@tomleslie 

Thanks for prompt reply. Can you specify the procedure for option 1 and 3 suggested above.

Thanks and Regards,

Nilesh

@vv 

I would try to use evalr with isdefinite function to know the concavity in subinterval. If any problem arise, then will let you know .

Thanks,

Nilesh

@vv 

Hi Sir,

As you shown that first function is not concave at a specific value of decision variables. If I specify an interval for variables, for eg. 

T =0..4, E=0..30000, W=0..30000, p=0..200 and ask maple to get a subrange ( subdomain) of these variables where function is concave.

I am sure that for some subdomain of these range, this function is concave. I just want to search those subrange. 

Can we specify a range in "isdefinite" or any other command in maple to know concavity in any subdomain of this range. Or we have to run a for loop for repetitive computation at each possible point ( specific value of four variables) in the provided range. Whether any efficient method is available for this?

Thanks for your pseudo concavity concept. I would also try to understand that.

Thanks and regards,

Nilesh

@acer 

Thanks for providing solutions.

I solved ap6 by hand and got ap7 so both are equal.i prefer ap7.

In ap7 there is a term (m-T)^2 term so can I tell maple to rearrange ap6 in a form in which this term is present.

Regards,

Nilesh

@vv

 Ok Sir. Thanks for help. 

Regards,

Nilesh

@John Fredsted 

Hi Sir,

Thanks a lot for your help. Sir I incorrectly used Diff. Sir, I have two queries-

1) In question 3 , equation ap4, can I get expression in such a way, that I just get t1 instead of t1(T). Although I want to say maple that t1 is a function of T T but for display purpose, it just show t1.

2) in question 5, in equation ap7, can I tell maple to get answer which should have a factor of W^(1-beta) - E^(1-beta).

Thanks you very much Sir as all my questions are answered. Please guide whether any documents/ manual available in which I get information for getting desired form of output as I think I would encounter many such problems in my research work.

Thanks and regards,

Nilesh

@vv 

Hi Sir,

You are right that I need maximum solution but still proving concavity is necessary due to following reasons-

1) For highly nonlinear function, any global optimization package  (or solver) may give local or near optimal result but we can not say with any guarantee that it is  a global optimal output. To ensure it is a global optimal solution, concavity is required. Because the function may be multimodal and contains many local optima. The package produces which maxima, we can't say.

2) To publish an article in a research journal, it is very difficult to convince the reviewer that it is a global optimal solution without concavity proof.

Hope you understand my problem.

Thanks and Regards,

Nilesh

@vv 

Thanks Sir for your reply.

Can you give some clarification/procedure how I can check concavity of the function with Direct Search method. Any other global optimization package availlable in maple/mathematica/matlab?

Thanks and Regards,

Nilesh

@Ramakrishnan 

Thanks Sir for your early reply. As per your advice, I have computed Hessian Metrix of my objective function with respect to all four variables by following maple command-

Hess_TP1 := Hessian(rhs(sol2), [T, E, W, p])

After that, I executed following maple command to know whether this hessian is negative semidefinite ( to prove its concavity) through following command-

IsDefinite(Hess_TP1, 'query' = 'negative_semidefinite')

but maple is not able to give any answer and continously shows as evaluating.

Sir, I want to prove the concavity of objective function with respect to all four decision variables jointly and not separately so I have not computed individual second order derivate of TP with respect to each variable separately. Instead I computed Hessian of objective function for four decision variable at once. 

Is there any alternative method/package/techniques/tricks/software available to prove concavity of a multivariate nonlinear function because through above method maple doesn't show any result.

The updated maple worksheet is enclosed.

concavity_proof_question.mw

Thanks for your reply.

Nilesh

@Axel Vogt 

Hi Sir,

Sorry I didn't notice it. It may be a typo. It should be multiplication (*) instead of (.).

Thanks Sir for rectifying mistake.

Regards,

Nilesh

@Axel Vogt 

Dear Sir,

I can't find a.b term in my maple worksheet in sol1 equations.

Nilesh

@Markiyan Hirnyk 

Hello Sir,

I have run direct search command as suggested by you. Although it takes few hours but the solution comes close to optimal one. Thanks for this precious help.

I am also trying to use this method with my other problem, and if any problem comes in that, I would definitely discuss it with you.

Thanks Sir for devoting your time for it.

Thanks and regards,

Nilesh

@Markiyan Hirnyk 

Sorry Sir that I could not reply to your previous mail due to some reasons. Your recommended solution worked perfectly fine and I got the expected answer in that problem. But the same thing is not worked with this problem so I posted this question. I would try to solve through your above recommended procedure and let you know the results.

Thanks and regards,

Nilesh

Ok. Thanks Sir for this help. I would try to solve it through direct search method as you did.

Regards,

Nilesh

Thanks Sir for your prompt help. One thing I want to ask  whether there is any trick to get all the roots of these simultaneous equations through either fsolve, solve or any other command in maple.

if not, then is it available either in mathematica or matlab. I am asking it because, I have another set of three simultaneous equations with three unknown variables and fsolve doesn't give all the roots. Solve command is also just showing evaluating and takes lot of time and provides no anaser.

Regards,

Nilesh