A new auxiliary function approach for inequality constrained global optimization problems

In this study, we deal with the nonlinear constrained global optimization problems. First, we introduce a new smooth exact penalty function for solvingconstrained optimization problems. We combine the exact penalty functionwith the auxiliary function in regard to constrained global optimization. Wepresent a new auxiliary function approach and the adapted algorithm in orderto solve non-linear inequality constrained global optimization problems. Finally, we illustrate the efficiency of the algorithm on some numerical examples.

PDF

___

[1] Ling, B.W.K., Wu, C.Z., Teo, K.L. & Rehbock, V. (2013). Global optimal design of IIR filters via constraint transcription and filled function methods. Circuits, Systems, and Signal Processing, 32, 1313–1334.
[2] Evirgen, F. (2016). Analyze the optimal solutions of optimization problems by means of fractional gradient based system using VIM. An International Journal of Optimization and Control: Theories & Applications (IJOCTA). 6(2), 75–83.
[3] Akteke-Ozkurt, B., Weber, G.W. and Koksal, G. (2017). Optimization of generalized desirability functions under model uncertainty. Optimization, 66(12), 2157–2169.
[4] Ozmen, A., Kropat, E. & Weber, G.W. (2017). Robust optimization in spline regression models for multi-model regulatory networks under polyhedral uncertainty. Optimization, 66 (12), 2135–2155.
[5] Nocedal, J. and Wright, S.J. (2006). Numerical Optimization. 2nd Edition, Springer, New York.
[6] Di Pillo, G. & Grippo, L. (1989). Exact penalty functions in constrained optimization, SIAM Journal on Control and Optimization, 27(6), 1333–1360.
[7] Zheng, F.Y. & Zhang, L.S. (2012). New simple exact penalty function for constrainedoptimization. Applied Mathematics and Mechanics, 33(7), 951–962.
[8] Rao, S.S. (2009). Engineering Optimization: Theory and Practice. 4th Edition, John Wiley & Sons, New Jersey.
[9] Zangwill, W.I. (1967). Nonlinear programing via penalty functions. Management Science, 13, 344–358.
[10] Bertsekas, D. (1975). Nondifferentiable optimization via approximation. Mathematical Programming Study, 3, 1–25.
[11] Chen, C., & Mangasarian, O.L. (1996). A class of smoothing functions for nonlinear and mixed complementarity problem. Computational Optimization and Applications, 5, 97– 138.
[12] Xavier, A.E. (2010). The hyperbolic smoothing clustering method. Pattern Recognition, 43, 731–737.
[13] Chen, X. (2012). Smoothing methods for nonsmooth, nonconvex minimzation. Mathematical Programming Ser. B, 134 71–99.
[14] Grossmann, C. (2016). Smoothing techniques for exact penalty function methods. Contemporary Mathematics, 658 , 249–265.
[15] Zang, I. (1980). A smoothing out technique for min-max optimization. Mathematical Programming, 19, 61–77.
[16] Bagirov, A.M., Nuamiat, A. Al & Sultanova, N. (2013). Hyperbolic smoothing functions for non-smooth minimization. Optimization, 62(6), 759-782.
[17] Pinar, M.C. & Zenios, S. (1994). On smoothing exact penalty functions for convex constrained optimization. SIAM Journal of Optimization, 4, 468–511.
[18] Meng, Z., Dang, C., Jiang, M. & Shen R. (2011). A smoothing objective penalty function algorithm for inequality constrained optimization problems. Journal Numerical Functional Analysis and Optimization, 32, 806– 820.
[19] Lian, S.J. (2012). Smoothing approximation to l1 exact penalty for inequality constrained optimization. Applied Mathematics and Computation, 219, 3113–3121 (2012).
[20] Sahiner, A., Kapusuz, G. & Yilmaz, N. (2016). A new smoothing approach to exact penalty functions for inequality constrained optimization problems. Numerical Algebra, Control and Optimization, 6(2), 161–173.
[21] Sahiner, A., Yilmaz, N. & Kapusuz, G. (2019). A novel modeling and smoothing technique in global optimization. Journal of Industrial and Management Optimization, 15 (1), 113–139.
[22] Lin, H., Wang, Y., Gao, Y. & Wang, X. (2018). A filled function method for global optimization with inequality constraints. Computational and Applied Mathematics, 37(2), 1524–1536.
[23] Horst, R. & Pardalos, P.M. (eds.). (1995). Handbook of Global Optimization. Kluwer Academic Publishers, Dordrecht.
[24] Locatelli, M. & Schoen, F. (2013). Global Optimization: Theory, Algorithms, and Applications. SIAM, Philadelphia.
[25] Sergeyev, Y.D., Strongin, R.G. & Lera, D. (2013). Introduction to Global Optimization Exploiting Space-Filling Curves. Springer, New York.
[26] Zhigljavsky, A. & Zilinskas, A. (2008). Stochastic Global Optimization, Springer, New York.
[27] Levy, A.V. & Montalvo, A. (1985). The tunneling algorithm for the global minimizationof functions. SIAM Journal on Scientific and Statistical Computing, 6 (2), 15–29.
[28] Ge, R.P. (1990). A filled function method for finding global minimizer of a function of several variables. Mathematical Programming, 46, 191–204.
[29] Zhang, L.-S., Ng, C.-K., Li, D. & Tian, W.- W. (2004). A new filled function method for global optimization. Journal of Global Optimization, 28, 17–43.
[30] Ng, C.K., Li, D. & Zhang, L.S. (2010). Global descent method for global optimization. SIAM Journal of Optimization, 20(6), 3161–3184.
[31] Wang, Y., Fang, W. & Wu, T. (2009). A cutpeak function method for global optimization. Journal of Computational and Applied Mathematics, 230, 135–142.
[32] Liu, J., Zhang, S., Wu, C., Liang, J., Wang, X. & Teo, K.L. (2016). A hybrid approach to constrained global optimization. Applied Soft Computing, 47, 281–294.
[33] Sahiner, A., Yilmaz, N. & Kapusuz, G. (2017). A descent global optimization method based on smoothing techniques via Bezier curves. Carpathian Journal of Mathematics, 33(3), 373–380.
[34] Xu, X., Meng, Z., Sun, J. & Shen, R. (2011). A penalty function method based on smoothing lower order penalty function. Journal of Computational and Applied Mathematics, 235, 4047–4058.
[35] Zhang, Y., Xu, Y. & Zhang, L. (2009). A filled function method applied to nonsmooth constrained global optimization. Journal of Computational and Applied Mathematics, 232, 415–426.
[36] Wu, Z.Y., Bai, F.S., Lee, H.W.J. & Yang, Y.J. (2007). A filled function method for constrained global optimization. Journal of Global Optimization, 39, 495–507.
[37] Gao, Y., Yang, Y. & You, M. (2015). A new filled function method for global optimization. Applied Mathematics and Computation, 268, 685–695.
[38] Jones, D.R., Perttunen, C.D. & Stuckman, B.E. (1993). Lipschitzian optimization without the Lipschitz constant. Journal of Optimization Theory and Applications, 79, 157– 181.
[39] Jones, D.R., Schonlau, M. & Welch, W.J. (1998). Efficient global optimization of expensive black-box functions. Journal of Global Optimization, 13, 455–492.
[40] Kirkpatrick, S., Gelatt, C.D. & Vecci, P.M. (1983). Optimization by simulated annealing. Science, 220, 671–680.
[41] Kennedy, J. & Eberhart, R. (1997). Particle swarm optimization. IEEE International Conference on Neural Networks, 1, 1942– 1948.