On the Positive Definiteness of Limiting Coderivative for Set-Valued Mappings

This paper concerns the interconnection between the positive definiteness of limiting coderivative and the local strong maximal monotonicity of set-valued mappings suspected in Mordukhovich and Nghia (SIAM J. Optim.26, 1032-1059,2016, Conjecture 3.6). We disprove the conjecture by a counterexample and provide some special classes at which it is true. However, the positive definiteness of limiting coderivative characterizes a new property called nearly strong monotonicity. Consequently, we show that the strong metric regularity of set-valued mappings could be obtained under the positive definiteness of limiting coderivative.

Automated summary

This paper explores the connection between the positive definiteness of limiting coderivative and the local strong maximal monotonicity of set-valued mappings. The conjecture proposed in a previous study was disproved with a counterexample, but special cases where it holds true were identified. Additionally, a new property called nearly strong monotonicity was introduced, and it was shown that strong metric regularity of set-valued mappings can be achieved under the positive definiteness of limiting coderivative.