On Minimax Theorems for Lower Semicontinuous Functions in Hilbert Spaces

Ewa Bednarczuk; Monika Syga

doi:10.68381/jca25027

Abstract

We prove minimax theorems for lower semicontinuous functions defined on a Hilbert space. The main tools are the theory of Φ-convex functions and sufficient and necessary conditions for the minimax equality for general Φ-convex functions. The conditions we propose are expressed in terms of abstract Φ-subgradients.

Author information

Contact details are reproduced from the original publication and may be historical.

Ewa Bednarczuk

System Research Institute, Polish Academy of Sciences, Newelska 6, 01-447 Warsaw, Poland
and: Faculty of Mathematics and Information Science, Warsaw University of Technology, ul. Koszykowa 75, 00-662 Warsaw, Poland

Ewa.Bednarczuk@ibspan.waw.pl

Monika Syga

Faculty of Mathematics and Information Science, Warsaw University of Technology, ul. Koszykowa 75, 00-662 Warsaw, Poland

M.Syga@mini.pw.edu.pl

Keywords

Abstract convexity
minimax theorems
intersection property
abstract Phi-subdifferential
abstract Phi-subgradient

Mathematics Subject Classification

32F17, 49J52, 49K27, 49K35, 52A01

References

E. M. Bednarczuk, M. Syga: Minimax theorems for Φ-convex functions with applications, Control and Cybernetics 43 (2014) 421–437.Google Scholar
F. Bernard, L. Thibault: Prox-regular functions in Hilbert spaces, J. Math. Anal. Appl. 303 (2005) 1–14.https://doi.org/10.1016/j.jmaa.2004.06.003 Google Scholar
A. Brøndsted, R. T. Rockafellar: On the subdifferentiability of convex functions, Proc. Amer. Math. Soc. 16 (1965) 605–611.https://doi.org/10.1090/S0002-9939-1965-0178103-8 Google Scholar
S. Dolecki, S. Kurcyusz: On Φ-convexity in extremal problems, SIAM J. Control Optimization 16 (1978) 277–300.https://doi.org/10.1137/0316018 Google Scholar
I. Ekeland, R. Temam: Convex Analysis and Variational Problems, North-Holland, Amsterdam (1976).Google Scholar
K. Fan: Minimax theorems, Proc. Nat. Acad. Sci. 39 (1953) 42–47.https://doi.org/10.1073/pnas.39.1.42 Google Scholar
A. D. Ioffe, A. M. Rubinov: Abstract convexity and nonsmooth analysis: global aspects, Advances in Mathematical Economics 4, Springer, Tokyo (2002) 1–23.Google Scholar
A. D. Ioffe, V. M. Tihomirov: Theory of Extremal Problems, North-Holland, Amsterdam (1979).Google Scholar
D. Pallaschke, S. Rolewicz: Foundations of Mathematical Optimization, Kluwer Academic, Dordrecht (1997).https://doi.org/10.1007/978-94-017-1588-1 Google Scholar
R. T. Rockafellar, R. J.-B.Wets: Variational Analysis, Grundlehren der Mathematischen 317, Springer-Verlag, Berlin (1998).https://doi.org/10.1007/978-3-642-02431-3 Google Scholar
S. Rolewicz: On a globalization property, Applicationes Mathematicae 22(1) (1993) 69–73.https://doi.org/10.4064/am-22-1-69-73 Google Scholar
A. M. Rubinov: Abstract Convexity and Global Optimization, Kluwer Academic, Dordrecht (2000).https://doi.org/10.1007/978-1-4757-3200-9 Google Scholar
W. Schirotzek: Nonsmooth Analysis, Springer-Verlag, Berlin (2007).https://doi.org/10.1007/978-3-540-71333-3 Google Scholar
S. Simons: Minimax theorems and their proofs, in: Minimax and Applications, D.-Z. Du, P.M. Pardalos (eds.), Kluwer Academic Publishers (1995) 1–23.https://doi.org/10.1007/978-1-4613-3557-3_1 Google Scholar
E. Sharikov: Conditions for Global Minimum through Abstract Convexity, LAP LAMBERT Academic Publishing (2009).Google Scholar
M. Syga: Minimax Theorems for Φ-convex Functions with Applications to Optimization (Polish), PhD thesis, Warsaw (2015).Google Scholar
M. Syga: Minimax theorems for Φ-convex functions: sufficient and necessary conditions, Optimization 65(3) (2016) 635–649.https://doi.org/10.1080/02331934.2015.1062010 Google Scholar
M. Syga: Minimax theorems for extended real-valued Φ-convex functions, preprint: arXive:1606.08389 (2016).Google Scholar

Suggested citation

Published by Heldermann Verlag, 2018. Rights now held by Banach Press.

[1] E. M. Bednarczuk, M. Syga: Minimax theorems for Φ-convex functions with applications, Control and Cybernetics 43 (2014) 421–437.Google Scholar

[2] F. Bernard, L. Thibault: Prox-regular functions in Hilbert spaces, J. Math. Anal. Appl. 303 (2005) 1–14.https://doi.org/10.1016/j.jmaa.2004.06.003 Google Scholar

[3] A. Brøndsted, R. T. Rockafellar: On the subdifferentiability of convex functions, Proc. Amer. Math. Soc. 16 (1965) 605–611.https://doi.org/10.1090/S0002-9939-1965-0178103-8 Google Scholar

[4] S. Dolecki, S. Kurcyusz: On Φ-convexity in extremal problems, SIAM J. Control Optimization 16 (1978) 277–300.https://doi.org/10.1137/0316018 Google Scholar

[5] I. Ekeland, R. Temam: Convex Analysis and Variational Problems, North-Holland, Amsterdam (1976).Google Scholar

[6] K. Fan: Minimax theorems, Proc. Nat. Acad. Sci. 39 (1953) 42–47.https://doi.org/10.1073/pnas.39.1.42 Google Scholar

[7] A. D. Ioffe, A. M. Rubinov: Abstract convexity and nonsmooth analysis: global aspects, Advances in Mathematical Economics 4, Springer, Tokyo (2002) 1–23.Google Scholar

[8] A. D. Ioffe, V. M. Tihomirov: Theory of Extremal Problems, North-Holland, Amsterdam (1979).Google Scholar

[9] D. Pallaschke, S. Rolewicz: Foundations of Mathematical Optimization, Kluwer Academic, Dordrecht (1997).https://doi.org/10.1007/978-94-017-1588-1 Google Scholar

[10] R. T. Rockafellar, R. J.-B.Wets: Variational Analysis, Grundlehren der Mathematischen 317, Springer-Verlag, Berlin (1998).https://doi.org/10.1007/978-3-642-02431-3 Google Scholar

[11] S. Rolewicz: On a globalization property, Applicationes Mathematicae 22(1) (1993) 69–73.https://doi.org/10.4064/am-22-1-69-73 Google Scholar

[12] A. M. Rubinov: Abstract Convexity and Global Optimization, Kluwer Academic, Dordrecht (2000).https://doi.org/10.1007/978-1-4757-3200-9 Google Scholar

[13] W. Schirotzek: Nonsmooth Analysis, Springer-Verlag, Berlin (2007).https://doi.org/10.1007/978-3-540-71333-3 Google Scholar

[14] S. Simons: Minimax theorems and their proofs, in: Minimax and Applications, D.-Z. Du, P.M. Pardalos (eds.), Kluwer Academic Publishers (1995) 1–23.https://doi.org/10.1007/978-1-4613-3557-3_1 Google Scholar

[15] E. Sharikov: Conditions for Global Minimum through Abstract Convexity, LAP LAMBERT Academic Publishing (2009).Google Scholar

[16] M. Syga: Minimax Theorems for Φ-convex Functions with Applications to Optimization (Polish), PhD thesis, Warsaw (2015).Google Scholar

[17] M. Syga: Minimax theorems for Φ-convex functions: sufficient and necessary conditions, Optimization 65(3) (2016) 635–649.https://doi.org/10.1080/02331934.2015.1062010 Google Scholar

[18] M. Syga: Minimax theorems for extended real-valued Φ-convex functions, preprint: arXive:1606.08389 (2016).Google Scholar