Mathematical Problems in Engineering
Volume 2012 (2012), Article ID 871741, 16 pages
Research Article

Global Sufficient Optimality Conditions for a Special Cubic Minimization Problem

1Department of Applied Mathematics, Shanghai University of Finance and Economics, Shanghai 200433, China
2School of Economics and Management, Tongji University, Shanghai 200092, China

Received 9 February 2012; Accepted 11 June 2012

Academic Editor: Jianming Shi

Copyright © 2012 Xiaomei Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We present some sufficient global optimality conditions for a special cubic minimization problem with box constraints or binary constraints by extending the global subdifferential approach proposed by V. Jeyakumar et al. (2006). The present conditions generalize the results developed in the work of V. Jeyakumar et al. where a quadratic minimization problem with box constraints or binary constraints was considered. In addition, a special diagonal matrix is constructed, which is used to provide a convenient method for justifying the proposed sufficient conditions. Then, the reformulation of the sufficient conditions follows. It is worth noting that this reformulation is also applicable to the quadratic minimization problem with box or binary constraints considered in the works of V. Jeyakumar et al. (2006) and Y. Wang et al. (2010). Finally some examples demonstrate that our optimality conditions can effectively be used for identifying global minimizers of the certain nonconvex cubic minimization problem.