September  2020, 10(3): 455-470. 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  September 2020 Early access  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 and Related Fields, 2020, 10 (3) : 455-470. doi: 10.3934/mcrf.2020006
References:
[1]

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

[2]

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

[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.

[4]

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

[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.

[6]

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

[7]

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

[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.

[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.

[10]

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

[11]

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

[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.

[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.

[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.
[15]

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

[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.

[17]

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

[18]

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

[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.

[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.

[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. 

[22]

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

[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.

[24]

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

[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. 

[26]

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

[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.

[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.

[29]

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

[30]

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

[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.

[32]

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

[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.

[34]

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

[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.

show all references

References:
[1]

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

[2]

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

[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.

[4]

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

[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.

[6]

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

[7]

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

[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.

[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.

[10]

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

[11]

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

[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.

[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.

[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.
[15]

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

[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.

[17]

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

[18]

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

[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.

[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.

[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. 

[22]

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

[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.

[24]

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

[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. 

[26]

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

[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.

[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.

[29]

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

[30]

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

[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.

[32]

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

[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.

[34]

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

[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.

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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