a***n
发帖数: 3633 | 1 F(x,y)
定义 G(x)=min_y F(x,y)
是否有min_x G(x)= min_{x,y} F(x,y) ?
如果无，光滑函数有这样的性质么？什么书会讲这个问题？谢谢。 | s*****e
发帖数: 115 | 2 If all of the minima used in your definitions actually exist, then the
answer is positive even if the function F(x,y) is discontinuous.
1. Suppose min_x G(x) = G(x'), then
min_x G(x) = G(x') = min_y F(x',y) >= min_{x,y} F(x,y)
2. Suppose min_{x,y} F(x,y) = F(x*,y*), then
min_{x,y} F(x,y) = F(x*,y*) >= min_y F(x*,y) = G(x*) >= min_x G(x)
It follows that min_x G(x) = min_{x,y} F(x,y). | l*****a
发帖数: 119 | 3 min_{x,y} F(x,y) = F(x*,y*) >= min_y F(x*,y)
不等号方向反了吧?
LZ的说的方法叫 alternating direction methods 自己找找 我不熟悉 只知道
in general是肯定不成立的 什么情况成立就要查文献了 | s*****e
发帖数: 115 | 4
F(x*,y*) >= min_y F(x*,y)
Don't see why this is true? Just think again!
The statement is TRUE even if "min" is replaced by "inf", and F(x,y) can be
any (even discontinuous) function, i.e.,
inf_{x,y} F(x,y) = inf_y { inf_x F(x,y) } = inf_x { inf_y F(x,y) }
The proof will be somewhat different from the "min" case since there may not
exist (x*,y*) such that F(x*,y*) = inf_{x,y} F(x,y). But still, this is just a very
simple exercise for "inf/sup" in intro to real analysis.
【在 l*****a 的大作中提到】
: min_{x,y} F(x,y) = F(x*,y*) >= min_y F(x*,y)
: 不等号方向反了吧?
: LZ的说的方法叫 alternating direction methods 自己找找 我不熟悉 只知道
: in general是肯定不成立的 什么情况成立就要查文献了
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