A Differential Characterisation of the Minimax Inequality

作者：Mosconi Sunra J N^*

来源：Journal of Convex Analysis, 2012, 19(1): 185-199.

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

We prove the following result: let K subset of R-N be convex with nonempty interior, X a topological space and f :K x X --%26gt; 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 %26lt;br%26gt;sup(lambda is an element of K) inf(x is an element of X) f(lambda, x) + g(lambda) %26lt; inf(x is an element of X) sup(lambda is an element of K) f(lambda, x) + g(lambda), %26lt;br%26gt;for some continuous concave g : K --%26gt; R, is equivalent to the following condition on superdifferentials: if F(lambda) = infx f (lambda, x), for some lambda is an element of (K) over circle %26lt;br%26gt;partial derivative F(lambda) \ boolean OR(x is an element of X f(lambda, x) = F(lambda)) partial derivative f(lambda, x) not equal empty set %26lt;br%26gt;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.

• 出版日期2012