June  2022, 21(6): 1917-1951. doi: 10.3934/cpaa.2021076

Hadamard semidifferential, oriented distance function, and some applications

Département de mathématiques et de statistique and Centre de recherches mathématiques, Université de Montréal, CP 6128, succ. Centre-ville, Montréal (Qc), Canada H3C 3J7

Received  December 2020 Revised  March 2021 Published  June 2022 Early access  April 2021

Fund Project: This research was supported by the Natural Sciences and Engineering research Council of Canada through Discovery Grants RGPIN-05279-2017 and a Grant from the Collaborative research and Training Experience (CREATE) program in Simulation-based Engineering Science

The Hadamard semidifferential calculus preserves all the operations of the classical differential calculus including the chain rule for a large family of non-differentiable functions including the continuous convex functions. It naturally extends from the $ n $-dimensional Euclidean space $ \operatorname{\mathbb R}^n $ to subsets of topological vector spaces. This includes most function spaces used in Optimization and the Calculus of Variations, the metric groups used in Shape and Topological Optimization, and functions defined on submanifolds.

Certain set-parametrized functions such as the characteristic function $ \chi_A $of a set $ A $, the distance function $ d_A $ to $ A $, and the oriented (signed) distance function $ b_A = d_A-d_{ \operatorname{\mathbb R}^n\backslash A} $ can be used to identify a space of subsets of $ \operatorname{\mathbb R}^n $ with a metric space of set-parametrized functions. Many geometrical properties of domains (convexity, outward unit normal, curvatures, tangent space, smoothness of boundaries) can be expressed in terms of the analytical properties of $ b_A $ and a simple intrinsic differential calculus is available for functions defined on hypersurfaces without appealing to local bases or Christoffel symbols.

The object of this paper is to extend the use of the Hadamard semidifferential and of the oriented distance function from finite to infinite dimensional spaces with some selected illustrative applications from shapes and geometries, plasma physics, and optimization.

Citation: Michel C. Delfour. Hadamard semidifferential, oriented distance function, and some applications. Communications on Pure and Applied Analysis, 2022, 21 (6) : 1917-1951. doi: 10.3934/cpaa.2021076
References:
[1]

N. Aronszajn, Differentiability of Lipschitzian mappings between Banach spaces, Stud. Math., 57 (1976), 147-190.  doi: 10.4064/sm-57-2-147-190.

[2]

J. P. Aubin and H. Frankowska, Set-Valued Analysis, Birkhäuser, Boston, 1990.

[3]

D. P. Bertsekas, Control of Uncertain Systems with a Set-Membership Description of Uncertainty, Ph. D. Thesis, Massachusetts Institute of Technology, Cambridge, MA.

[4]

D. P. Bertsekas, Nonlinear Programming, 2$^{nd}$ edition, Athena Scientific, Belmont, MA, 2004.

[5]

J. M. Borwein and S. Fitzpatrick, Existence of nearest points in Banach spaces, Can. J. Math., 41 (1989), 702-720.  doi: 10.4153/CJM-1989-032-7.

[6]

J. M. Borwein and R. Giles, The proximal normal formula in Banach spaces, Trans. American Math. Soc., 302 (1989), 371-381.  doi: 10.2307/2000915.

[7]

J. M. Borwein and A. S. Lewis, Convex Analysis and Nonlinear Optimization. Theory and Examples, Springer-Verlag, New York, 2000. doi: 10.1007/978-1-4757-9859-3.

[8]

J. M. Borwein and H. M. Strojwas, Primal analysis and boundaries of closed sets in Banach spaces part I: theory, Can. J. Math., 38 (1986), 431-452.  doi: 10.4153/CJM-1986-022-4.

[9]

J. M. Borwein and H. M. Strojwas, Primal analysis and boundaries of closed sets in Banach spaces part II: applications, Can. J. Math., 39 (1987), 428–472.,

[10]

G. Bouligand, Sur les surfaces dépourvues de points hyperlimités, Ann. Soc. Polon. Math., 9 (1930), 32-41. 

[11]

P. Cannarsa and C. Sinestrari, Nonlinear Differential Equations and Their Applications, Birkhäuser Boston, MA, 2004.

[12]

F. H. Clarke, Necessary Conditions for Nonsmooth Problems in Optimal Control and the Calculus of Variations, Ph.D. thesis, Univ. of Washington, Seattle, WA, 1973.

[13]

F. H. Clarke, Optimization and Nonsmooth Analysis, John Wiley and Sons, New York, Chichester, Brisbane, Toronto, Singapore, 1983.

[14]

J. M. Danskin, The theory of max-min, with applications, SIAM J. Appl. Math., 14 (1966), 641-664.  doi: 10.1137/0114053.

[15]

M. C. Delfour, Tangential differential calculus and functional analysis on a $C^{1, 1}$ submanifold, in Differential-Geometric Methods in the Control of Partial Differential Equations, Amer. Math. Soc., Providence, RI, 2000. doi: 10.1090/conm/268/04309.

[16]

M. C. Delfour, Metrics spaces of shapes and geometries from set parametrized functions, in New Trends in Shape Optimization, Birkhäuser, Basel, 2016. doi: 10.1007/978-3-319-17563-8_4.

[17]

M. C. Delfour, Differentials and semidifferentials for metric spaces of shapes and geometries, in System Modeling and Optimization, AICT Series, Springer, 2017.

[18]

M. C. Delfour, Topological derivative: a semidifferential via the Minkowski content, J. Con. Anal., 25 (2018), 957-982. 

[19]

M. C. Delfour, Differentials and semidifferentials for metric spaces of shapes and geometries, in Shape Optimization, Homogenization and Optimal Control, Birkäuser, Cham, Switzerland, 2018. doi: 10.1137/1.9781611972153.

[20]

M. C. Delfour, Control, shape, and topological derivatives via minimax differentiability of Lagrangians, in Numerical Methods for Optimal Control Problems, Springer, Cham, Switzerland AG, 2018.

[21]

M. C. Delfour, Introduction to Optimization and Hadamard Semidifferential Calculus, 2$^{nd}$ edition, MOS-SIAM series on Optimization 27, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2020.

[22]

M. C. Delfour, Hadamard semidifferential of functions on an unstructured subset of a TVS, J. Pure Appl. Funct. Anal., 5 (2020), 1039-1072. 

[23]

M. C. Delfour and J. P. Zolésio, Shape analysis via oriented distance functions, J. Funct. Anal., 123 (1994), 129-201.  doi: 10.1006/jfan.1994.1086.

[24]

M. C. Delfour and J. P. Zolésio, Shapes and Geometries: Analysis, Differential Calculus and Optimization, Advances in Design and Control 4, SIAM, Philadelphia, PA 2001.

[25]

M. C. Delfour and J. P. Zolésio, Oriented distance function and its evolution equation for initial sets with thin boundary, SIAM J. Control and Optim., 42 (2004), 2286-2304.  doi: 10.1137/S0363012902411945.

[26]

M. C. Delfour and J. P. Zolésio, The new family of cracked sets and the image segmentation problem revisited, Commun. Inf. Syst., 4 (2004), 29-52. 

[27]

M. C. Delfour and J. P. Zolésio, Shapes and Geometries: Metrics, Analysis, Differential Calculus and Optimization, 2$^{nd}$ edition, Advances in Design and Control 22, SIAM, Philadelphia, PA 2011. doi: 10.1137/1.9780898719826.

[28]

M. Edelstein, Weakly proximinal sets, J. Approx. Theory, 18 (1976), 1-8.  doi: 10.1016/0021-9045(76)90115-5.

[29]

I. Ekeland and R. Temam, Convex Analysis and Variational Problems, SIAM, Philadelphia, PA, 1999.

[30]

L. C. Evans and R. F. Gariepy, Measure Theory and the Properties of Functions, CRC Press, Boca Raton, FL, 1992.

[31]

H. Federer, Curvature measures, Trans. Amer. Math. Soc., 93 (1959), 418-419.  doi: 10.2307/1993504.

[32]

J. Gauvin, A necessary and sufficient regularity condition to have bounded multipliers in nonconvex programming, Math. Program., 12 (1977), 136-138.  doi: 10.1007/BF01593777.

[33]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin 1983. doi: 10.1007/978-3-642-61798-0.

[34]

H. GradP. N. Hu and D. C. Stevens, Adiabatic evolution of plasma equilibrium, Proc. Nat. Acad., 72 (1975), 3789-3793. 

[35]

J. Hadamard, La notion de différentielle dans l'enseignement, Math. Gazette, 19 (1935), 341-342. 

[36]

J. Horváth, Topological Vector Spaces and Distributions, Addison-Wesley, Reading, MA, 1966.

[37]

F. John, Extremum problems with inequalities as subsidiary conditions, in Studies and Essays Presented to R. Courant on his 60th Birthday, January 8, 1948, Interscience Publishers, N.Y., 1948.

[38]

C. Mercier, The magneto-hydrodynamic approach to the problem of plasma confinement in closed magnetic configurations, Publication of EURATOM C. E. A., Luxembourg, 1974.

[39]

A. M. Micheletti, Metrica per famiglie di domini limitati e proprietà generiche degli autovalori, Ann. Scuola Norm. Sup. Pisa, 26 (1972), 683-694. 

[40]

F. Mignot, Contrôle dans les inéquations variationnelles elliptiques, J. Funct. Anal., 22 (1976), 130-185.  doi: 10.1016/0022-1236(76)90017-3.

[41]

B. S. Mordukhovich, Variational Analysis and Applications, Springer, Cham, Switzerland, 2018 doi: 10.1007/978-3-319-92775-6.

[42]

J. Mossino, Application des inéquations quasi-variationelles à quelques problèmes non linéaire de la physique des plasmas, Israel J. Math., 30 (1978), 14-50.  doi: 10.1007/BF02760826.

[43]

J. Mossino, Some nonlinear problems involving a free boundary in plasma physics, J. Differ. Equ., 34 (1979), 114-138.  doi: 10.1016/0022-0396(79)90021-4.

[44]

J. Mossino and J. P. Zolésio, Solution variationnelle d'un problème non linéaire de la physique des plasmas, C. R. Acad. Sci. Paris Sér. A-B, 285 (1977), A1033–A1036.

[45]

A. A. Novotny and J. Sokołowski, Topological Derivatives in Shape Optimization, Springer, Heidelberg, New York, 2013. doi: 10.1007/978-3-642-35245-4.

[46]

A. A. Novotny, J. Sokołowski, and A. Zȯchowski, Applications of the Topological Derivative Method, Springer, 2019. doi: 10.1007/978-3-030-05432-8.

[47]

J. P. Penot, Calcul sous-différentiel et optimisation, J. Funct. Anal., 27 (1978), 248-276.  doi: 10.1016/0022-1236(78)90030-7.

[48]

J. B. Poly and G. Raby, Fonction distance et singularités, Bull. Sci. Math., 108 (1984), 187-195. 

[49]

R. T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, NJ, 1972.

[50]

J. Sokołowski and A. Zȯchowski, On the topological derivative in shape optimization, SIAM J. Control Optim., 37 (1999), 1251-1272.  doi: 10.1137/S0363012997323230.

[51]

R. Temam, Nonlinear boundary value problems arising in physics, in Differential Equations and Applications, North-Holland, Amsterdam-New York, 1978.

[52]

R. Temam, Monotone rearrangement of a function and the Grad-Mercier equation of plasmaphysics, in Proc. International Meeting on Recent Methods in Nonlinear Analysis, Pitagora, Bologna, 1979.

[53]

J. P. Zolésio, Solution variationnelle d'un problème non linéaire de la physique des plasmas, C. R. Acad. Sci. Paris Sér. A-B, 285 (1977), A1033–A1036.

[54]

J. P. Zolésio, Solution variationnelle d'un problème de valeur propre non linéaire et frontière libre en physique des plasmas, C. R. Acad. Sci. Paris Sér. A-B, 288 (1979), A911–A913.

[55]

J. P. Zolésio, Identification de domaines par déformation, Thèse de doctorat d'état, Université de Nice, France, 1979.

show all references

References:
[1]

N. Aronszajn, Differentiability of Lipschitzian mappings between Banach spaces, Stud. Math., 57 (1976), 147-190.  doi: 10.4064/sm-57-2-147-190.

[2]

J. P. Aubin and H. Frankowska, Set-Valued Analysis, Birkhäuser, Boston, 1990.

[3]

D. P. Bertsekas, Control of Uncertain Systems with a Set-Membership Description of Uncertainty, Ph. D. Thesis, Massachusetts Institute of Technology, Cambridge, MA.

[4]

D. P. Bertsekas, Nonlinear Programming, 2$^{nd}$ edition, Athena Scientific, Belmont, MA, 2004.

[5]

J. M. Borwein and S. Fitzpatrick, Existence of nearest points in Banach spaces, Can. J. Math., 41 (1989), 702-720.  doi: 10.4153/CJM-1989-032-7.

[6]

J. M. Borwein and R. Giles, The proximal normal formula in Banach spaces, Trans. American Math. Soc., 302 (1989), 371-381.  doi: 10.2307/2000915.

[7]

J. M. Borwein and A. S. Lewis, Convex Analysis and Nonlinear Optimization. Theory and Examples, Springer-Verlag, New York, 2000. doi: 10.1007/978-1-4757-9859-3.

[8]

J. M. Borwein and H. M. Strojwas, Primal analysis and boundaries of closed sets in Banach spaces part I: theory, Can. J. Math., 38 (1986), 431-452.  doi: 10.4153/CJM-1986-022-4.

[9]

J. M. Borwein and H. M. Strojwas, Primal analysis and boundaries of closed sets in Banach spaces part II: applications, Can. J. Math., 39 (1987), 428–472.,

[10]

G. Bouligand, Sur les surfaces dépourvues de points hyperlimités, Ann. Soc. Polon. Math., 9 (1930), 32-41. 

[11]

P. Cannarsa and C. Sinestrari, Nonlinear Differential Equations and Their Applications, Birkhäuser Boston, MA, 2004.

[12]

F. H. Clarke, Necessary Conditions for Nonsmooth Problems in Optimal Control and the Calculus of Variations, Ph.D. thesis, Univ. of Washington, Seattle, WA, 1973.

[13]

F. H. Clarke, Optimization and Nonsmooth Analysis, John Wiley and Sons, New York, Chichester, Brisbane, Toronto, Singapore, 1983.

[14]

J. M. Danskin, The theory of max-min, with applications, SIAM J. Appl. Math., 14 (1966), 641-664.  doi: 10.1137/0114053.

[15]

M. C. Delfour, Tangential differential calculus and functional analysis on a $C^{1, 1}$ submanifold, in Differential-Geometric Methods in the Control of Partial Differential Equations, Amer. Math. Soc., Providence, RI, 2000. doi: 10.1090/conm/268/04309.

[16]

M. C. Delfour, Metrics spaces of shapes and geometries from set parametrized functions, in New Trends in Shape Optimization, Birkhäuser, Basel, 2016. doi: 10.1007/978-3-319-17563-8_4.

[17]

M. C. Delfour, Differentials and semidifferentials for metric spaces of shapes and geometries, in System Modeling and Optimization, AICT Series, Springer, 2017.

[18]

M. C. Delfour, Topological derivative: a semidifferential via the Minkowski content, J. Con. Anal., 25 (2018), 957-982. 

[19]

M. C. Delfour, Differentials and semidifferentials for metric spaces of shapes and geometries, in Shape Optimization, Homogenization and Optimal Control, Birkäuser, Cham, Switzerland, 2018. doi: 10.1137/1.9781611972153.

[20]

M. C. Delfour, Control, shape, and topological derivatives via minimax differentiability of Lagrangians, in Numerical Methods for Optimal Control Problems, Springer, Cham, Switzerland AG, 2018.

[21]

M. C. Delfour, Introduction to Optimization and Hadamard Semidifferential Calculus, 2$^{nd}$ edition, MOS-SIAM series on Optimization 27, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2020.

[22]

M. C. Delfour, Hadamard semidifferential of functions on an unstructured subset of a TVS, J. Pure Appl. Funct. Anal., 5 (2020), 1039-1072. 

[23]

M. C. Delfour and J. P. Zolésio, Shape analysis via oriented distance functions, J. Funct. Anal., 123 (1994), 129-201.  doi: 10.1006/jfan.1994.1086.

[24]

M. C. Delfour and J. P. Zolésio, Shapes and Geometries: Analysis, Differential Calculus and Optimization, Advances in Design and Control 4, SIAM, Philadelphia, PA 2001.

[25]

M. C. Delfour and J. P. Zolésio, Oriented distance function and its evolution equation for initial sets with thin boundary, SIAM J. Control and Optim., 42 (2004), 2286-2304.  doi: 10.1137/S0363012902411945.

[26]

M. C. Delfour and J. P. Zolésio, The new family of cracked sets and the image segmentation problem revisited, Commun. Inf. Syst., 4 (2004), 29-52. 

[27]

M. C. Delfour and J. P. Zolésio, Shapes and Geometries: Metrics, Analysis, Differential Calculus and Optimization, 2$^{nd}$ edition, Advances in Design and Control 22, SIAM, Philadelphia, PA 2011. doi: 10.1137/1.9780898719826.

[28]

M. Edelstein, Weakly proximinal sets, J. Approx. Theory, 18 (1976), 1-8.  doi: 10.1016/0021-9045(76)90115-5.

[29]

I. Ekeland and R. Temam, Convex Analysis and Variational Problems, SIAM, Philadelphia, PA, 1999.

[30]

L. C. Evans and R. F. Gariepy, Measure Theory and the Properties of Functions, CRC Press, Boca Raton, FL, 1992.

[31]

H. Federer, Curvature measures, Trans. Amer. Math. Soc., 93 (1959), 418-419.  doi: 10.2307/1993504.

[32]

J. Gauvin, A necessary and sufficient regularity condition to have bounded multipliers in nonconvex programming, Math. Program., 12 (1977), 136-138.  doi: 10.1007/BF01593777.

[33]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin 1983. doi: 10.1007/978-3-642-61798-0.

[34]

H. GradP. N. Hu and D. C. Stevens, Adiabatic evolution of plasma equilibrium, Proc. Nat. Acad., 72 (1975), 3789-3793. 

[35]

J. Hadamard, La notion de différentielle dans l'enseignement, Math. Gazette, 19 (1935), 341-342. 

[36]

J. Horváth, Topological Vector Spaces and Distributions, Addison-Wesley, Reading, MA, 1966.

[37]

F. John, Extremum problems with inequalities as subsidiary conditions, in Studies and Essays Presented to R. Courant on his 60th Birthday, January 8, 1948, Interscience Publishers, N.Y., 1948.

[38]

C. Mercier, The magneto-hydrodynamic approach to the problem of plasma confinement in closed magnetic configurations, Publication of EURATOM C. E. A., Luxembourg, 1974.

[39]

A. M. Micheletti, Metrica per famiglie di domini limitati e proprietà generiche degli autovalori, Ann. Scuola Norm. Sup. Pisa, 26 (1972), 683-694. 

[40]

F. Mignot, Contrôle dans les inéquations variationnelles elliptiques, J. Funct. Anal., 22 (1976), 130-185.  doi: 10.1016/0022-1236(76)90017-3.

[41]

B. S. Mordukhovich, Variational Analysis and Applications, Springer, Cham, Switzerland, 2018 doi: 10.1007/978-3-319-92775-6.

[42]

J. Mossino, Application des inéquations quasi-variationelles à quelques problèmes non linéaire de la physique des plasmas, Israel J. Math., 30 (1978), 14-50.  doi: 10.1007/BF02760826.

[43]

J. Mossino, Some nonlinear problems involving a free boundary in plasma physics, J. Differ. Equ., 34 (1979), 114-138.  doi: 10.1016/0022-0396(79)90021-4.

[44]

J. Mossino and J. P. Zolésio, Solution variationnelle d'un problème non linéaire de la physique des plasmas, C. R. Acad. Sci. Paris Sér. A-B, 285 (1977), A1033–A1036.

[45]

A. A. Novotny and J. Sokołowski, Topological Derivatives in Shape Optimization, Springer, Heidelberg, New York, 2013. doi: 10.1007/978-3-642-35245-4.

[46]

A. A. Novotny, J. Sokołowski, and A. Zȯchowski, Applications of the Topological Derivative Method, Springer, 2019. doi: 10.1007/978-3-030-05432-8.

[47]

J. P. Penot, Calcul sous-différentiel et optimisation, J. Funct. Anal., 27 (1978), 248-276.  doi: 10.1016/0022-1236(78)90030-7.

[48]

J. B. Poly and G. Raby, Fonction distance et singularités, Bull. Sci. Math., 108 (1984), 187-195. 

[49]

R. T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, NJ, 1972.

[50]

J. Sokołowski and A. Zȯchowski, On the topological derivative in shape optimization, SIAM J. Control Optim., 37 (1999), 1251-1272.  doi: 10.1137/S0363012997323230.

[51]

R. Temam, Nonlinear boundary value problems arising in physics, in Differential Equations and Applications, North-Holland, Amsterdam-New York, 1978.

[52]

R. Temam, Monotone rearrangement of a function and the Grad-Mercier equation of plasmaphysics, in Proc. International Meeting on Recent Methods in Nonlinear Analysis, Pitagora, Bologna, 1979.

[53]

J. P. Zolésio, Solution variationnelle d'un problème non linéaire de la physique des plasmas, C. R. Acad. Sci. Paris Sér. A-B, 285 (1977), A1033–A1036.

[54]

J. P. Zolésio, Solution variationnelle d'un problème de valeur propre non linéaire et frontière libre en physique des plasmas, C. R. Acad. Sci. Paris Sér. A-B, 288 (1979), A911–A913.

[55]

J. P. Zolésio, Identification de domaines par déformation, Thèse de doctorat d'état, Université de Nice, France, 1979.

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