Abstract
In this paper, we focus on deriving some sufficient conditions for global solutions to cubic minimization problems with box constraints. Our main tool is an extension of the global subdifferential, L-normal cone approach, developed by Jeyakumar et al. (J. Glob. Optim., 2007; Math. Program. Ser. A 110, 2007), and underestimator functions. By applying these tools to characteristic global solutions, we provide some sufficient conditions for cubic programming problem with box constraints. An example is given to demonstrate that the sufficient conditions can be used effectively for identifying global minimizers of certain cubic minimization problems with box constraints.
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Canfield, R.A.: Multipoint cubic surrogate function for sequential approximate optimization. Struct. Multidiscip. Optim. 27, 326–336 (2004)
Nesterov, Y.: Accelerating the cubic regularization of Newton’s method on convex problems. Math. Program. 112(1), 159–181 (2008)
Lin, C.-S., Chang, P.-R., Luh, J.Y.S.: Formulation and optimization of cubic polynomial joint trajectories for industrial robots. IEEE Trans. Autom. Control 28(12), 1066–1074 (1983)
Jeyakumar, V., Rubinov, A.M., Wu, Z.Y.: Nonconvex quadratic minimization with quadratic constraints: global optimality conditions. Math. Program. Ser. A 110(3), 521–541 (2007)
Bertsekas, D.P., Nedic, A., Ozdaglar, A.E.: Convex Analysis and Optimization. Athena Scientific and Tsinghua University Press, Belmont (2006)
Hiriart-Urruty, J.B., Lemarechal, C.: Convex Analysis and Minimization Algorithms. Springer, Berlin (1993)
Wang, Y., Liang, Z.: Global optimality conditions for cubic minimization problem with box or binary constraints. J. Glob. Optim. 47(4), 583–595 (2010)
Jeyakumar, V., Huy, N.Q.: Global minimization of difference of quadratic and convex functions over box or binary constraints. Optim. Lett. 2, 223–238 (2008)
Acknowledgements
This research was supported by NSFC (11271243), Innovation Program of Shanghai Municipal Education Commission (12ZZ071), and Shanghai Pujiang Program (11PJC059).
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Wang, Y., Liang, Z., Shen, L. (2015). Global Sufficient Conditions for Nonconvex Cubic Minimization Problem with Box Constraints. In: Gao, D., Ruan, N., Xing, W. (eds) Advances in Global Optimization. Springer Proceedings in Mathematics & Statistics, vol 95. Springer, Cham. https://doi.org/10.1007/978-3-319-08377-3_4
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DOI: https://doi.org/10.1007/978-3-319-08377-3_4
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