The discussion revolves around the convexity of a specific function defined as f(x)=ln(|x1|+1)+(-2x1^2 + 3x2^2 + 2x3^3) + sin(x1 + x2 + x3). Participants are exploring the use of the Hessian matrix to determine convexity and discussing the implications of certain terms within the function.
The discussion is active, with participants sharing insights on the Hessian and its role in determining convexity. There are multiple interpretations regarding the convexity of specific terms, such as the sin function and the cubic term involving x3. Some participants have provided guidance on potential approaches, but no consensus has been reached.
Participants note that the problem is part of a computer science class, and there is an acknowledgment of the need for a proof regarding convexity. The discussion includes considerations of specific variables that may affect the overall convexity of the function.
Only because the practice question is this function in particular, but thank you, and yes my apologies it is for a computer science class so I placed it here but note taken for next time.pbuk said:Where a Hessian exists then that it is positive semi-definite is a necessary and sufficient condition for convexity, why do you ask about this function in particular?
Note to mentors: probably better in calculus and beyond.
sin is not convex, but I have to include a proof so I chose to do it by doing the hessian.Office_Shredder said:I think computing the hessian and checking it is good practice, so try it if you've never done it.
Often though you can do something quicker by inspection. Is there any single variable that stands out as looking particularly non convex here?
I was actually thinking ##x_3^3## which has a huge section where it is very much not convex. Restricting to ##x_1=_2=0## you get a function in ##x_3## which is not convex (can prove by taking the second derivative easily enough) and hence the whole thing is not convex. ##sin## is not convex, but if you add it to stuff that is convex enough, you canstill end up with something convex overall. E.g. ##1000000x^2+\sin(x)## has second derivative ##2000000-\sin(x)## which gives a convex function. You really want to keep your eye out for stuff which is like, unboundedly not convex.ver_mathstats said:sin is not convex, but I have to include a proof so I chose to do it by doing the hessian.
You can use the ' Report' button on the lower left to contact the mentors.pbuk said:Where a Hessian exists then that it is positive semi-definite is a necessary and sufficient condition for convexity, why do you ask about this function in particular?
Note to mentors: probably better in calculus and beyond.
I did, that's how it got moved here.WWGD said:You can use the ' Report' button on the lower left to contact the mentors.