On Saddle Points in Semidefinite Optimization via Separation Scheme

Hezhi Luo (), Huixian Wu () and Jianzhen Liu ()
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Hezhi Luo: Zhejiang University of Technology
Huixian Wu: Hangzhou Dianzi University
Jianzhen Liu: Hangzhou Dianzi University

Journal of Optimization Theory and Applications, 2015, vol. 165, issue 1, No 6, 113-150

Abstract: Abstract This paper aims at investigating saddle point conditions for augmented Lagrangian functions for semidefinite optimization problems. By means of the image space analysis, the existence of a saddle point is shown to be equivalent to a regular weak nonlinear separation of two suitable subsets in the image space (IS) associated with the given problem. Especially, three classes of augmented Lagrangians based on smooth spectral penalty functions can be derived, as particular cases, from a nonlinear separation scheme in the IS. Without requiring the strict complementarity, it is proved that, under strong second-order sufficiency conditions, all these augmented Lagrangian functions admit a local saddle point, and their Hessians become positive definite in a neighborhood of a local optimal point of the original problem. The existence of global saddle points is then obtained under additional assumptions that do not require the compactness of the feasible set.

Keywords: Semidefinite optimization; Regular weak nonlinear separation; Augmented Lagrangian function; Saddle point; Strong second-order sufficiency condition; 90C22; 65K05; 49J52 (search for similar items in EconPapers)
Date: 2015
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Citations: View citations in EconPapers (3)

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DOI: 10.1007/s10957-014-0634-3

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