Abstract
In this paper, we employ the image space analysis (for short, ISA) to investigate vector quasi-equilibrium problems (for short, VQEPs) with a variable ordering relation, the constrained condition of which also consists of a variable ordering relation. The quasi relatively weak VQEP (for short, qr-weak VQEP) are defined by introducing the notion of the quasi relative interior. Linear separation for VQEP (res., qr-weak VQEP) is characterized by utilizing the quasi interior of a regularization of the image and the saddle points of generalized Lagrangian functions. Lagrangian type optimality conditions for VQEP (res., qr-weak VQEP) are then presented. Gap functions for VQEP (res., qr-weak VQEP) are also provided and moreover, it is shown that an error bound holds for the solution set of VQEP (res., qr-weak VQEP) with respect to the gap function under strong monotonicity.
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The authors greatly appreciate anonymous referees and Dr. G. Mastroeni for their valuable suggestions, which have helped to improve an early version of the paper.
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This work was supported by NSC 100-2221-E-182-072-MY2, the Natural Science Foundation of China (60804065), the Key Project of Chinese Ministry of Education (211163) and Sichuan Youth Science and Technology Foundation (2012JQ0032).
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Guu, SM., Li, J. Vector quasi-equilibrium problems: separation, saddle points and error bounds for the solution set. J Glob Optim 58, 751–767 (2014). https://doi.org/10.1007/s10898-013-0073-y
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DOI: https://doi.org/10.1007/s10898-013-0073-y