# Question:How do I solve non linear optimization problem and proving it concavity

## Question:How do I solve non linear optimization problem and proving it concavity

Maple

I want to maximize a total profit (TP) function which is dependent on five independent variables (E,W,T, theta, tp). All these five variables can have non negative values. The TP function is given below- ( first TP is directly copied from maple worksheet and than copied again as a picture for clear viewing).

TP = (p1*(Q-q)+p1*(1-theta)*(q-E)+s*E-c*Q-o-h*((1/6)*alpha*W^beta*a*p1^(-b)*tp^3/m-(1/2)*alpha*W^beta*a*p1^(-b)*tp^2+Q*tp)-(t1-tp)*h*((1/2*(-(2/3)*t1+m-(1/3)*tp))*W^beta*a*alpha*(t1-tp)*(-p1*(-1+theta))^(-b)+W*m)/m-(1/2)*h*(W+E)*(T-t1))/T-u*W

I am trying the maximize TP with respect to above five independent variables. I tried to solve  five equations ( representing first order partial derivative of TP with respect to each of the independent variables equated to zero) simultaneously by "solve" and "fsolve" command but both these commands fail to give any output. I have also tried three other commands in optimization package ( QPSolve, NLPSolve, Maximize) but all these three commands also doesn't give any output. I want to prove the concavity of TP function with respect to five independent variables, please guide how it can be done. ( I have computed the Hessian matrix but since five first order equations doesn't give output ( through fsolve command) so I am unable to compute Hessian at these first order optimiality condition solution.). The values of the paramters in the TP equation are -

[alpha = 50, beta = .7, c = 20, h = 4, m = .4, o = 10, p1 = 40, s = 10, u = 5, a = 15000, b = 2]

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