海塞矩阵和函数凸性

是 否 +2 论坛币

k人 参与回答

经管之家送您一份

应届毕业生专属福利!

求职就业群

赵安豆老师微信：zhaoandou666

经管之家联合CDA

送您一个全额奖学金名额~ !

立即领取

感谢您参与论坛问题回答

经管之家送您两个论坛币！

+2 论坛币

用海塞矩阵证明函数凸性时，需要保证海塞矩阵的顺序主子式全部半正定吗？
另外什么文献可以找到海塞矩阵半正定推出函数为凸函数的证明，谢谢

扫码加我 拉你入群

请注明：姓名-公司-职位

以便审核进群资格，未注明则拒绝

分享0 收藏2 回帖

关键词：凸函数 正定

If you want to show that a function is convex (not strictly convex), only LEADING principle minors >=0 is not enough. You need to show all principle minors >=0.

Google Hessian and principle minors.

pingguagain 发表于 2014-10-19 22:37
If you want to show that a function is convex (not strictly convex), only LEADING principle minors > ...

thank you very much and I have another question .let a set D1 is a convex set and D2 is the other one,then is the set D1-D2  also a convex set ? what's more ,how to proof.   

【炊烟】 发表于 2014-10-20 00:26
thank you very much and I have another question .let a set D1 is a convex set and D2 is the other  ...

Maybe I have understood the problem.But i am  wondering whether we can use Hessian  matrix to show a function is convex when  independent  variables in a  small interval . For it is too difficult to proof it by using the definition of convex set.

【炊烟】 发表于 2014-10-20 00:26
thank you very much and I have another question .let a set D1 is a convex set and D2 is the other  ...

The answer is no. Think about two circles and one contains the other one. The difference is not convex any more.

【炊烟】 发表于 2014-10-20 00:54
Maybe I have understood the problem.But i am  wondering whether we can use Hessian  matrix to show ...

I didn't get your question. What do you mean "independent variables in a small interval". It is not a complete sentence.

pingguagain 发表于 2014-10-20 04:48
I didn't get your question. What do you mean "independent variables in a small interval". It is no ...

For example,the function z=xy is not convex  when x or y is not limited.But when x>0 and y>0,the function is a convex one.SO the question is whether Hessian matrix can slove both of the two questions.
AND the function f=xy-yz is whether a convex one when x>0,y>0,z>0.

【炊烟】 发表于 2014-10-20 09:35
For example,the function z=xy is not convex  when x or y is not limited.But when x>0 and y>0,the f ...

I don't think f=xy is convex when both variables are positive.

pingguagain 发表于 2014-10-20 10:32
I don't think f=xy is convex when both variables are positive.

DO you mean function f=xy is concave? SO I  am wondering that whether concave function is a kind of convex with different orientation.

【炊烟】 发表于 2014-10-20 10:53
DO you mean function f=xy is concave? SO I  am wondering that whether concave function is a kind o ...

The Hessian of f=xy is
0 1
1 0
Its determinant is -1<0. Function f=xy is neither concave nor convex.