# A strict minimax inequality criterion and some of its consequences

A strict minimax inequality criterion and some of its consequences In this paper, we point out a very flexible scheme within which a strict minimax inequality occurs. We then show the fruitfulness of this approach presenting a series of various consequences. Here is one of them: Let Y be a finite-dimensional real Hilbert space, J : Y → R a C 1 function with locally Lipschitzian derivative, and $${\varphi : Y \to [0, + \infty[}$$ a C 1 convex function with locally Lipschitzian derivative at 0 and $${\varphi^{-1}(0) = \{0\}}$$ . Then, for each $${x_0 \in Y}$$ for which J′(x 0) ≠ 0, there exists δ > 0 such that, for each $${r \in ]0, \delta[}$$ , the restriction of J to B(x 0, r) has a unique global minimum u r which satisfies $$J(u_r)\leq J(x)-\varphi(x-u_r)$$ for all $${x \in B(x_0, r)}$$ , where $${B(x_0, r) = \{x \in Y : \|x-x_0\|\leq{r}\}.}$$ http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

# A strict minimax inequality criterion and some of its consequences

, Volume 16 (3) – Mar 4, 2012
16 pages

/lp/springer_journal/a-strict-minimax-inequality-criterion-and-some-of-its-consequences-WSiJi3fQrR
Publisher
Springer Journals
Subject
Mathematics; Operator Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics; Potential Theory; Fourier Analysis
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-012-0164-x
Publisher site
See Article on Publisher Site

### Abstract

In this paper, we point out a very flexible scheme within which a strict minimax inequality occurs. We then show the fruitfulness of this approach presenting a series of various consequences. Here is one of them: Let Y be a finite-dimensional real Hilbert space, J : Y → R a C 1 function with locally Lipschitzian derivative, and $${\varphi : Y \to [0, + \infty[}$$ a C 1 convex function with locally Lipschitzian derivative at 0 and $${\varphi^{-1}(0) = \{0\}}$$ . Then, for each $${x_0 \in Y}$$ for which J′(x 0) ≠ 0, there exists δ > 0 such that, for each $${r \in ]0, \delta[}$$ , the restriction of J to B(x 0, r) has a unique global minimum u r which satisfies $$J(u_r)\leq J(x)-\varphi(x-u_r)$$ for all $${x \in B(x_0, r)}$$ , where $${B(x_0, r) = \{x \in Y : \|x-x_0\|\leq{r}\}.}$$

### Journal

PositivitySpringer Journals

Published: Mar 4, 2012

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