I have two objective function which I want to prove concave with respect to four independent variables jointly (simultaneously).
`TP1 is the first objective function which is defined in sol1 equation. TP2 is the second objective function which is defined in sol3 equation. I want to maximize both function. There are four decision variables (independent variables) in both the objective function expression- T,E,W,p. Ten parameters used in both the equations are- alpha, beta, c, h, m, o, s, u, a, b.
1) My first question is - How can I prove both the function as a concave function with respect to four decision variables jointly. Can I specify some range of parameters in which both these function would behave like a concave function. The feasible range of these parameters are-
`[alpha>0, 0<= beta<1, c>0, h>0, 0<= m<=1, o>0, s>0, u>0, a>0] [ b>0 for objective function TP1 and b>1 for objective function TP2].
Some other restriction on parameters and variables are- p>c , T<=m , T>=0, E>=0, W>=0, W>=E.
2) My second question is if I simplify my first question and specify some specific values of parameters as I done for both objective function, then I got two new objective function in sol2 and sol4 equation. Now can I prove these two objective function (sol2 & sol4) as concave with respect to four decision variables T,E,W,p jointly. I want to prove concavity for these two objective function because if it is proved concave then The first order optimality solution would give me the global optimal solution.
Maple worksheet is enclosed.
Thanks and Regards,