The paper is devoted to an analysis of optimality conditions for nonsmooth multidimensional problems of the calculus of variations with various types of constraints, such as additional constraints at the boundary and isoperimetric constraints. To derive optimality conditions, we study generalised concepts of differentiability of nonsmooth functions called codifferentiability and quasidifferentiability. Under some natural and easily verifiable assumptions we prove that a nonsmooth integral functional defined on the Sobolev space is continuously codifferentiable and compute its codifferential and quasidifferential. Then we apply general optimality conditions for nonsmooth optimisation problems in Banach spaces to obtain optimality conditions for nonsmooth problems of the calculus of variations. Through a series of simple examples we demonstrate that our optimality conditions are sometimes better than existing ones in terms of various subdifferentials, in the sense that our optimality conditions can detect the non-optimality of a given point, when subdifferential-based optimality conditions fail to disqualify this point as non-optimal.
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Keywords: Nonsmooth analysis, calculus of variations, optimality conditions, codifferential, quasidifferential
@article{COCV_2021__27_1_A81_0,
author = {Dolgopolik, Maksim},
title = {Constrained nonsmooth problems of the calculus of variations},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
year = {2021},
publisher = {EDP-Sciences},
volume = {27},
doi = {10.1051/cocv/2021074},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2021074/}
}
TY - JOUR AU - Dolgopolik, Maksim TI - Constrained nonsmooth problems of the calculus of variations JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2021 VL - 27 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2021074/ DO - 10.1051/cocv/2021074 LA - en ID - COCV_2021__27_1_A81_0 ER -
%0 Journal Article %A Dolgopolik, Maksim %T Constrained nonsmooth problems of the calculus of variations %J ESAIM: Control, Optimisation and Calculus of Variations %D 2021 %V 27 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/cocv/2021074/ %R 10.1051/cocv/2021074 %G en %F COCV_2021__27_1_A81_0
Dolgopolik, Maksim. Constrained nonsmooth problems of the calculus of variations. ESAIM: Control, Optimisation and Calculus of Variations, Tome 27 (2021), article no. 79. doi: 10.1051/cocv/2021074
[1] , Sobolev Spaces. Academic Press, New York (1975).
[2] and , Set-Valued Analysis. Birkhäuser, Boston (1990).
[3] , , and , Nonsmooth optimization of hydrothermal problems. J. Comput. Appl. Math. 192 (2006) 11–19.
[4] , , and , A constrained and non-smooth hydrothermal problem. Appl. Math. Comput. 209 (2009) 10–18.
[5] , , and , A hydrothermal problem with non-smooth Lagrangian. J. Ind. Manag. Optim. 10 (2014) 761–776.
[6] , Contributions à l’optimisation multicritère, Ph.D. thesis, Université de Bourgogne, Laboratoire Analyse Appliquée et Optimisation, Dijon, France (2003). Available at: https://tel.archives-ouvertes.fr/file/index/docid/46039/filename/tel-00004421.pdf.
[7] , and , Dualization of the Euler and Hamiltonian inclusions. Nonlinear Anal. 43 (2001) 861–882.
[8] , Vol. I of Measure Theory. Springer-Verlag, Berlin, Heidelberg (2007).
[9] and , The validity of the Euler-Lagrange equation. Discret. Contin. Dyn. Syst. 28 (2010) 511–517.
[10] , The Euler equation in the multiple integrals calculus of variations. SIAM J. Control Optim. 51 (2013) 1047–1062.
[11] , Necessary Conditions in Dynamic Optimization. American Mathematical Society, Providence, Rhode Island (2005).
[12] , Functional Analysis, Calculus of Variations and Optimal Control. Springer-Verlag, London (2013).
[13] , The Euler-Lagrange differential inclusion. J. Differ. Equ. 19 (1975) 80–90.
[14] , The generalized problem of Bolza. SIAM J. Control Optim. 14 (1976) 682–699.
[15] , Multiple integrals of Lipschitz functions in the calculus of variations. Proc. Am. Math. Soc. 64 (1977) 260–264.
[16] , The Erdmann condition and Hamiltonian inclusions in optimal control and the calculus of variations. Can. J. Math. 32 (1980) 494–509.
[17] , Optimization and Nonsmooth Analysis. Wiley–Interscience, New York (1983).
[18] , A decoupling principle in the calculus of variations. J. Math. Anal. Appl. 172 (1993) 92–105.
[19] and , The nonsmooth maximum principle. Control Cybern. 38 (2009) 1151–1167.
[20] , , and , Nonsmooth Analysis and Control Theory. Springer-Verlag, New York (1998).
[21] , and , Necessary conditions and non-existence results for autonomous nonconvex variational problems. J. Differ. Equ. 243 (2007) 329–348.
[22] , Direct Methods in the Calculus of Variations. Springer-Verlag, New York (2008).
[23] , Continuous generalized gradients for nonsmooth functions, in Optimization, Parallel Processing and Applications, edited by , and . Springer Berlin, Heidelberg (1988) 24–27.
[24] , On codifferentiable functions. Vestn. Leningr. Univ., Math. 2 (1988) 22–26.
[25] , Smoothness of nonsmooth functions, in Nonsmooth Optimization and Related Topics, edited by , and . Springer, Boston (1989) 79–88.
[26] and , Quasidifferential Calculus. Springer Berlin, Heidelberg (1986).
[27] and , Constructive Nonsmooth Analysis. Peter Lang, Frankfurt am Main (1995).
[28] and , Quasidifferentiability and Related Topics. Kluwer Academic Publishers, Dordrecht (2000).
[29] , , and , Quasidifferentiability and Nonsmooth Modelling in Mechanics, Engineering and Economics. Kluwer Academic Publishers, Dordrecht (1996).
[30] , Codifferential calculus in normed spaces. J. Math. Sci. 173 (2011) 441–462.
[31] , Nonsmooth problems of calculus of variations via codifferentiation. ESAIM: COCV 20 (2014) 1153–1180.
[32] , Abstract convex approximations of nonsmooth functions. Optim 64 (2015) 1439–1469.
[33] , A convergence analysis of the method of codifferential descent. Comput. Optim. Appl. 71 (2018) 879–913.
[34] , Metric regularity of quasidifferentiable mappings and optimality conditions for nonsmooth mathematical programming problems. Set-Valued Var. Anal. 28 (2019) 427–449.
[35] , A new constraint qualification and sharp optimality conditions for nonsmooth mathematical programming problems in terms of quasidifferentials. SIAM J. Optim. 30 (2020) 2603–2627.
[36] and , Linear Operators Part 1: General Theory. John Wiley & Sons, New Jersey (1958).
[37] and , Convex Analysis and Variational Problems. SIAM, Philadelphia (1999).
[38] , Real Analysis. Modern Techniques and Their Applications. Interscience Publishers, New York (1984).
[39] , On the minimal quasidifferential in the one-dimensional case. Soochow J. Math. 24 (1998) 211–218.
[40] , A common understanding or a common misunderstanding? Numer. Funct. Anal. Optim. 16 (1995) 1359–1363.
[41] , and , On the reduction of pairs of bounded closed convex sets. Studia Math. 189 (2008) 1–12.
[42] , and , On the amount of minimal pairs of convex sets. Optim. Methods Softw. 25 (2010) 89–96.
[43] and , Minimal pairs of bounded closed convex sets. Studia Math. 126 (1997) 95–99.
[44] and , Three criteria of minimality for pairs of compact convex sets. Optim 55 (2006) 569–576.
[45] , On equivalent quasidifferentials in the two-dimensional case. Optim 20 (1989) 37–43.
[46] , Euler-Lagrange and Hamiltonian formalisms in dynamic optimization. Trans. Am. Math. Soc. 349 (1997) 2871–2900.
[47] , On necessary conditions for a minimum. J. Math. Sci. 217 (2016) 751–772.
[48] , On generalized Bolza problems and its application to dynamic optimization. J. Optim. Theory Appl. 182 (2019) 285–309.
[49] and , The Euler and Weierstrass conditions for nonsmooth variational problems. Calc. Var. Partial Differ. Equ. 4 (1996) 59–87.
[50] and , Theory of Extremal Problems. North-Holland Publishing Company, Amsterdam (1979).
[51] , Lagrangian and Hamiltonian necessary conditions for the generalized Bolza problem and applications. J. Nonlinear Convex Anal. 10 (2009) 437–454.
[52] and , Approximate subdifferential and metric regularity: the finite-dimensional case. Math. Program. 47 (1990) 203–218.
[53] , A characterization of continuously codifferentiable functions and some consequences. Optim 22 (1991) 539–547.
[54] , A First Course in Sobolev spaces. American Mathematical Society, Providence, RI (2009).
[55] , Optimal Control via Nonsmooth Analysis. American Mathematical Society, Providence, Rhode Island (1993).
[56] and , The adjoint arc in nonsmooth optimization. Trans. Am. Math. Soc. 325 (1991) 39–72.
[57] and , Optimal control of unbounded differential inclusions. SIAM J. Control Optim. 32 (1994) 442–470.
[58] and , New necessary conditions for the generalized problem of Bolza. SIAM J. Control Optim. 34 (1996) 1496–1511.
[59] and , Bolza problem with general time constraints. SIAM J. Control Optim. 35 (1997) 2050–2069.
[60] , Variational problems with nonconvex, noncoercive, highly discontinuous integrands: characterization and existence of minimizers. SIAM J. Control Optim. 40 (2002) 1473–1490.
[61] , Necessary and sufficient conditions for optimality of nonconvex, noncoercive autonomous variational problems with constraints. Trans. Am. Math. Soc. 360 (2008) 5201–5227.
[62] , and , Remarks on necessary conditions for minimizers of one-dimensional variational problems. Math. Prepr. Arch. 2001 (2001) 1145–1163.
[63] , Approximation Methods in Problems of Optimization and Control. Nauka, Moscow (1988). [in Russian].
[64] , Discrete approximation and refined Euler-Lagrange conditions for nonconvex differential inclusions. SIAM J. Control Optim. 33 (1995) 882–915.
[65] , Variational Analysis and Generalized Differentiation I: Basic Theory. Springer-Verlag, Berling, Heidelberg (2006).
[66] , Variational Analysis and Generalized Differentiation II: Applications. Springer-Verlag, Berling, Heidelberg (2006).
[67] and , Multidimensional variational functionals with subsmooth integrands. Eurasian Math. J. 6 (2015) 54–75.
[68] and , Some criteria for the minimality of pairs of compact convex sets. ZOR — Methods Models Oper. Res. 37 (1993) 129–150.
[69] and , Quasidifferentiable calculus and minimal pairs of compact convex sets. Schedae Informaticae 21 (2012) 107–125.
[70] , Differential inclusions with unbounded right-hand side and necessary optimality conditions. Proc. Stekov Inst. Math. 291 (2015) 237–252.
[71] , Necessary optimality conditions for the Mayer problem with unbounded differential inclusion. IFAC-PapersOnline 51 (2018) 521–524.
[72] , Pontryagin’s direct method for optimization problems with differential inclusions. Proc. Stekov Inst. Math. 304 (2019) 241–256.
[73] , Necessary Conditions for an Extremum. Marcel Dekker, New York (1971).
[74] , Conjugate convex functions in optimal control and the calculus of variations. J. Math. Anal. Appl. 32 (1970) 174–222.
[75] , Generalized Hamiltonian equations for convex problems of Lagrange. Pac. J. Math. 33 (1970) 411–427.
[76] , Existence and duality theorems for convex problems of Bolza. Trans. Am. Math. Soc. 159 (1971) 1–40.
[77] , Dualization of subgradient conditions for optimality. Nonlinear Anal. 20 (1993) 627–646.
[78] , Minimal pairs of convex bodies in two dimensions Mathematika 39 (1992) 267–273.
[79] , and , Generalized Euler-Lagrange equation for nonsmooth calculus of variations. Nonlinear Dyn. 75 (2014) 85–100.
[80] and , The extended Euler-Lagrange condition for nonconvex variational problems. SIAM J. Control Optim. 35 (1997) 56–77.
[81] , Optimal Control. Birkhäuser, Boston (2000).
[82] , Codifferentiable mappings with applications to vector optimality. Pilska Studia Mathematica Bulgarica 12 (1998) 255–266.
[83] , Continuous approximations, codifferentiable functions and minimization methods, in Quasidifferentiability and related Topics, edited by and . Kluwer Academic Publishers, Dordrecht (2000) 361–391.
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The results presented in this article were supported by the President of Russian Federation grant for the support of young Russian scientists (grant number MK-3621.2019.1).
