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doi: 10.3934/naco.2021061
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## A new relaxation method for optimal control of semilinear elliptic variational inequalities obstacle problems

 1 Laboratory of Fundamental and Numerical Mathematics, University Ferhat Abbas of Setif 1, Setif 19000, Algeria 2 INSA Rennes, CNRS, IRMAR-UMR 6625, University of Rennes, F-35000 Rennes, France

* Corresponding author: el-hassene.osmani@insa-rennes.fr

Received  March 2021 Revised  November 2021 Early access December 2021

In this paper, we investigate optimal control problems governed by semilinear elliptic variational inequalities involving constraints on the state, and more precisely the obstacle problem. Since we adopt a numerical point of view, we first relax the feasible domain of the problem, then using both mathematical programming methods and penalization methods we get optimality conditions with smooth Lagrange multipliers. Some numerical experiments using the Interior Point Optimizer (IPOPT), Nonlinear Interior point Trust Region Optimization (KNITRO) and Sequential Quadratic Optimization Technique (SNOPT) are presented to verify the efficiency of our approach.

Citation: EL Hassene Osmani, Mounir Haddou, Naceurdine Bensalem. A new relaxation method for optimal control of semilinear elliptic variational inequalities obstacle problems. Numerical Algebra, Control and Optimization, doi: 10.3934/naco.2021061
##### References:
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##### References:
 [1] A. Akkouche, A. Maidi and A. Aidene, Optimal control of partial differential equations based on the variational iteration method, Computers & Mathematics with Applications, 68 (2014), 622-631.  doi: 10.1016/j.camwa.2014.07.007. [2] AMPL Modeling Language for Mathematical Programming, Available from: http://www.ampl.com. [3] V. Barbu, Optimal Control of Variational Inequalities, Boston, Pitman Advanced Pub. Program, 1984. doi: 9780080958767. [4] V. Barbu, Analysis and Control of Non Linear Infinite Dimensional Systems, Mathematics in Science and Engineering, Academic Press, 1993. doi: 978-0-12-078145-4. [5] M. Bergounioux, Optimal control of semilinear elliptic obstacle problems, Journal of Nonlinear and Convex Analysis, 31 (2002), 25-39. [6] M. Bergounioux, Optimality Conditions For Optimal Control of Elliptic Problems Governed by Variational Inequalities, Rapport de Recherche, Université d'Orléans, 1995. [7] M. Bergounioux, Optimal control of an obstacle problem, Applied Mathematics and Optimization, 36 (1997), 147-172.  doi: 10.1007/s002459900058. [8] M. Bergounioux and F. Mignot, Control of variational inequalities and Lagrange multipliers, ESAIM, COCV, 5 (2000), 45-70.  doi: 10.1051/cocv:2000101. [9] M. Bergounioux and M. Haddou, A SQP-Augmented Lagrangian method for optimal control of semilinear elliptic variational inequalities, ISNM International Series of Numerical Mathematics, 143 (2003), 57-72.  doi: 10.1007/978-3-0348-8001-5_4. [10] M. Bergounioux and D. Tiba, General optimality conditions for constrained convex control problems, SIAM Journal on Control and Optimization, 34 (1994), 698-711.  doi: 10.1137/S0363012994261987. [11] M. Bergounioux, Optimal control of variational inequalities: A mathematical programming approach, International Federation for Information Processing. Springer, Boston, MA, (1996), 123–130. doi: 10.1007/978-0-387-34922-0_11. [12] A. Bermudez and C. Saguez, Optimal control of variational inequalities: Optimality conditions and numerical methods, Free boundary problems: applicatons and theory, Research Notes in Mathematics, Pitman, Boston, 121 (1988), 478–487. [13] S. I. Birbil, S. H. Fang and J. Han, An entropic regularization approach for mathematical programs with equilibrium constraints, Computer and Operations Research, 31 (2004), 2249-2262.  doi: 10.1016/S0305-0548(03)00176-X. [14] A. Friedman, Variational Principles and Free-Boundray Problems, John Wiley & Sons, Inc. New York, 1982. [15] P. E. Gill, W. Murray, M. A. Sanders, A. Drud and E. Kalvelagen, GAMS/SNOPT: An SQP Algorithm for large-scale constrained optimization, 2000, Available from: http://www.gams.com/docs/solver/snopt. doi: 10.1137/S1052623499350013. [16] M. Haddou, A new class of smoothing methods for mathematical programs with equilibrium constraints, An International Journal Pacific Journal of Optimization, 5 (2009), 87-95. [17] J. H. He and H. Latifizadeh, A general numerical algorithm for nonlinear differential equations by the variational iteration method, International Journal of Numerical Methods for Heat and Fluid Flow, 30 (2020), 4797-4810.  doi: 10.1108/HFF-01-2020-0029. [18] J. H. He, Lagrange crisis and generalized variational principle for 3D unsteady flow, International Journal of Numerical Methods for Heat and Fluid Flow, 30 (2020), 1189-1196. [19] M. A. Krasnosel'skii and Y. B. Rutickii, Convex Functions and Orlicz Spaces, Noordhoff, Groningen, 1996. [20] F. Mignot and J. P. Puel, Optimal control in some variational inequalities, SIAM Journal on Control and Optimization, 22 (1984), 466-476.  doi: 10.1137/0322028. [21] H. R. Byrd, J. Nocedal and A. R. Waltz, KNITRO: An integrated package for nonlinear optimization, Large-Scale Nonlinear Optimization, 83 (2006), 35-59.  doi: 10.1007/0-387-30065-1_4. [22] F. Troltzsch, Optimality Conditions for Parabolic Control Problems and Applications, Leipzig, 1984. [23] A. Wächter and T. Biegler, On the implementation of a primal-dual interior point filter line search algorithm for large-scale nonlinear programming, Mathematical Programming, 106 (2006), 25-57.  doi: 10.1007/s10107-004-0559-y. [24] J. Zowe and S. Kurcyusz, Regularity and stability for the mathematical programming problem in Banach spaces, Applied Mathematics and Optimization, 5 (1979), 49-62.  doi: 10.1007/BF01442543.
Data of the considered example
Optimal solution with IPOPT solver using the $\theta_{\alpha}^{1}$, N = 20, $\alpha = 10^{-3}$, and $\varepsilon = 10^{-3}$
Example 2 using $\theta^{1}_{\alpha}, N = 15$ and $\alpha = 10^{-3}$
Using the $\theta^{1}_{\alpha}$ smoothing function -Example 1- N = 20
 $\alpha$ $\mid\mid Ay-g(y)-f-v-\xi \mid\mid_{2}$ $\left( y-\psi, \xi \right)_{2}$ $\text{J}$ 1.e-1 4.19213e-06 0.00554001 6.4520064e-01 1.e-2 9.73059e-06 3.90692e-05 6.4808311e-01 1.e-3 1.66885e-05 3.15736e-07 6.4906597e-01 1.e-4 7.20318e-06 3.45708e-09 6.4906596e-01
 $\alpha$ $\mid\mid Ay-g(y)-f-v-\xi \mid\mid_{2}$ $\left( y-\psi, \xi \right)_{2}$ $\text{J}$ 1.e-1 4.19213e-06 0.00554001 6.4520064e-01 1.e-2 9.73059e-06 3.90692e-05 6.4808311e-01 1.e-3 1.66885e-05 3.15736e-07 6.4906597e-01 1.e-4 7.20318e-06 3.45708e-09 6.4906596e-01
Using the $\theta_{\alpha}^{\text{log}}$ smoothing function -Example 1- N = 20
 $\alpha$ $\mid\mid Ay-g(y)-f-v-\xi \mid\mid_{2}$ $\left( y-\psi, \xi \right)_{2}$ $\text{J}$ 1.e-1 4.77128e-06 0.00275542 6.4585951e-01 1.e-2 1.00013e-05 2.29846e-05 6.4810490e-01 1.e-3 1.67039e-05 2.54095e-07 6.4906099e-01 1.e-4 3.05604e-07 3.97519e-09 6.4906099e-01
 $\alpha$ $\mid\mid Ay-g(y)-f-v-\xi \mid\mid_{2}$ $\left( y-\psi, \xi \right)_{2}$ $\text{J}$ 1.e-1 4.77128e-06 0.00275542 6.4585951e-01 1.e-2 1.00013e-05 2.29846e-05 6.4810490e-01 1.e-3 1.67039e-05 2.54095e-07 6.4906099e-01 1.e-4 3.05604e-07 3.97519e-09 6.4906099e-01
Using the $\theta_{\alpha}^{\text{1}}$ smoothing function -Example 1- N = 20 where $\alpha = 10^{-2}$ is fixed
 $\varepsilon$ $\mid\mid Ay-g(y)-f-v-\xi \mid\mid_{2}$ $\left( y-\psi, \xi \right)_{2}$ $\text{J}$ 1.e-1 0.000190242 6.60632e-05 6.4772424e-01 1.e-2 2.49746e-05 5.1126e-05 6.4776774e-01 1.e-3 9.73059e-06 3.90692e-05 6.4808311e-01 1.e-4 2.50301e-06 3.77642e-05 6.4810870e-01
 $\varepsilon$ $\mid\mid Ay-g(y)-f-v-\xi \mid\mid_{2}$ $\left( y-\psi, \xi \right)_{2}$ $\text{J}$ 1.e-1 0.000190242 6.60632e-05 6.4772424e-01 1.e-2 2.49746e-05 5.1126e-05 6.4776774e-01 1.e-3 9.73059e-06 3.90692e-05 6.4808311e-01 1.e-4 2.50301e-06 3.77642e-05 6.4810870e-01
Using the $\theta_{\alpha}^{1}$ smoothing function -Example 1- N = 20
 Solver $\mid\mid Ay-g(y)-f-v-\xi \mid\mid_{2}$ $\left( y-\psi, \xi \right)_{2}$ J Nb.Iter SNOPT 3.73674e-09 9.04096e-07 6.47795123e-1 46346 KNITRO 7.72611e-13 9.05794e-05 6.4779048e-01 64 IPOPT 9.73059e-06 3.90692e-05 6.4808311e-01 478
 Solver $\mid\mid Ay-g(y)-f-v-\xi \mid\mid_{2}$ $\left( y-\psi, \xi \right)_{2}$ J Nb.Iter SNOPT 3.73674e-09 9.04096e-07 6.47795123e-1 46346 KNITRO 7.72611e-13 9.05794e-05 6.4779048e-01 64 IPOPT 9.73059e-06 3.90692e-05 6.4808311e-01 478
Using the $\theta_{\alpha}^{1}$ smoothing function -Example 2- N = 15
 $\alpha$ $\mid\mid Ay-g(y)-f-v-\xi \mid\mid_{2}$ $\left( y-\psi, \xi \right)_{2}$ $\mid\mid y-y^{*} \mid\mid_{2}$ $\mid\mid v-v^{*} \mid\mid_{2}$ $|J-J^{*}|$ 1.e-1 3.63549e-08 0.00420714 0.00029353 3.63549e-07 2.24511e-05 1.e-2 4.61188e-08 4.35654e-05 3.21283e-06 4.60966e-07 3.93248e-05 1.e-3 3.51106e-13 4.39172e-07 2.54572e-08 5.07363e-07 3.94924e-05 1.e-4 3.93201e-13 1.19934e-09 2.01061e-08 1.72508e-06 3.94944e-05
 $\alpha$ $\mid\mid Ay-g(y)-f-v-\xi \mid\mid_{2}$ $\left( y-\psi, \xi \right)_{2}$ $\mid\mid y-y^{*} \mid\mid_{2}$ $\mid\mid v-v^{*} \mid\mid_{2}$ $|J-J^{*}|$ 1.e-1 3.63549e-08 0.00420714 0.00029353 3.63549e-07 2.24511e-05 1.e-2 4.61188e-08 4.35654e-05 3.21283e-06 4.60966e-07 3.93248e-05 1.e-3 3.51106e-13 4.39172e-07 2.54572e-08 5.07363e-07 3.94924e-05 1.e-4 3.93201e-13 1.19934e-09 2.01061e-08 1.72508e-06 3.94944e-05
Using the $\theta_{\alpha}^{1}$ smoothing function -Example 2- $\alpha = 10^{-3}$
 $N$ $|J-J^{*}|$ 3 0.009068814 6 0.000722747 9 0.000270778 12 5.37292e-05 15 3.94924e-05 18 9.16493e-06 21 8.82412e-06 24 8.85534e-07
 $N$ $|J-J^{*}|$ 3 0.009068814 6 0.000722747 9 0.000270778 12 5.37292e-05 15 3.94924e-05 18 9.16493e-06 21 8.82412e-06 24 8.85534e-07
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