Abstract

We prove the following result: let K⊆RNK\subseteq \mathbb{R}^N be convex with nonempty interior, XX a topological space and f ⁣:K×X→Rf\colon K\times X\to\mathbb{R} be concave and u.s.c. in the first variable and coercive and l.s.c. in the second. Then the (perturbed) strict minimax inequality sup⁡λ∈Kinf⁡x∈Xf(λ,x)+g(λ)<inf⁡x∈Xsup⁡λ∈Kf(λ,x)+g(λ),\sup_{\lambda\in K}\inf_{x\in X}f(\lambda,x)+g(\lambda)<\inf_{x\in X} \sup_{\lambda\in K}f(\lambda,x)+g(\lambda), for some continuous concave g ⁣:K→Rg\colon K\to\mathbb{R}, is equivalent to the following condition on superdifferentials: if F(λ)=inf⁡Xf(λ,x)F(\lambda)=\inf_X f(\lambda, x), for some λ∈K˚\lambda\in\mathring{K} ∂F(λ)∖⋃x∈Xf(λ,x)=F(λ)∂f(λ,x)≠∅.\partial F(\lambda)\setminus \bigcup_{\substack{x\in X\\ f(\lambda, x) =F(\lambda)}}\partial f(\lambda, x)\neq\emptyset. As an application of this differential characterisation we prove a generalised version of a theorem of Ricceri, a criterion of regularity for marginal functions, and the fact that to check whether some perturbed minimax inequality holds, one can test with affine perturbation only.

Author information

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Sunra J. N. Mosconi

Dip. di Matematica e Informatica, Università di Catania, Viale A. Doria 6, 95125 Catania, Italy

mosconi@dmi.unict.it

Keywords

  • Minimax inequality
  • concave functions
  • marginal functions
  • multiple solutions to variational problems
  • nonlinear eigenvalues

References

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Suggested citation

S. J. N. Mosconi. “A Differential Characterisation of the Minimax Inequality.” Journal of Convex Analysis 19 (2012), No. 1, 185–199. https://doi.org/10.68381/jca19011

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Published by Heldermann Verlag, 2012. Rights now held by Banach Press LLC.