Finding Optimal Solutions to a Class of Parametric Optimization Problems in Terms of Parameter Values by using Multilayer Neural Networks

Document Type : Original Paper


Department of Mathematics, Tabriz Branch, Islamic azad University, Tabriz, Iran.


‎‎In this paper, parametric optimization problems are investigated. ‎In a‎ ‎parametric ‎optimization ‎problem ‎we ‎assume ‎‏‎$‎‎‎‎‎‏‎‎lambda‎in‎mathbb{R}^n‎$‎‎ ‎is ‎the ‎vector ‎of ‎the ‎parameters ‎and ‎‎$‎‎x^*$ ‎is ‎the ‎optimal ‎answer ‎corresponding ‎to ‎it. ‎The ‎purpose ‎of ‎this ‎paper ‎is ‎to ‎determine a‎ ‎function ‎such ‎as ‎‎$‎‎psi$ ‎so ‎that ‎we ‎have ‎‎$‎‎psi(‎lambda‎)=x^*$.‎ To do this, first for each ‎$‎‎‎lambda‎$‎, the corresponding optimal answer is calculated. In this way, a set of data bases consisting of parameters and the corresponding optimal values are obtained. A multilayer network of data base is trained to obtain the function ‎$‎‎psi$‎ in a domain. In fact, the function ‎$‎‎psi$‎ for each value of the parameter specifies the corresponding answer by the trained multilayer network.‎‎ Finally, we conduct several numerical examples to test our method.


Main Subjects

[1] A. Azizi and M. Masdarian and A. Hassanzadeh and Z. Bahri and T. Niedoba and A. Surowiak, Parametric optimization in rougher Flotation performance of a aulfidized mixed Copper, Ore. Minerals 4 (5) (2020) 1–19.
[2] A. Fotiou and A. P. Rostalski and P. A. Parrilo and M. Morari, Parametric optimization and optimal control using algebric geometry methods, International Journal of Control. 79 (11) (2007) 1340–1358.
[3] B. Bank and J. Guddat and D. Klatte and B. Kummer and K. Tammer, Non-linear Parametric Optimization, Springer Fachmedien Wieshaden GmbH, 1993.
[4] B. Bereanu, On stochastic linear programming II: distribution problems non stochastical matrix, Review Roumain Mathematical Pures Applications 11 (1966) 713–725 (II).
[5] C. Bazgan and A. Herzel and S. Ruzika and C. Thielen and D. Vanderpooten, An approximation algorithm for a general class of parametric optimization problems, Journal of Combinatorial Optimization, (2020) 1–31.
[6] C. Bazgan and A. Herzel and S. Ruzika and C. Thielen and D. Vanderpooten, An FPTAS for a general class of parametric optimization problems, The German Academic Exchange Service 186 (2019) 25–37.
[7] D. A. Sprecher, On the structure of continuous functions of several variables, Transactions American Mathematical Society 115 (1965) 340–355.
[8] D. Devarasiddappa and M. Chandrasekaran and R. Arunachalam, Experimental investigation and parametric optimization for minimizing surface roughness during WEDM of Ti6AL4V alloy using modified TLBO algorithm, Journal of the Brazilian Society of Mechanical Sciences and Engineering 42 (128) (2020) 1–18.
[9] D. G. Luenberger, Introduction to Linear and Nonlinear Programming, Reading MA: Addision Wesley, Second Edition , 1984.
[10] D. T. K. Huyen and J. C. Yao and N. D. Yen, The stationary point set map in general parametric optimization problems, Set-Valued and Variational Analysis, (2020) 1–23.
[11] E. Simons, Note on Parametric Linear Programming, INFORMS 8 (3) (1962) 355–358.
[12] G. Cybenko, Approximation by superpositions of a sigmoidal function, Mathematics of Control, Signals, and Systems 2 (4) (1989) 304–314.
[13] J. Guddat and F. Guerra Vasquez and H. Th. Jongen, Parametric Optimization: Singularities, Pathfollowing and Jumps, Springer Fachmedien Wiesbaden GmbH, 1990.
[14] J. M. Weaver-Rosen and P. B. C. Leal and D. J. Hartl and R. J. Malak Jr., Parametric optimization for morphing structure design: application to morphing wings adapting to changing flight conditions, Structural and Multidisciplinary Optimization 62 (6) (2020) 2995–3007.
[15] J. M. Weaver-Rosen and R. J. Malak Jr., Efficient parametric optimization for expensive single objective problems, Journal of Mechanical Design 143 (3) (2021) 1–9.
[16] K. P. Bennett and E. J. Bredensteiner, A parametric optimization method for machin learning, Informs Journal on Computing. 9 (3) (1997) 311–318.
[17] L. Schrage, Optimization Modelling with LINGO, LINDO Systems Inc., 2003.
[18] M. M. Gupta and L. Jin and N. Homma, Static and Dynamic Neural Networks, IEEE, A John Wiley & Sons, INC. Publication, 2003.
[19] M. S. Bazaraa and C. M. Shetty, Nonlinear Programming Theory and Applications, Wiley and Sons New York, 1990.
[20] M. S. Bazaraa and J. Jarvis and D. Sherali, Linear Programming and Network Falows, John Wiley and Sons, 1992.
[21] M. Zeleny, Linear Multiobjective Programming, Springer-Verlog, Berlin-Heildelberg, New York, 1974 (III).
[22] R. Oberdieck and N. A. Diangelakis and M. M. Papathenasiou and I. Nascu and N. E. N. Pistikopoulos, PoP-Parametric optimization toolbax, Industrial & Engineering Chemistry Research 55 (33) (2016) 8979–8991.
[23] S. Avraamidou and E. N. Pistokopoulos, A multi-parametric optimization approach for bilevel mixedinteger
linear and quadratic programming problems, Computers and Chemical Engineering, 125 (2019) 98–113.
[24] S. Avraamidou and E. N. Pistokopoulos, B-POP: Bi-level parametric optimization Toolbox, Computers and Chemical Engineering 122 (4) (2018) 193–202.
[25] S. Effati and M. Jafarzadeh, A new nonlinear newral network for solving a class of constrained parametric
optimization problems, Applied Mathematics and Computation 186 (1) (2007) 814–819.
[26] S. S. Rao, Engineering Optimization, Theory and Practice, Purdue University, West Lafayette, Hndiana, 1996.
[27] V. Kungurtsev and J. Jaschke, A predictor-corrector path-following algorithm for doua-degenerate parametric optimization problems, Society for Industrial and Applied Mathematics 27 (1) (2017) 538–564.
[28] W. M. Haddad and V. Chelboina, Nonlinear Dynamical Systems and Control, Princeton University Press, Princeton & Qxford, 2008.
[29] W. Xu and J. X. Xu and D. He and K. C. Tan, An evolutionary constraint-handling technique for parametric optimization of a cancer immunotherapy model, IEEE Transactions on Emerging Topics in Computational Intelligence 3 (2) (2018) 151–162.
[30] Y. Yang and Y. Gao, A new newral network for solving nonlinear covex programs with linear constraints, Neurocomputing 74 (2011) 3079–3083.