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

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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

Abstract
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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 and Continuous Dynamical Systems, 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), 147-190.
[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), 29-39, Actes du Deuxième Colloque d'Analyse Fonctionnelle de Bordeaux (Univ. Bordeaux, 1973), I, 29-39.
[3]	F. H. Clarke, On the inverse function theorem, Pacific J. Math., 64 (1976), 97-102.
[4]	F. H. Clarke, "Optimization and Nonsmooth Analysis," John Wiley & Sons, Inc., New York, 1983.
[5]	H. Halkin, Interior mapping theorem with set-valued derivatives, J. Analyse Math., 30 (1976), 200-207. doi: doi:10.1007/BF02786714.
[6]	H. Halkin, Mathematical programming without differentiability, in "Calculus of Variations and Control Theory" (Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1975), Academic Press, New York, (1976), 279-287.
[7]	A. D. Ioffe, Nonsmooth analysis: Differential calculus of nondifferentiable mappings, Trans. Amer. Math. Soc., 266 (1981), 1-56.
[8]	H. Th. Jongen and D. Pallaschke, On linearization and continuous selections of functions, Optimization, 19 (1988), 343-353. doi: doi:10.1080/02331938808843350.
[9]	S. Kaplan, On the second dual of the space of continuous functions, Trans. Amer. Math. Soc., 86 (1957), 70-90.
[10]	D. Klatte and B. Kummer, Nonsmooth equations in optimization, in "Regularity, Calculus, Methods and Applications," Nonconvex Optimization and its Applications, vol. 60, Kluwer Academic Publishers, Dordrecht, 2002.
[11]	L. Kuntz and S. Scholtes, Structural analysis of nonsmooth mappings, inverse functions, and metric projections, J. Math. Anal. Appl., 188 (1994), 346-386. doi: doi:10.1006/jmaa.1994.1431.
[12]	L. Kuntz and S. Scholtes, Qualitative aspects of the local approximation of a piecewise differentiable function, Nonlinear Anal., 25 (1995), 197-215. doi: doi:10.1016/0362-546X(94)00202-S.
[13]	G. Lebourg, Generic differentiability of Lipschitzian functions, Trans. Amer. Math. Soc., 256 (1979), 125-144.
[14]	B. S. Mordukhovich, Metric approximations and necessary conditions for optimality for general classes of nonsmooth extremal problems, Dokl. Akad. Nauk SSSR, 254 (1980), 1072-1076.
[15]	B. S. Mordukhovich, Generalized differential calculus for nonsmooth and set-valued mappings, J. Math. Anal. Appl., 183 (1994), 250-288. doi: doi:10.1006/jmaa.1994.1144.
[16]	B. S. Mordukhovich, Coderivatives of set-valued mappings: Calculus and applications, proceedings of the "Second World Congress of Nonlinear Analysts," Part 5 (Athens, 1996), vol. 30 (1997), 3059-3070.
[17]	B. S. Mordukhovich, "Variational Analysis and Generalized Differentiation, I. Basic Theory," Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 330, Springer-Verlag, Berlin, 2006.
[18]	J.-S. Pang and D. Ralph, Piecewise smoothness, local invertibility, and parametric analysis of normal maps, Math. Oper. Res., 21 (1996), 401-426. doi: doi:10.1287/moor.21.2.401.
[19]	Zs. Páles and V. Zeidan, Generalized Jacobian for functions with infinite dimensional range and domain, Set-Valued Anal., 15 (2007), 331-375. doi: doi:10.1007/s11228-007-0043-y.
[20]	Zs. Páles and V. Zeidan, Infinite dimensional Clarke generalized Jacobian, J. Convex Anal., 14 (2007), 433-454.
[21]	Zs. Páles and V. Zeidan, Infinite dimensional generalized Jacobian: Properties and calculus rules, J. Math. Anal. Appl., 344 (2008), 55-75. doi: doi:10.1016/j.jmaa.2008.02.044.
[22]	Zs. Páles and V. Zeidan, The core of the infinite dimensional generalized Jacobian, J. Convex Anal., 16 (2009), 321-349.
[23]	Zs. Páles and V. Zeidan, Co-Jacobian for Lipschitzian maps, Set-Valued and Variational Anal., 18 (2010), 57-78. doi: doi:10.1007/s11228-009-0130-3.
[24]	D. Ralph and S. Scholtes, Sensitivity analysis of composite piecewise smooth equations, Math. Programming Ser. B, 76 (1997), 593-612. doi: doi:10.1007/BF02614400.
[25]	D. Ralph and H. Xu, Implicit smoothing and its application to optimization with piecewise smooth equality constraints, J. Optim. Theory Appl., 124 (2005), 673-699. doi: doi:10.1007/s10957-004-1180-1.
[26]	R. T. Rockafellar, A property of piecewise smooth functions, Comput. Optim. Appl., 25 (2003), 247-250, A tribute to Elijah (Lucien) Polak. doi: doi:10.1023/A:1022921624832.
[27]	S. Scholtes, "Introduction to Piecewise Differentiable Equations," Habilitation thesis, University of Karlsruhe, Karlsruhe, Germany, 1994.
[28]	T. H. Sweetser, A minimal set-valued strong derivative for vector-valued Lipschitz functions, J. Optimization Theory Appl., 23 (1977), 549-562. doi: doi:10.1007/BF00933296.
[29]	L. Thibault, Subdifferentials of compactly Lipschitzian vector-valued functions, Ann. Mat. Pura Appl. (4), 125 (1980), 157-192. doi: doi:10.1007/BF01789411.
[30]	L. Thibault, On generalized differentials and subdifferentials of Lipschitz vector-valued functions, Nonlinear Anal., 6 (1982), 1037-1053. doi: doi:10.1016/0362-546X(82)90074-8.
[31]	J. Warga, Derivative containers, inverse functions, and controllability, in "Calculus of Variations and Control Theory" (Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1975; dedicated to Laurence Chisholm Young on the occasion of his 70th birthday), Academic Press, New York, (1976), 13-45; errata, p. 46. Math. Res. Center, Univ. Wisconsin, Publ. No. 36.
[32]	J. Warga, Fat homeomorphisms and unbounded derivate containers, J. Math. Anal. Appl., 81 (1981), 545-560; (Errata: ibid. 82 (1982), 582-583).

show all references

References:

[1]	N. Aronszajn, Differentiability of Lipschitzian mappings between Banach spaces, Studia Math., 57 (1976), 147-190.
[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), 29-39, Actes du Deuxième Colloque d'Analyse Fonctionnelle de Bordeaux (Univ. Bordeaux, 1973), I, 29-39.
[3]	F. H. Clarke, On the inverse function theorem, Pacific J. Math., 64 (1976), 97-102.
[4]	F. H. Clarke, "Optimization and Nonsmooth Analysis," John Wiley & Sons, Inc., New York, 1983.
[5]	H. Halkin, Interior mapping theorem with set-valued derivatives, J. Analyse Math., 30 (1976), 200-207. doi: doi:10.1007/BF02786714.
[6]	H. Halkin, Mathematical programming without differentiability, in "Calculus of Variations and Control Theory" (Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1975), Academic Press, New York, (1976), 279-287.
[7]	A. D. Ioffe, Nonsmooth analysis: Differential calculus of nondifferentiable mappings, Trans. Amer. Math. Soc., 266 (1981), 1-56.
[8]	H. Th. Jongen and D. Pallaschke, On linearization and continuous selections of functions, Optimization, 19 (1988), 343-353. doi: doi:10.1080/02331938808843350.
[9]	S. Kaplan, On the second dual of the space of continuous functions, Trans. Amer. Math. Soc., 86 (1957), 70-90.
[10]	D. Klatte and B. Kummer, Nonsmooth equations in optimization, in "Regularity, Calculus, Methods and Applications," Nonconvex Optimization and its Applications, vol. 60, Kluwer Academic Publishers, Dordrecht, 2002.
[11]	L. Kuntz and S. Scholtes, Structural analysis of nonsmooth mappings, inverse functions, and metric projections, J. Math. Anal. Appl., 188 (1994), 346-386. doi: doi:10.1006/jmaa.1994.1431.
[12]	L. Kuntz and S. Scholtes, Qualitative aspects of the local approximation of a piecewise differentiable function, Nonlinear Anal., 25 (1995), 197-215. doi: doi:10.1016/0362-546X(94)00202-S.
[13]	G. Lebourg, Generic differentiability of Lipschitzian functions, Trans. Amer. Math. Soc., 256 (1979), 125-144.
[14]	B. S. Mordukhovich, Metric approximations and necessary conditions for optimality for general classes of nonsmooth extremal problems, Dokl. Akad. Nauk SSSR, 254 (1980), 1072-1076.
[15]	B. S. Mordukhovich, Generalized differential calculus for nonsmooth and set-valued mappings, J. Math. Anal. Appl., 183 (1994), 250-288. doi: doi:10.1006/jmaa.1994.1144.
[16]	B. S. Mordukhovich, Coderivatives of set-valued mappings: Calculus and applications, proceedings of the "Second World Congress of Nonlinear Analysts," Part 5 (Athens, 1996), vol. 30 (1997), 3059-3070.
[17]	B. S. Mordukhovich, "Variational Analysis and Generalized Differentiation, I. Basic Theory," Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 330, Springer-Verlag, Berlin, 2006.
[18]	J.-S. Pang and D. Ralph, Piecewise smoothness, local invertibility, and parametric analysis of normal maps, Math. Oper. Res., 21 (1996), 401-426. doi: doi:10.1287/moor.21.2.401.
[19]	Zs. Páles and V. Zeidan, Generalized Jacobian for functions with infinite dimensional range and domain, Set-Valued Anal., 15 (2007), 331-375. doi: doi:10.1007/s11228-007-0043-y.
[20]	Zs. Páles and V. Zeidan, Infinite dimensional Clarke generalized Jacobian, J. Convex Anal., 14 (2007), 433-454.
[21]	Zs. Páles and V. Zeidan, Infinite dimensional generalized Jacobian: Properties and calculus rules, J. Math. Anal. Appl., 344 (2008), 55-75. doi: doi:10.1016/j.jmaa.2008.02.044.
[22]	Zs. Páles and V. Zeidan, The core of the infinite dimensional generalized Jacobian, J. Convex Anal., 16 (2009), 321-349.
[23]	Zs. Páles and V. Zeidan, Co-Jacobian for Lipschitzian maps, Set-Valued and Variational Anal., 18 (2010), 57-78. doi: doi:10.1007/s11228-009-0130-3.
[24]	D. Ralph and S. Scholtes, Sensitivity analysis of composite piecewise smooth equations, Math. Programming Ser. B, 76 (1997), 593-612. doi: doi:10.1007/BF02614400.
[25]	D. Ralph and H. Xu, Implicit smoothing and its application to optimization with piecewise smooth equality constraints, J. Optim. Theory Appl., 124 (2005), 673-699. doi: doi:10.1007/s10957-004-1180-1.
[26]	R. T. Rockafellar, A property of piecewise smooth functions, Comput. Optim. Appl., 25 (2003), 247-250, A tribute to Elijah (Lucien) Polak. doi: doi:10.1023/A:1022921624832.
[27]	S. Scholtes, "Introduction to Piecewise Differentiable Equations," Habilitation thesis, University of Karlsruhe, Karlsruhe, Germany, 1994.
[28]	T. H. Sweetser, A minimal set-valued strong derivative for vector-valued Lipschitz functions, J. Optimization Theory Appl., 23 (1977), 549-562. doi: doi:10.1007/BF00933296.
[29]	L. Thibault, Subdifferentials of compactly Lipschitzian vector-valued functions, Ann. Mat. Pura Appl. (4), 125 (1980), 157-192. doi: doi:10.1007/BF01789411.
[30]	L. Thibault, On generalized differentials and subdifferentials of Lipschitz vector-valued functions, Nonlinear Anal., 6 (1982), 1037-1053. doi: doi:10.1016/0362-546X(82)90074-8.
[31]	J. Warga, Derivative containers, inverse functions, and controllability, in "Calculus of Variations and Control Theory" (Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1975; dedicated to Laurence Chisholm Young on the occasion of his 70th birthday), Academic Press, New York, (1976), 13-45; errata, p. 46. Math. Res. Center, Univ. Wisconsin, Publ. No. 36.
[32]	J. Warga, Fat homeomorphisms and unbounded derivate containers, J. Math. Anal. Appl., 81 (1981), 545-560; (Errata: ibid. 82 (1982), 582-583).

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