I got utterly confused as I don’t know the result I should trust. I calculated the hessian by two different tools and the result was what I showed. I don’t know why Maple shows this result.
My current problem is solving the second problem, which we are talking about that I wrote in 8th post.
But now after transforming the problem to what I’ve shown in the 18th post, I cannot write p(1)v(1) <= s(1) in a way that conforms DCP rules. The alternative way is to convert p(1)/t(1) <= s(1). I solved the problem of multiplication in the constraint p(1)/t(1)<=s(1) by using geo_mean([s(1),t(1)]) but If I write sqrt(p(1)) I would encounter the error concave - concave.
Do you know any way to rewrite one of these constraints?
The eigenvalues of the Hessian you posted agree within roundoff to what I posted. Two negative eigenvalues, two positive eigenvalues, so Hessian is indefinite. So if your constraint relates to this NON-CONVEX problem, then forget about it - NON-CONVEX. If that is not the problem, please state clearly in one place everything needed to make the problem clear.
Ok, it seems that you are sure about the nonconvexity of problem.
In the 14th post you mentioned that restricting the parameter values may make it convex.
What is your advice to make it possible for this problem?
For example, can using binary paramater instead of p change the problem as mosek relaxes binary in CVX?
Or if we assume the denominator equals t and the nonconvex problem is minimization of p/t could the result of p^2/t be equivalent to main problem?
It is your problem. not mine. it is non-convex, at least in the generality you originally presented. It is non-convex in my simplified version. I think the simplified form is convex if a = b, so that is a special case of parameter values.
If the problem you really want to solve is non-convex, then use a non-convex nonlinear optimizer. You may find YALMIP convenient for that, or formulate directly with native interfaces for FMINCON (local optimizer), BARON (global optimizer), or whatever.
In an earlier post, you wrote
I thought it could be convex as it was mentioned by a paper and its plot.
but you haven’t shown us the paper or the plot. Perhaps the paper is solving a specialized version of this problem? Or some relaxation or approximation was made? Or you have just misinterpreted something? Or the paper is wrong (it happens even in the most prestigious journals)?
Ok,
Thanks for your responses.