Maximum theorem states that f(x)=argmax_y { G(x,y } is a continuous function of x. So I would like to know if this function can be taken as lipschitz continuous or locally lipchitz continuous?
maximum theorem can be found here : http://en.wikipedia.org/wiki/Maximum_theorem
The maximum theorem of your Wikipedia reference makes heavy assumptions on G. If you make even stronger assumptions you may even get a Lipschitz continuous f. But, you seem not to understand this, it's all a matter of what you assume about G.
Thank you very much for your massage. so if we have this Question f(x)=argmax_{y\in C(y)} { G(x,y) } then what assumption on G(x,y) and C(X) may result in Lipschitz continuity of f(x)?
Do you know any reference that has discussed these properties of maximum theorem in a little more details than Wikipedia ? possibly with Frechet differentiability, Bouligand differentiability and maybe other differential properties.
Where do your functions finally come from? If it is an economics application in which all functions are interpolated from discrete data, Lifschitz continuity is trivially satisfied
since all derivatives exist and are are bounded. Your idea to look for more sophisticated differentiability concepts would than be completely inadequate.
my Idea come from multi-stage where followers equilibrium is a constraint for leaders game in a multi leader-follower game.
actually to satisfy constraint qualification in nash equilibrium of my leaders optimization problem, i need to satisfy the lipschitz continuity and differentiability of the equilibrium of follower game.
my specific follower game is equivalent to the argmax of G(x,y) which the G(X,Y) are continuous, convex, in C^2 class (second order differentiable) . all of the constraint of the follower game has the same properties of G(X,Y) which is demonstrated by the setvalued map y \in C(X).
As I said, I would not care about any regularity properties of functions since empirically meaningfull functions can always (sometimes in a bit artifical way, if for instance spikes have to be regularized) be considered maximally regular. In a computer implementation, when all your functions become finite lists this is more than obvious.
Since i am working on concrete math proof of the nash equilibrium existence of my problem i need some proposition like maximum theorem. do you have such math reference that has proof for regularity of the argmax?
As far as I can see, you will only have lower semicontinuity for the argmax (in the sense of set-valued maps), unless G is _strictly concave_ in x for each y (let's abbreviate this property as (SC)). In this case, the argmax defines a (single-valued) function. Assume (SC) for the moment:
Let G be C^2 (as in your case). This allows us to characterize the f(y):=argmax_x G(x,y)
by D_x G(f(y),y)=0. If D_{xx} G is non-degenerate the implicit function theorem yields the differentiability of f and - by the chain rule - :
D_{xx} G(f(y),y) Df(x) + D_{yx} G(f(y),y)=0.
In particular f is locally Lipschitz.
As to the set-valued case, I don't know enough about it (-I don't even know the proper definition of 'Lipschitz' in this context; yet I think, it might be worth while to look in the books by J. P. Aubin.-)
Thank you for your response, however ther is a subtle mistake in your justification.
implicit function theory is only applicable when x is not in interior set of the domain. actually the argmax even in the case that G is SC might result in a solution on the boundary of C(y). hence differentiability of f(y) might not be guaranteed in that case!
Article Lipschitz continuous policy functions for strongly concave o...
This might be helpgul.
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