Nonsmooth variational inequalities on Hadamard manifolds

ABSTRACT

Nonsmooth variational inequality problem (NVIP) and Minty nonsmooth variational inequality problem (MNVIP) in terms of a bifunction are considered in the setting of Hadamard manifolds. We first establish a Browder type fixed point theorem in the setting of Hadamard spaces, and then by using it we prove some existence results for solutions of NVIP and MNVIP. We give some relations among NVIP, MNVIP and nonsmooth optimization problem. The gap functions for NVIP and MNVIP are also studied in the setting of Hadamard manifolds.

Keywords:

• Nonsmooth variational inequalities
• Minty nonsmooth variational inequalities
• existence results
• Hadamard manifolds
• gap functions

AMS Subject Classifications:

• 49J40
• 49J52
• 47J20

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Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

In this research, the first author was supported by a research grant of DST-SERB No. EMR/2016/005124.