What the result of a Graduated non-convexity function，as you can see,the problem is non convex when B is large ,but cvx still can give me the result,can i believe it

1

target is
1
hessin matrix is
2
and when b goes bigger ,it will <0
but
3

屏幕截图 2023-07-11 104852
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can also give me result ,but if i use matlab global optimazation toolbox like ga or pso,the target result is smaller a lot than cvx(if parameters is small,cvx is two times than pso/ga,and when parameters goes big,cvx can 1e3-1e4 times)
so can i believe cvx?everyone say cvx only can solve convex problrm
thanks a lot

2

That is a convex optimization problem, presuming B_{i,t} \ge 0 . I don’t know what your Hessian calculations are, but apparently they are wrong. The formulation in your code using rel_entr proves the convexity. That objective function, in one variation or another, is very common on the CVX forum as you can see by searching posts.

If CVX status is solved, it should produce a globally optimal solution. However, your input data is rather extreme (many orders of magnitude from 1), which might make the solution unreliable. if you use Mosek as solve, perhaps it displays some warning messages about that.

ga and pso in the Mathworks Global Optimization Toolbox are not real global optimization solvers - they basically try some points and hope to get lucky. Given that you are maximizing, if they produce a smaller “optimal” value, they have not succeeded in finding the global optimum (unless the CVX solution is nit accurate due to extreme (badly scaled) input data.

3

Other than the double summation, your problem formulation basically uses CVBX acceptable formulations provided to you in your previous question Channel optimal problems - #7 by Mark_L_Stone . So I don’t know why you thought this problem is not convex.

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that is my Hessian calculations,and i see a lot of people think the sum of log function is non convex,and i calculate my Hessian calculations also see when B goes big ,i write rel_entr because that cvx cant get log ,so am i should recheck my calculations?and if cvx can do the problem ,the problem is convex?

5

The problem can be formulated in CVX, so it is convex.

log of concave (or affine) argument is concave. The sum of such terms is concave. Maximization of such a function is a convex optimization problem, presuming the constraints are convex. In your case, the individual terms in the objective function are concave perspeocive functions, and therefore their sum is concave, which can be maximized in a convex optimization problem.

I recommend you read

and read and solve some exercises in at least the first 5 chapters of Convex Optimization – Boyd and Vandenberghe, which is freely downloadable at the author’s website at the link provided.

6

thanks alot ，your help is very important