surveys 27(3), 137–192 (1972)

    Article  MathSciNet  Google Scholar 

  16. Kutateladze, S.S., Rubinov, A.M.: Minkowski Duality and its Applications (in Russian). Nauka, Novosibirsk (1976)

    Google Scholar 

  17. Uderzo, A.: Convex approximators, convexificators and exhausters: applications to constrained extremum problems. In: Demyanov, V.F., Rubinov, A.M. (eds.) Quasidifferentiability and Related Topics. Nonconvex Optimization and Its Applications, vol. 43, pp. 297–327. Kluwer Academic, Dordrecht (2000)

    Google Scholar 

  18. Gorokhovik, V.V., Starovoitava, M.A.: Characteristic properties of primal exhausters for various classes of positively homogeneous functions (in Russian). Proceedings of the Institute of Mathematics. National Academy of Sciences of Belarus 19(2), 12–25 (2011)

    MATH  Google Scholar 

  19. Gorokhovik, V.V., Trafimovich, M.A.: On methods for converting exhausters of positively homogeneous functions. Optimization 65(3), 589–608 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  20. Demyanov, V.F.: Exhausters and convexificators - new tools in nonsmooth analysis. In: Demyanov, V.F., Rubinov, A.M. (eds.) Quasidifferentiability and Related Topics. Nonconvex Optimization and Its Applications, vol. 43, pp. 85–137. Kluwer Academic, Dordrecht (2000)

    Google Scholar 

  21. Roshchina, V.A.: Relationships between upper exhausters and the basic subdifferentials. J. Math. Anal. Appl. 334(1), 261–272 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  22. Roshchina, V.A.: Reducing exhausters. J. Optim Theory Appl. 136(2), 261–273 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  23. Demyanov, V.F., Roshchina, V.A.: Exhausters and subdifferentials in nonsmooth analysis. Optimization 57(1), 41–56 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  24. Grzybowski, J., Pallaschke, D., Urbański, R.: Reduction of finite exhausters. J. Glob. Optim. 46(4), 589–601 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  25. Demyanov, V.F., Ryabova, J.A.: Exhausters, coexhausters and convertors in nonsmooth analysis. Discrete Contin. Dyn. Syst. 31(4), 1273–1292 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  26. Demyanov, V.F.: Proper exhausters and coexhausters in nonsmooth analysis. Optimization 61(11), 1347–1368 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  27. Grzybowski, J., Küçük, M., Küçük, Y., Urbański, R.: On minimal representations by a family of sublinear functions. J. Glob. Optim. 61(2), 279–289 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  28. Kruger, A.Y.: On Fréchet subdifferentials. J. Math. Sci. 116(3), 3325–3358 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  29. Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)

    MATH  Google Scholar 

  30. Michel, P., Penot, J.-P.: Calcul sous-différentiel pour des fonctions lipschitziennes et non-lipschitziennes. Comptes Rendus de l’Académie des Sciences de Paris 298, 684–687 (1984)

    MathSciNet  MATH  Google Scholar 

  31. Michel, P., Penot, J.-P.: A generalized derivative for calm and stable functions. Differ. Integral Equ. 5(2), 189–196 (1992)

    MathSciNet  MATH  Google Scholar 

  32. Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, Vol. I: Basic Theory. Springer, Berlin (2005)

    MATH  Google Scholar 

  33. Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, Vol. II: Applications. Springer, Berlin (2005)

    MATH  Google Scholar 

  34. Demyanov, V.F., Roshchina, V.A.: Optimality conditions in terms of upper and lower exhausters. Optimization 55(5–6), 525–540 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  35. Demyanov, V.F., Roshchina, V.A.: Exhausters, optimality conditions and related problems. J. Glob. Optim. 40(1), 71–85 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  36. Abbasov, M.E., Demyanov, V.F.: Proper and adjoint exhausters in nonsmooth analysis: optimality conditions. J. Glob. Optim. 56(2), 569–585 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  37. Abbasov, M.E., Demyanov, V.F.: Adjoint coexhausters in nonsmooth analysis and extremality conditions. J. Optim. Theory Appl. 156(3), 535–553 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  38. Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, Berlin (1998)

    Book  MATH  Google Scholar 

  39. Gorokhovik, V.V., Gorokhovik, S.Ya.: A criterion of the global epi-Lipschitzian property of sets (in Russian). Izvestiya Natsionalnoj Akademii Nauk Belarusi. Seriya Fiziko-Matematicheskikh Nauk, Issue 1, 118–120 (1995)

  40. Tammer, Chr, Zălinescu, C.: Lipschitz properties of the scalarization function and applications. Optimization 59(2), 305–319 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  41. Aliprantis, ChD, Border, K.C.: Infinite Dimensional Analysis. Springer, Berlin (2006)

    MATH  Google Scholar 

  42. Bauschke, H.H., Combetters, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer, New York (2011)

    Book  Google Scholar 

  43. Stein, J.D.: Functions satisfying lipschitz conditions. Michigan Math. J. 16(4), 385–396 (1968)

    MathSciNet  MATH  Google Scholar 

  44. Castellani, M.: A dual representation for proper positively homogeneous functions. J. Glob. Optim. 16(4), 393–400 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  45. Castellani, M.: Dual representation of classes of positively homogeneous functions. In: Demyanov, V.F., Rubinov, A.M. (eds.) Quasidifferentiability and Related Topics. Nonconvex Optimization and Its Applications, vol. 43, pp. 73–84. Kluwer Academic, Dordrecht (2000)

    Google Scholar 

  46. Castellani, M., Uderzo, A.: Dual representations for positively homogeneous functions: an application to constrained extremum problems. In: Crespi, G.P. et al. (eds.) Optimization in Economics, Finance and Industry. Proceedings of the Workshop, held in Verona, June14/15th 2001, pp. 129–143. Datanova (2001)

  47. Demyanov, V.F., Rubinov, A.M.: On quasidifferentiable functionals (in Russian). Doklady USSR Acad. Sci. 250(1), 21–25 (1980)

    MathSciNet  MATH  Google Scholar 

  48. Demyanov, V.F., Polyakova, L.N.: Minimization of a quasi-differentiable function in a quasi-differentiable set. USSR Comput. Math. Math. Phys. 20(4), 34–43 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  49. Polyakova, L.N.: Necessary conditions for extremum od quasidifferentiable functions (in Russian). Vestnik Leningradskogo Univ. 13, 57–62 (1980)

    MathSciNet  MATH  Google Scholar 

  50. Polyakova, L.N.: On global properties of d.c. functions. In: Gilbert, R.P., Panagiotopoulos, P.D., Pardalos, P.M. (eds.) From Convexity to Nonconvexity. Nonconvex Optimization and Its Applications, vol. 55, pp. 223–231. Kluwer Academic Publishers, Dordrecht (2001)

    Chapter  Google Scholar 

  51. Gorokhovik, V.V.: On quasidifferentiability of real-valued functions (in Russian). Doklady USSR Acad. Sci. 266(6), 1294–1298 (1982)

    MathSciNet  MATH  Google Scholar 

  52. Gorokhovik, V.V.: Quasidifferentiability of real-valued functions and local extremum conditions. Sibirian Math. J. 25(3), 62–70 (1984)

    MathSciNet  MATH  Google Scholar 

  53. Gorokhovik, V.V.: \(\varepsilon \)-Quasidifferentiability of real-valued functions and conditions in extremal problems. In: Demyanov, V.F., Dixon, L.C.W. (eds.) Quasidifferential Calculus. Mathematical Programming Studies, vol. 29, pp. 203–218. North-Holland, Amsterdam (1986)

    Chapter  Google Scholar 

  54. Gorokhovik, V.V.: Convex and non-smooth vector optimization problem (in Russian). Navuka i Tekhnika, Minsk (1990)

    MATH  Google Scholar 

  55. Pallaschke, D., Urbański, R.: Minimal pairs of convex sets, with applications to quasidifferential calculus. In: Demyanov, V.F., Rubinov, A.M. (eds.) Quasidifferentiability and Related Topics. Nonconvex Optimization and Its Applications, vol. 43, pp. 173–213. Kluwer Academic, Dordrecht (2000)

    Google Scholar 

  56. Pallaschke, D., Recht, P., Urbański, R.: On locally Lipschitz quasidifferentiable functions in Banach spaces. Optimization 17(3), 287–295 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  57. Pallaschke, D., Rolewicz, S.: Foundations of Mathematical Optimization. Kluwer Academic Publisher, Dordrecht (1997)

    Book  MATH  Google Scholar 

  58. Pallaschke, D., Urbański, R.: Reduction of quasidifferentials and minimal representations. Math. Program. 66(1), 161–180 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  59. Pallaschke, D., Urbański, R.: Pairs of Compact Convex Sets, Fractional Arithmetic with Convex Sets. Kluwer Acadamic Publisher, Dordrecht (2002)

    Book  MATH  Google Scholar 

  60. Uderzo, A.: Quasi-multiplier rules for quasidifferentiable extremum problems. Optimization 51(6), 761–795 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  61. Uderzo, A.: Convex difference criteria for the quantitative stability of parametric quasidifferentiable systems. Set-Valued Anal. 15(1), 81–104 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  62. Kripfhanz, A., Schulze, R.: Piecewise-affine functions as the difference of two convex functions. Optimization 29(1), 23–29 (1987)

    Article  MathSciNet  Google Scholar 

  63. Melzer, D.: On the expressibility of piecewise linear continuous functions as the difference of two piecewise linear convex functions. In: Demyanov, V.F., Dixon, L.C.W. (eds.) Quasidifferential Calculus. Mathematical Programming Studies, vol. 29, pp. 118–134. North-Holland, Amsterdam (1986)

    Chapter  Google Scholar 

  64. Gorokhovik, V.V., Zorko, O.I.: Piecewise affine functions and polyhedral sets. Optimization 31(2), 209–221 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  65. Shapiro, A.: Quasidifferential calculus and first-order optimality conditions in nonsmooth optimization. In: Demyanov, V.F., Dixon, L.C.W. (eds.) Quasidifferential Calculus. Mathematical Programming Studies, vol. 29, pp. 59–68. North-Holland, Amsterdam (1986)

    Google Scholar 

  66. Alexandrov, A.D.: Surfaces represented as the difference of convex functions (in Russian). USSR Acad. Sci. Doklady 74, 613–616 (1950)

    Google Scholar 

  67. Roberts, A.W., Varberg, D.E.: Convex Functions. Academic Press Inc., New York (1973)

    MATH  Google Scholar 

  68. Hiriart-Urruty, J.-B.: Generalized differentiability, duality and optimization for problems dealing with differences of convex functions. In: Ponstein, J. (ed.) Convexity and Duality in Optimization. Lecture Notes in Economics and Mathematical Systems, vol. 256, pp. 37–70. Springer, Berlin (1985)

    Chapter  Google Scholar 

  69. Tuy, H.: Convex Analysis and Global Optimization. Kluwer Academic Publisher, Dordrecht (1998)

    Book  MATH  Google Scholar 

  70. Vesely, L., Zajiček, L.: Delta-convex mappings between Banach spaces and applications. Dissertationes Mathematicae. Vol. 289. Instytut Matematyczny Polskiej Akademii Nauk, Warszawa (1989)

  71. Bačák, M., Borwein, J.M.: On difference convexity of locally Lipschitz functions. Optimization 60(8–9), 961–978 (2011)

    MathSciNet  MATH  Google Scholar 

  72. Ha, T.X.D.: Banach spaces of d.c. functions and quasidifferentiable functions. Acta Math. Vietnam. 13(2), 55–70 (1988)

    MathSciNet  MATH  Google Scholar 

  73. Cegrell, U.: On the space of delta-convex functions and its dual. BUL. MATH. de la Soc. Sci. Math. de la R. S. de Roumaine 22(70), 133–139 (1978)

    MathSciNet  Google Scholar 

  74. Bartels, S.G., Pallaschke, D.: Some remarks on the space of differences of sublinear functions. Appl. Math. 22(3), 419–426 (1994)

    MathSciNet  MATH  Google Scholar 

  75. Polovinkin, E.S.: Set-Valued Analysis and Differential Inclusions (in Russian). Fizmatlit, Moscow (2014)

    Google Scholar 

  76. Grzybowski, J., Pallaschke, D., Urbański, R.: Demyanov difference in infinite dimensional spaces. In: Demyanov, V.F., Pardalos, P.M., Batsyn, M. (eds.) Constructive Nonsmooth Analysis and Related Topics. Springer Optimization and Its Applications, vol. 87, pp. 13–24. Springer, New York (2014)

    Google Scholar 

  77. Diamond, P., Kloeden, P., Rubinov, A., Vladimirov, A.: Three metrics in the space of compact convex sets. Research Report 4/97, School of Information Technology and Mathematical Sciences, University of Ballarat (1997)

  78. Diamond, P., Kloeden, P., Rubinov, A., Vladimirov, A.: Comparative properties of three metrics in the space of compact convex sets. Set-Valued Anal. 5(3), 267–289 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  79. Rzezuchowski, T.: The Demyanov metric and some other metrics in the family of convex sets. Cent. Eur. J. Math. 10(6), 2229–2239 (2012)

    MathSciNet  MATH  Google Scholar 

  80. Grzybowski, J., Przybycień, H., Urbański, R.: Decomposition of Minkowski-Rådström-Hörmander space to the direct sum of symmetric and asymmetric subspaces. Set-Valued Variational Anal. 21, 201–216 (2013)

    Article  MATH  Google Scholar 

  81. Bank, B., Guddat, J., Klatte, D., Kummer, B., Tammer, K.: Non-linear Parametric Optimization. Birkhauser, Basel (1983)

    MATH  Google Scholar 

  82. Robinson, S.: Some continuity properties of polyhedral multifunctions. In: König, H., Korte, B., Ritter, K. (eds.) Mathematical Programming at Oberwolfach. Mathematical Programming Studies, vol. 14, pp. 206–214. Springer, Berlin (1981)

    Chapter  Google Scholar 

  83. Aliprantis, ChD, Tourky, R.: Cones and Duality. American Mathematical Society, Providence (2007)

    Book  MATH  Google Scholar 

  84. Ovchinnikov, S.: Max-Min representation of piecewise linear functions. Contrib. Algebra Geom. 43(1), 297–302 (2002)

    MathSciNet  MATH  Google Scholar 

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Acknowledgments

The authors wish to acknowledge the very thorough refereeing of the article and express their gratitude to the two anonymous referees for valuable and helpful comments and suggestions.

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  1. Institute of Mathematics, National Academy of Sciences of Belarus, ul. Surganova, 11, 220072, Minsk, Belarus

    Valentin V. Gorokhovik

  2. Faculty of Mathematics and Mechanics, Belorussian State University, Nezavisimosty Avenue, 4, 220030, Minsk, Belarus

    Marina Trafimovich

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  1. Valentin V. Gorokhovik
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Gorokhovik, V.V., Trafimovich, M. Positively Homogeneous Functions Revisited. J Optim Theory Appl 171, 481–503 (2016). https://doi.org/10.1007/s10957-016-0891-4

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