A Mountain Pass Lemma and its implications regarding the uniqueness of constrained minimizers

Mascarenhas, Walter F.

doi:10.1080/02331934.2010.527973

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“…While this condition is sufficient in one dimension, Tarone and Gruenhage (1975) gives a counterexample in multiple dimensions. A variety of papers, including Mäkeläinen et al (1981), Demidenko (2008), and Mascarenhas (2010) supplement the Hessian condition with additional regularity conditions to prove uniqueness of the minimizer.…”

Section: Introductionmentioning

confidence: 99%

Almost Sure Uniqueness of a Global Minimum Without Convexity

Cox¹

2018

Preprint

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This paper establishes the argmin of a random objective function to be unique almost surely. This paper first formulates a general result that proves almost sure uniqueness without convexity of the objective function. The general result is then applied to a variety of applications in statistics. Four applications are discussed, including uniqueness of M-estimators, both classical likelihood and penalized likelihood estimators, and two applications of the argmin theorem, threshold regression and weak identification.

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