Testing for Convexity for a function

Question

Please any one can help figure out if this funcion is concave or convex, any help is greatly appriciated. Any links on how to test fo convexity for such a function is also greatly appriciated. I tried to find the Hessian and I have some terms zero,so I am confused.

$f(p,q) = 1-p1*q1-p2*q2-p3*q3$

where all p and q are probabilities, i.e 0 $\le$ $p_i$ $\le$ 1 and 0 $\le$ $q_i$ $\le$ for i=1,2,3.

Thank you in advace.

You can add a constant to a function and it doesn't change convexity/concavity, so really you just want to know the convexity/concavity of $f(p,q) = - p \cdot q$. — user2566092, Sep 01 '13 at 23:22
Are the p-probabilities constrained to sum to unity? Are the q-probabilities so constrained also? — Alecos Papadopoulos, Sep 01 '13 at 23:34
@user2566092, I am really not sure, I understand what you said — user92636, Sep 01 '13 at 23:48

score 0 · Accepted Answer · answered Sep 01 '13 at 23:48

0

A positive linear combination of concave/convex functions is again concave/convex. For you, the base case is when you consider a single coordinate $f(p,q) = -p_iq_i$. This function is concave for all $p_i,q_i$ in the interval $[0,1]$, so your overall function is concave.

answered Sep 01 '13 at 23:48

user2566092

26,142

and how, do you tell that −piqi is concave, sorry if it is a supid question – user92636 Sep 01 '13 at 23:52
The Hessian is $H = \left({\begin{array}[cc] 00 & -1 \ -1 & 0 \end{array}}\right)$ so $(x,y)H(x,y)^T = -2xy \leq 0$ if $x,y \geq 0$. – user2566092 Sep 01 '13 at 23:56
I probably should read some,more because for the H I, thought I get all zeros. – user92636 Sep 02 '13 at 00:00
mybad,i get the H part of it, but what about the H transpose, what is that you doing? and please if not too much to ask, can you send me some good links? I find the det. and I got -1 – user92636 Sep 02 '13 at 00:02

Testing for Convexity for a function

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