American Institute of Mathematical Sciences

Advanced
April  2011, 29(2): 623-646. doi: 10.3934/dcds.2011.29.623

$V$-Jacobian and $V$-co-Jacobian for Lipschitzian maps

Zsolt Páles 1, and  Vera Zeidan 2,

1. 

Institute of Mathematics, University of Debrecen, H-4010 Debrecen, Pf. 12, Hungary
2. 

Department of Mathematics, Michigan State University, East Lansing, MI 48824, United States

Received  September 2009 Revised  March 2010 Published  October 2010

The notions of $V$-Jacobian and $V$-co-Jacobian are introduced for locally Lipschitzian functions acting between arbitrary normed spaces $X$ and $Y$, where $V$ is a subspace of the dual space $Y^*$. The main results of this paper provide a characterization, calculus rules and also the computation of these Jacobians of piecewise smooth functions.
Keywords: Sum rule, Characterization theorem, Co-Jacobian, Generalized Jacobian, Continuous selection., Chain rule.
Mathematics Subject Classification: 49J52, 49A52, 58C2.
Citation: Zsolt Páles, Vera Zeidan. $V$-Jacobian and $V$-co-Jacobian for Lipschitzian maps. Discrete & Continuous Dynamical Systems - A, 2011, 29 (2) : 623-646. doi: 10.3934/dcds.2011.29.623
References:
[1]

N. Aronszajn, Differentiability of Lipschitzian mappings between Banach spaces,, Studia Math., 57 (1976), 147. Google Scholar

[2]

J. P. R. Christensen, Measure theoretic zero sets in infinite dimensional spaces and applications to differentiability of Lipschitz mappings,, Publ. Dép. Math. (Lyon), 10 (1973), 29. Google Scholar

[3]

F. H. Clarke, On the inverse function theorem,, Pacific J. Math., 64 (1976), 97. Google Scholar

[4]

F. H. Clarke, "Optimization and Nonsmooth Analysis,", John Wiley & Sons, (1983). Google Scholar

[5]

H. Halkin, Interior mapping theorem with set-valued derivatives,, J. Analyse Math., 30 (1976), 200. doi: doi:10.1007/BF02786714. Google Scholar

[6]

H. Halkin, Mathematical programming without differentiability,, in, (1976), 279. Google Scholar

[7]

A. D. Ioffe, Nonsmooth analysis: Differential calculus of nondifferentiable mappings,, Trans. Amer. Math. Soc., 266 (1981), 1. Google Scholar

[8]

H. Th. Jongen and D. Pallaschke, On linearization and continuous selections of functions,, Optimization, 19 (1988), 343. doi: doi:10.1080/02331938808843350. Google Scholar

[9]

S. Kaplan, On the second dual of the space of continuous functions,, Trans. Amer. Math. Soc., 86 (1957), 70. Google Scholar

[10]

D. Klatte and B. Kummer, Nonsmooth equations in optimization,, in, 60 (2002). Google Scholar

[11]

L. Kuntz and S. Scholtes, Structural analysis of nonsmooth mappings, inverse functions, and metric projections,, J. Math. Anal. Appl., 188 (1994), 346. doi: doi:10.1006/jmaa.1994.1431. Google Scholar

[12]

L. Kuntz and S. Scholtes, Qualitative aspects of the local approximation of a piecewise differentiable function,, Nonlinear Anal., 25 (1995), 197. doi: doi:10.1016/0362-546X(94)00202-S. Google Scholar

[13]

G. Lebourg, Generic differentiability of Lipschitzian functions,, Trans. Amer. Math. Soc., 256 (1979), 125. Google Scholar

[14]

B. S. Mordukhovich, Metric approximations and necessary conditions for optimality for general classes of nonsmooth extremal problems,, Dokl. Akad. Nauk SSSR, 254 (1980), 1072. Google Scholar

[15]

B. S. Mordukhovich, Generalized differential calculus for nonsmooth and set-valued mappings,, J. Math. Anal. Appl., 183 (1994), 250. doi: doi:10.1006/jmaa.1994.1144. Google Scholar

[16]

B. S. Mordukhovich, Coderivatives of set-valued mappings: Calculus and applications,, proceedings of the, 30 (1997), 3059. Google Scholar

[17]

B. S. Mordukhovich, "Variational Analysis and Generalized Differentiation, I. Basic Theory,", Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 330 (2006). Google Scholar

[18]

J.-S. Pang and D. Ralph, Piecewise smoothness, local invertibility, and parametric analysis of normal maps,, Math. Oper. Res., 21 (1996), 401. doi: doi:10.1287/moor.21.2.401. Google Scholar

[19]

Zs. Páles and V. Zeidan, Generalized Jacobian for functions with infinite dimensional range and domain,, Set-Valued Anal., 15 (2007), 331. doi: doi:10.1007/s11228-007-0043-y. Google Scholar

[20]

Zs. Páles and V. Zeidan, Infinite dimensional Clarke generalized Jacobian,, J. Convex Anal., 14 (2007), 433. Google Scholar

[21]

Zs. Páles and V. Zeidan, Infinite dimensional generalized Jacobian: Properties and calculus rules,, J. Math. Anal. Appl., 344 (2008), 55. doi: doi:10.1016/j.jmaa.2008.02.044. Google Scholar

[22]

Zs. Páles and V. Zeidan, The core of the infinite dimensional generalized Jacobian,, J. Convex Anal., 16 (2009), 321. Google Scholar

[23]

Zs. Páles and V. Zeidan, Co-Jacobian for Lipschitzian maps,, Set-Valued and Variational Anal., 18 (2010), 57. doi: doi:10.1007/s11228-009-0130-3. Google Scholar

[24]

D. Ralph and S. Scholtes, Sensitivity analysis of composite piecewise smooth equations,, Math. Programming Ser. B, 76 (1997), 593. doi: doi:10.1007/BF02614400. Google Scholar

[25]

D. Ralph and H. Xu, Implicit smoothing and its application to optimization with piecewise smooth equality constraints,, J. Optim. Theory Appl., 124 (2005), 673. doi: doi:10.1007/s10957-004-1180-1. Google Scholar

[26]

R. T. Rockafellar, A property of piecewise smooth functions,, Comput. Optim. Appl., 25 (2003), 247. doi: doi:10.1023/A:1022921624832. Google Scholar

[27]

S. Scholtes, "Introduction to Piecewise Differentiable Equations,", Habilitation thesis, (1994). Google Scholar

[28]

T. H. Sweetser, A minimal set-valued strong derivative for vector-valued Lipschitz functions,, J. Optimization Theory Appl., 23 (1977), 549. doi: doi:10.1007/BF00933296. Google Scholar

[29]

L. Thibault, Subdifferentials of compactly Lipschitzian vector-valued functions,, Ann. Mat. Pura Appl. (4), 125 (1980), 157. doi: doi:10.1007/BF01789411. Google Scholar

[30]

L. Thibault, On generalized differentials and subdifferentials of Lipschitz vector-valued functions,, Nonlinear Anal., 6 (1982), 1037. doi: doi:10.1016/0362-546X(82)90074-8. Google Scholar

[31]

J. Warga, Derivative containers, inverse functions, and controllability,, in, (1976), 13. Google Scholar

[32]

J. Warga, Fat homeomorphisms and unbounded derivate containers,, J. Math. Anal. Appl., 81 (1981), 545. Google Scholar

show all references

References:
[1]

N. Aronszajn, Differentiability of Lipschitzian mappings between Banach spaces,, Studia Math., 57 (1976), 147. Google Scholar

[2]

J. P. R. Christensen, Measure theoretic zero sets in infinite dimensional spaces and applications to differentiability of Lipschitz mappings,, Publ. Dép. Math. (Lyon), 10 (1973), 29. Google Scholar

[3]

F. H. Clarke, On the inverse function theorem,, Pacific J. Math., 64 (1976), 97. Google Scholar

[4]

F. H. Clarke, "Optimization and Nonsmooth Analysis,", John Wiley & Sons, (1983). Google Scholar

[5]

H. Halkin, Interior mapping theorem with set-valued derivatives,, J. Analyse Math., 30 (1976), 200. doi: doi:10.1007/BF02786714. Google Scholar

[6]

H. Halkin, Mathematical programming without differentiability,, in, (1976), 279. Google Scholar

[7]

A. D. Ioffe, Nonsmooth analysis: Differential calculus of nondifferentiable mappings,, Trans. Amer. Math. Soc., 266 (1981), 1. Google Scholar

[8]

H. Th. Jongen and D. Pallaschke, On linearization and continuous selections of functions,, Optimization, 19 (1988), 343. doi: doi:10.1080/02331938808843350. Google Scholar

[9]

S. Kaplan, On the second dual of the space of continuous functions,, Trans. Amer. Math. Soc., 86 (1957), 70. Google Scholar

[10]

D. Klatte and B. Kummer, Nonsmooth equations in optimization,, in, 60 (2002). Google Scholar

[11]

L. Kuntz and S. Scholtes, Structural analysis of nonsmooth mappings, inverse functions, and metric projections,, J. Math. Anal. Appl., 188 (1994), 346. doi: doi:10.1006/jmaa.1994.1431. Google Scholar

[12]

L. Kuntz and S. Scholtes, Qualitative aspects of the local approximation of a piecewise differentiable function,, Nonlinear Anal., 25 (1995), 197. doi: doi:10.1016/0362-546X(94)00202-S. Google Scholar

[13]

G. Lebourg, Generic differentiability of Lipschitzian functions,, Trans. Amer. Math. Soc., 256 (1979), 125. Google Scholar

[14]

B. S. Mordukhovich, Metric approximations and necessary conditions for optimality for general classes of nonsmooth extremal problems,, Dokl. Akad. Nauk SSSR, 254 (1980), 1072. Google Scholar

[15]

B. S. Mordukhovich, Generalized differential calculus for nonsmooth and set-valued mappings,, J. Math. Anal. Appl., 183 (1994), 250. doi: doi:10.1006/jmaa.1994.1144. Google Scholar

[16]

B. S. Mordukhovich, Coderivatives of set-valued mappings: Calculus and applications,, proceedings of the, 30 (1997), 3059. Google Scholar

[17]

B. S. Mordukhovich, "Variational Analysis and Generalized Differentiation, I. Basic Theory,", Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 330 (2006). Google Scholar

[18]

J.-S. Pang and D. Ralph, Piecewise smoothness, local invertibility, and parametric analysis of normal maps,, Math. Oper. Res., 21 (1996), 401. doi: doi:10.1287/moor.21.2.401. Google Scholar

[19]

Zs. Páles and V. Zeidan, Generalized Jacobian for functions with infinite dimensional range and domain,, Set-Valued Anal., 15 (2007), 331. doi: doi:10.1007/s11228-007-0043-y. Google Scholar

[20]

Zs. Páles and V. Zeidan, Infinite dimensional Clarke generalized Jacobian,, J. Convex Anal., 14 (2007), 433. Google Scholar

[21]

Zs. Páles and V. Zeidan, Infinite dimensional generalized Jacobian: Properties and calculus rules,, J. Math. Anal. Appl., 344 (2008), 55. doi: doi:10.1016/j.jmaa.2008.02.044. Google Scholar

[22]

Zs. Páles and V. Zeidan, The core of the infinite dimensional generalized Jacobian,, J. Convex Anal., 16 (2009), 321. Google Scholar

[23]

Zs. Páles and V. Zeidan, Co-Jacobian for Lipschitzian maps,, Set-Valued and Variational Anal., 18 (2010), 57. doi: doi:10.1007/s11228-009-0130-3. Google Scholar

[24]

D. Ralph and S. Scholtes, Sensitivity analysis of composite piecewise smooth equations,, Math. Programming Ser. B, 76 (1997), 593. doi: doi:10.1007/BF02614400. Google Scholar

[25]

D. Ralph and H. Xu, Implicit smoothing and its application to optimization with piecewise smooth equality constraints,, J. Optim. Theory Appl., 124 (2005), 673. doi: doi:10.1007/s10957-004-1180-1. Google Scholar

[26]

R. T. Rockafellar, A property of piecewise smooth functions,, Comput. Optim. Appl., 25 (2003), 247. doi: doi:10.1023/A:1022921624832. Google Scholar

[27]

S. Scholtes, "Introduction to Piecewise Differentiable Equations,", Habilitation thesis, (1994). Google Scholar

[28]

T. H. Sweetser, A minimal set-valued strong derivative for vector-valued Lipschitz functions,, J. Optimization Theory Appl., 23 (1977), 549. doi: doi:10.1007/BF00933296. Google Scholar

[29]

L. Thibault, Subdifferentials of compactly Lipschitzian vector-valued functions,, Ann. Mat. Pura Appl. (4), 125 (1980), 157. doi: doi:10.1007/BF01789411. Google Scholar

[30]

L. Thibault, On generalized differentials and subdifferentials of Lipschitz vector-valued functions,, Nonlinear Anal., 6 (1982), 1037. doi: doi:10.1016/0362-546X(82)90074-8. Google Scholar

[31]

J. Warga, Derivative containers, inverse functions, and controllability,, in, (1976), 13. Google Scholar

[32]

J. Warga, Fat homeomorphisms and unbounded derivate containers,, J. Math. Anal. Appl., 81 (1981), 545. Google Scholar

[1]

Isabelle Déchène. On the security of generalized Jacobian cryptosystems. Advances in Mathematics of Communications, 2007, 1 (4) : 413-426. doi: 10.3934/amc.2007.1.413
[2]

Shingo Takeuchi. The basis property of generalized Jacobian elliptic functions. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2675-2692. doi: 10.3934/cpaa.2014.13.2675
[3]

Jingzhi Tie, Qing Zhang. An optimal mean-reversion trading rule under a Markov chain model. Mathematical Control & Related Fields, 2016, 6 (3) : 467-488. doi: 10.3934/mcrf.2016012
[4]

Pedro Teixeira. Dacorogna-Moser theorem on the Jacobian determinant equation with control of support. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 4071-4089. doi: 10.3934/dcds.2017173
[5]

Bernard Dacorogna, Olivier Kneuss. Multiple Jacobian equations. Communications on Pure & Applied Analysis, 2014, 13 (5) : 1779-1787. doi: 10.3934/cpaa.2014.13.1779
[6]

Kevin Kuo, Phong Luu, Duy Nguyen, Eric Perkerson, Katherine Thompson, Qing Zhang. Pairs trading: An optimal selling rule. Mathematical Control & Related Fields, 2015, 5 (3) : 489-499. doi: 10.3934/mcrf.2015.5.489
[7]

Piotr Jaworski, Marcin Pitera. The 20-60-20 rule. Discrete & Continuous Dynamical Systems - B, 2016, 21 (4) : 1149-1166. doi: 10.3934/dcdsb.2016.21.1149
[8]

Andrew D. Lewis, David R. Tyner. Geometric Jacobian linearization and LQR theory. Journal of Geometric Mechanics, 2010, 2 (4) : 397-440. doi: 10.3934/jgm.2010.2.397
[9]

Ronen Peretz, Nguyen Van Chau, L. Andrew Campbell, Carlos Gutierrez. Iterated images and the plane Jacobian conjecture. Discrete & Continuous Dynamical Systems - A, 2006, 16 (2) : 455-461. doi: 10.3934/dcds.2006.16.455
[10]

Neil S. Trudinger. On the local theory of prescribed Jacobian equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (4) : 1663-1681. doi: 10.3934/dcds.2014.34.1663
[11]

Caifang Wang, Tie Zhou. The order of convergence for Landweber Scheme with $\alpha,\beta$-rule. Inverse Problems & Imaging, 2012, 6 (1) : 133-146. doi: 10.3934/ipi.2012.6.133
[12]

Regina S. Burachik, C. Yalçın Kaya. An update rule and a convergence result for a penalty function method. Journal of Industrial & Management Optimization, 2007, 3 (2) : 381-398. doi: 10.3934/jimo.2007.3.381
[13]

Andreas Asheim, Alfredo Deaño, Daan Huybrechs, Haiyong Wang. A Gaussian quadrature rule for oscillatory integrals on a bounded interval. Discrete & Continuous Dynamical Systems - A, 2014, 34 (3) : 883-901. doi: 10.3934/dcds.2014.34.883
[14]

Hoi Tin Kong, Qing Zhang. An optimal trading rule of a mean-reverting asset. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1403-1417. doi: 10.3934/dcdsb.2010.14.1403
[15]

Jingzhi Tie, Qing Zhang. Switching between a pair of stocks: An optimal trading rule. Mathematical Control & Related Fields, 2018, 8 (3&4) : 965-999. doi: 10.3934/mcrf.2018042
[16]

Vladimir Georgiev, Koichi Taniguchi. On fractional Leibniz rule for Dirichlet Laplacian in exterior domain. Discrete & Continuous Dynamical Systems - A, 2019, 39 (2) : 1101-1115. doi: 10.3934/dcds.2019046
[17]

João Correia-da-Silva, Joana Pinho. The profit-sharing rule that maximizes sustainability of cartel agreements. Journal of Dynamics & Games, 2016, 3 (2) : 143-151. doi: 10.3934/jdg.2016007
[18]

Ayla Sayli, Ayse Oncu Sarihan. Statistical query-based rule derivation system by backward elimination algorithm. Discrete & Continuous Dynamical Systems - S, 2015, 8 (6) : 1341-1356. doi: 10.3934/dcdss.2015.8.1341
[19]

Frederik Riis Mikkelsen. A model based rule for selecting spiking thresholds in neuron models. Mathematical Biosciences & Engineering, 2016, 13 (3) : 569-578. doi: 10.3934/mbe.2016008
[20]

Shiliang Weng, Xiang Zhang. Integrability of vector fields versus inverse Jacobian multipliers and normalizers. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 6539-6555. doi: 10.3934/dcds.2016083

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (9)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences