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.
Continuous optimization is a mathematical technique used to find the optimal solution for a problem with a continuous set of variables. It involves finding the minimum or maximum value of a function by adjusting the input variables.
The main difference between continuous and discrete optimization is that continuous optimization deals with problems where the variables can take on any real value, while discrete optimization deals with problems where the variables can only take on specific values, usually integers.
A convex function is a function where the line segment between any two points on the graph of the function lies above or on the graph. In other words, the function is always curving upwards and does not have any "dips".
A problem is considered convex if the objective function and the constraints are all convex functions. This means that the function is always curving upwards and the constraints do not create any "dips" in the graph.
Knowing if a problem is convex is important because it allows us to use efficient algorithms to find the optimal solution. Convex problems have a unique global minimum, which means that any local minimum is also the global minimum. This makes it easier to find the optimal solution without getting stuck in local minima.