doi: 10.3934/mcrf.2020006

Implicit parametrizations and applications in optimization and control

1. 

Institute of Mathematics, Romanian Academy, P.O. BOX 1-764, 014700 Bucharest, Romania

2. 

Academy of Romanian Scientists, Splaiul Independenţei 54, 050094 Bucharest, Romania

Dedicated to Prof. Dr. Fréderic Bonnans on the occasion of his 60th birthday

Received  February 2018 Revised  July 2018 Published  December 2019

We discuss necessary conditions (with less Lagrange multipliers), perturbations and general algorithms in non convex optimization problems. Optimal control problems with mixed constraints, governed by ordinary differential equations, are also studied in this context. Our treatment is based on a recent approach to implicit systems, constructing parametrizations of the corresponding manifold, via iterated Hamiltonian equations.

Citation: Dan Tiba. Implicit parametrizations and applications in optimization and control. Mathematical Control & Related Fields, doi: 10.3934/mcrf.2020006
References:
[1]

V. Barbu, Optimal Control of Variational Inequalities, Research Notes in Mathematics, 100. Pitman, Boston, MA, 1984.  Google Scholar

[2]

D. P. Bertsekas, Nonlinear Programming, Third edition, Athena Scientific Optimization and Computation Series, Athena Scientific, Belmont, MA, 2016.  Google Scholar

[3]

J. F. Bonnans and A. Shapiro, Perturbation Analysis of Optimization Problems, Springer Series in Operations Research, Springer-Verlag, New York, 2000. doi: 10.1007/978-1-4612-1394-9.  Google Scholar

[4]

F. Bouchut and L. Desvillets, On two-dimensional Hamiltonian transport equations with continuous coefficients, Diff. Int. Eqns., 14 (2001), 1015-1024.   Google Scholar

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H. C. Chang, W. He and N. Prabhu, The analytic domain in the implicit function theorem, JIPAM. J. Inequal. Pure Appl. Math., 4 (2003), Art. 12, 5 pp.  Google Scholar

[6]

P. G. Ciarlet, Introduction to Numerical Linear Algebra and Optimization, Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge, 1989.  Google Scholar

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F. H. Clarke, Optimization and Nonsmooth Analysis, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1983.  Google Scholar

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F. Clarke and M. R. de Pinho, Optimal control problems with mixed constraints, SIAM J. Control Optim., 48 (2010), 4500-4524.  doi: 10.1137/090757642.  Google Scholar

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R. J. DiPerna and P.-L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math., 98 (1989), 511-547.  doi: 10.1007/BF01393835.  Google Scholar

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W. H. Fleming and R. W. Rishel, Deterministic and Stochastic Optimal Control, Applications of Mathematics, No. 1. Springer-Verlag, Berlin-New York, 1975.  Google Scholar

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P. Hartman, Ordinary Differential Equations, John Wiley & Sons, Inc., New York-London-Sydney, 1964.  Google Scholar

[12]

A. Henrot and M. Pierre, Variation et Optimization de Formes: Yne Analyse Geometrique, Mathématiques & Applications, 48. Springer, Berlin, 2005. doi: 10.1007/3-540-37689-5.  Google Scholar

[13]

C. Kublik and R. Tsai, Integration over curves and surfaces defined by the closest point mapping, Res. Math. Sci., 3 (2016), 17 pp, arXiv: 1504.05478v4. doi: 10.1186/s40687-016-0053-1.  Google Scholar

[14] J. B. Lasserre, An Introduction to Polynomial and Semi-Algebraic Optimization, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2015.  doi: 10.1017/CBO9781107447226.  Google Scholar
[15]

E. B. Lee and L. Markus, Foundations of Optimal Control Theory, John Wiley & Sons, Inc., New York-London-Sydney, 1967.  Google Scholar

[16]

J.-L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Die Grundlehren der mathematischen Wissenschaften, Band 170 Springer-Verlag, New York-Berlin, 1971.  Google Scholar

[17]

A. MitsosB. Chachuat and P. I. Barton, McCormick-based relaxations of algorithms, SIAM J. Optim., 20 (2009), 573-601.  doi: 10.1137/080717341.  Google Scholar

[18]

P. Neittaanmäki, J. Sprekels and D. Tiba, Optimization of Elliptic Systems. Theory and Applications, Springer Monographs in Mathematics, Springer, New York, 2006.  Google Scholar

[19]

P. Neittaanmäki and D. Tiba, Fixed domain approaches in shape optimization problems, Inverse Problems, 28 (2012), 093001, 35 pp. doi: 10.1088/0266-5611/28/9/093001.  Google Scholar

[20]

M. R. Nicolai and D. Tiba, Implicit functions and parametrizations in dimension three: Generalized solutions, Discrete Contin. Dyn. Syst., 35 (2015), 2701-2710.  doi: 10.3934/dcds.2015.35.2701.  Google Scholar

[21]

M. R. Nicolai, An algorithm for solving implicit systems in the critical case, Ann. Acad. Rom. Sci. Ser. Math. Appl., 7 (2015), 310-322.   Google Scholar

[22]

M. R. Nicolai, High dimensional applications of implicit parametrizations in nonlinear programming, Ann. Acad. Rom. Sci. Ser. Math. Appl., 8 (2016), 44-55.   Google Scholar

[23]

M. Patriksson, Nonlinear Programming and Variational Inequality Problems: A Unified Approach, Applied Optimization, 23. Kluwer Academic Publishers, Dordrecht, 1999. doi: 10.1007/978-1-4757-2991-7.  Google Scholar

[24]

P. Phien, Some quantitative results on Lipschitz inverse and implicit functions theorems, East-West J. Math., 13 (2011), 7–22, arXiv: 1204.4916v2.  Google Scholar

[25]

M. do R. de Pinho and M. H. A. Biswas, A nonsmooth maximum principle for optimal control problems with state and mixed constraints—convex case, Discrete Contin. Dyn. Syst., Dynamical Systems, Differential Equations and Applications. 8th AIMS Conference. Suppl., 1 (2011), 174-183.   Google Scholar

[26]

M. do R. de Pinho, On necessary conditions for implicit control systems, Pure Appl. Funct. Anal., 1 (2016), 185-196.   Google Scholar

[27]

J. Sokolowski and J.-P.Zolesio., Introduction to Shape Optimization. Shape Sensitivity Analysis, Springer Series in Computational Mathematics, 16. Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-642-58106-9.  Google Scholar

[28]

M. D. StuberJ. K. Scott and P. I. Barton, Convex and concave relaxations of implicit functions, Optimization Methods and Software, 30 (2015), 424-460.  doi: 10.1080/10556788.2014.924514.  Google Scholar

[29]

D. Tiba, Iterated Hamiltonian type systems and applications, J. Diff. Equations, 264 (2018), 5465-5479.  doi: 10.1016/j.jde.2018.01.003.  Google Scholar

[30]

D. Tiba, The implicit functions theorem and implicit parametrizations, Ann. Acad. Rom. Sci. Ser. Math. Appl., 5 (2013), 193-208.   Google Scholar

[31]

D. Tiba, Boundary observation in shape optimization, New Trends in Differential Equations, Control Theory, and optimization, World Sci. Publ., Hackensack, NJ, (2016), 301–314.  Google Scholar

[32]

D. Tiba, Some remarks on state constraints and mixed constraints, Ann. Acad. Rom. Sci. Ser. Math. Appl., 10 (2018), 25-40.   Google Scholar

[33]

D. Tiba, A penalization approach in shape optimization, Atti Accad. Peloritana Pericolanti Cl. Sci. Fis. Mat. Natur., 96 (2018), A8, 10 pp, http://dx.doi.org/10.1478/AAPP.961A8.  Google Scholar

[34]

D. Tiba and C. Zǎlinescu, On the necessity of some constraint qualification conditions in convex programming, J.Convex Anal., 11 (2004), 95-110.   Google Scholar

[35]

E. Zuazua, Log-Lipschitz regularity and uniqueness of the flow for a field in $(W^{n/p+1, p}_loc (R^n))^n$, C. R. Math. Acad. Sci. Paris, 335 (2002), 17-22.  doi: 10.1016/S1631-073X(02)02426-3.  Google Scholar

show all references

References:
[1]

V. Barbu, Optimal Control of Variational Inequalities, Research Notes in Mathematics, 100. Pitman, Boston, MA, 1984.  Google Scholar

[2]

D. P. Bertsekas, Nonlinear Programming, Third edition, Athena Scientific Optimization and Computation Series, Athena Scientific, Belmont, MA, 2016.  Google Scholar

[3]

J. F. Bonnans and A. Shapiro, Perturbation Analysis of Optimization Problems, Springer Series in Operations Research, Springer-Verlag, New York, 2000. doi: 10.1007/978-1-4612-1394-9.  Google Scholar

[4]

F. Bouchut and L. Desvillets, On two-dimensional Hamiltonian transport equations with continuous coefficients, Diff. Int. Eqns., 14 (2001), 1015-1024.   Google Scholar

[5]

H. C. Chang, W. He and N. Prabhu, The analytic domain in the implicit function theorem, JIPAM. J. Inequal. Pure Appl. Math., 4 (2003), Art. 12, 5 pp.  Google Scholar

[6]

P. G. Ciarlet, Introduction to Numerical Linear Algebra and Optimization, Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge, 1989.  Google Scholar

[7]

F. H. Clarke, Optimization and Nonsmooth Analysis, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1983.  Google Scholar

[8]

F. Clarke and M. R. de Pinho, Optimal control problems with mixed constraints, SIAM J. Control Optim., 48 (2010), 4500-4524.  doi: 10.1137/090757642.  Google Scholar

[9]

R. J. DiPerna and P.-L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math., 98 (1989), 511-547.  doi: 10.1007/BF01393835.  Google Scholar

[10]

W. H. Fleming and R. W. Rishel, Deterministic and Stochastic Optimal Control, Applications of Mathematics, No. 1. Springer-Verlag, Berlin-New York, 1975.  Google Scholar

[11]

P. Hartman, Ordinary Differential Equations, John Wiley & Sons, Inc., New York-London-Sydney, 1964.  Google Scholar

[12]

A. Henrot and M. Pierre, Variation et Optimization de Formes: Yne Analyse Geometrique, Mathématiques & Applications, 48. Springer, Berlin, 2005. doi: 10.1007/3-540-37689-5.  Google Scholar

[13]

C. Kublik and R. Tsai, Integration over curves and surfaces defined by the closest point mapping, Res. Math. Sci., 3 (2016), 17 pp, arXiv: 1504.05478v4. doi: 10.1186/s40687-016-0053-1.  Google Scholar

[14] J. B. Lasserre, An Introduction to Polynomial and Semi-Algebraic Optimization, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2015.  doi: 10.1017/CBO9781107447226.  Google Scholar
[15]

E. B. Lee and L. Markus, Foundations of Optimal Control Theory, John Wiley & Sons, Inc., New York-London-Sydney, 1967.  Google Scholar

[16]

J.-L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Die Grundlehren der mathematischen Wissenschaften, Band 170 Springer-Verlag, New York-Berlin, 1971.  Google Scholar

[17]

A. MitsosB. Chachuat and P. I. Barton, McCormick-based relaxations of algorithms, SIAM J. Optim., 20 (2009), 573-601.  doi: 10.1137/080717341.  Google Scholar

[18]

P. Neittaanmäki, J. Sprekels and D. Tiba, Optimization of Elliptic Systems. Theory and Applications, Springer Monographs in Mathematics, Springer, New York, 2006.  Google Scholar

[19]

P. Neittaanmäki and D. Tiba, Fixed domain approaches in shape optimization problems, Inverse Problems, 28 (2012), 093001, 35 pp. doi: 10.1088/0266-5611/28/9/093001.  Google Scholar

[20]

M. R. Nicolai and D. Tiba, Implicit functions and parametrizations in dimension three: Generalized solutions, Discrete Contin. Dyn. Syst., 35 (2015), 2701-2710.  doi: 10.3934/dcds.2015.35.2701.  Google Scholar

[21]

M. R. Nicolai, An algorithm for solving implicit systems in the critical case, Ann. Acad. Rom. Sci. Ser. Math. Appl., 7 (2015), 310-322.   Google Scholar

[22]

M. R. Nicolai, High dimensional applications of implicit parametrizations in nonlinear programming, Ann. Acad. Rom. Sci. Ser. Math. Appl., 8 (2016), 44-55.   Google Scholar

[23]

M. Patriksson, Nonlinear Programming and Variational Inequality Problems: A Unified Approach, Applied Optimization, 23. Kluwer Academic Publishers, Dordrecht, 1999. doi: 10.1007/978-1-4757-2991-7.  Google Scholar

[24]

P. Phien, Some quantitative results on Lipschitz inverse and implicit functions theorems, East-West J. Math., 13 (2011), 7–22, arXiv: 1204.4916v2.  Google Scholar

[25]

M. do R. de Pinho and M. H. A. Biswas, A nonsmooth maximum principle for optimal control problems with state and mixed constraints—convex case, Discrete Contin. Dyn. Syst., Dynamical Systems, Differential Equations and Applications. 8th AIMS Conference. Suppl., 1 (2011), 174-183.   Google Scholar

[26]

M. do R. de Pinho, On necessary conditions for implicit control systems, Pure Appl. Funct. Anal., 1 (2016), 185-196.   Google Scholar

[27]

J. Sokolowski and J.-P.Zolesio., Introduction to Shape Optimization. Shape Sensitivity Analysis, Springer Series in Computational Mathematics, 16. Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-642-58106-9.  Google Scholar

[28]

M. D. StuberJ. K. Scott and P. I. Barton, Convex and concave relaxations of implicit functions, Optimization Methods and Software, 30 (2015), 424-460.  doi: 10.1080/10556788.2014.924514.  Google Scholar

[29]

D. Tiba, Iterated Hamiltonian type systems and applications, J. Diff. Equations, 264 (2018), 5465-5479.  doi: 10.1016/j.jde.2018.01.003.  Google Scholar

[30]

D. Tiba, The implicit functions theorem and implicit parametrizations, Ann. Acad. Rom. Sci. Ser. Math. Appl., 5 (2013), 193-208.   Google Scholar

[31]

D. Tiba, Boundary observation in shape optimization, New Trends in Differential Equations, Control Theory, and optimization, World Sci. Publ., Hackensack, NJ, (2016), 301–314.  Google Scholar

[32]

D. Tiba, Some remarks on state constraints and mixed constraints, Ann. Acad. Rom. Sci. Ser. Math. Appl., 10 (2018), 25-40.   Google Scholar

[33]

D. Tiba, A penalization approach in shape optimization, Atti Accad. Peloritana Pericolanti Cl. Sci. Fis. Mat. Natur., 96 (2018), A8, 10 pp, http://dx.doi.org/10.1478/AAPP.961A8.  Google Scholar

[34]

D. Tiba and C. Zǎlinescu, On the necessity of some constraint qualification conditions in convex programming, J.Convex Anal., 11 (2004), 95-110.   Google Scholar

[35]

E. Zuazua, Log-Lipschitz regularity and uniqueness of the flow for a field in $(W^{n/p+1, p}_loc (R^n))^n$, C. R. Math. Acad. Sci. Paris, 335 (2002), 17-22.  doi: 10.1016/S1631-073X(02)02426-3.  Google Scholar

Figure 1.  The admissible set in Ex.2
Figure 2.  The geometry in Ex.2
Figure 3.  Admissible set of points
Figure 4.  Optimal trajectory
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