A note on the existence of saddle points of p-th power Lagrangian for constrained nonconvex optimization

Abstract

Li and Sun [D. Li and X.L. Sun, Existence of a saddle point in nonconvex constrained optimization, J. Global Optim. 21 (2001), pp. 39--50; D. Li and X.L. Sun, Convexification and existence of saddle point in a p-th-power reformulation for nonconvex constrained optimization, Nonlinear Anal. 47 (2001), pp. 5611--5622], present the existence of a global saddle point of the p-th power Lagrangian functions for constrained nonconvex optimization, under second-order sufficiency conditions and additional conditions that the feasible set is compact and the global solution of the primal problem is unique. In this article, it is shown that the same results can be obtained under additional assumptions that do not require the compactness of the feasible set and the uniqueness of global solution of the primal problem.

Keywords:

• constrained nonconvex optimization
• p-th power Lagrangian functions
• saddle points
• compactness and uniqueness

AMS Subject Classifications::

• 90C26
• 90C46

Acknowledgements

The authors are grateful to the anonymous referees for their valuable comments and constructive suggestions for improving this article. This work was partially supported by the National Natural Science Foundation of China under grant No. 11071219, the Zhejiang Provincial Natural Science Foundation of China under grant Nos. Y6090080 and Y7080184 and the Postdoctoral Key Research Foundation of China under grant No. 201003242.