海塞矩阵和函数凸性

【炊烟】

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收藏 2014-10-19

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

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pingguagain

2014-10-19 22:37:04

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.

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【炊烟】

2014-10-20 00:26:20

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.

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【炊烟】

2014-10-20 00:54:21

【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.

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pingguagain

2014-10-20 04:47:34

【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.

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pingguagain

2014-10-20 04:48:58

【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.

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【炊烟】

2014-10-20 09:35:43

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.

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pingguagain

2014-10-20 10:32:54

【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.

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【炊烟】

2014-10-20 10:53:26

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.

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pingguagain

2014-10-20 11:11:01

【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.

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【炊烟】

2014-10-20 12:13:55

pingguagain 发表于 2014-10-20 11:11
The Hessian of f=xy is
0 1
1 0

I am making a programming recently.sone constraints of the model contain conbination of x(i)*y(i),i=1,2..n.
SO how to make those constraints convex? could you give me some useful suggestions for it is very important for my work.thank you.

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【炊烟】

2014-10-20 15:55:16

pingguagain 发表于 2014-10-20 11:11
The Hessian of f=xy is
0 1
1 0

[x,y]=meshgrid(-1:0.1:1,-1:0.1:1);
z=x.*y;
contour(x,y,z,'g--')
Run these code on MAtLAB,you can see the area where x,y>0 is convex.I was confused.

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pingguagain

2014-10-20 21:04:58

【炊烟】发表于 2014-10-20 12:13
I am making a programming recently.sone constraints of the model contain conbination of x(i)*y(i), ...

Convex function and convex set are two different concepts although there is some connection between them. For the relationship, read Chapter 6 of Dixit's book, Optimization in Economic Theory. Function f=xy is neither convex nor concave when x>0,y>0. However, the set {(x,y)|x>0,y>0, and xy>0} is convex.

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【炊烟】

2014-10-20 21:33:06

pingguagain 发表于 2014-10-20 21:04
Convex function and convex set are two different concepts although there is some connection betwee ...

THANK YOU for your great enthusiasm.I want to make sure that whether the feasible zone of constraint sum(i,x(i)*y(i))>0,(x(i),y(i)>0) is convex.

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