non linear optimization

Question

How to solve this optimization problem using matlab or some other tool. I know that, this is a convex problem with non-linear constraint $\rho\geq \rho_{min}$ , so i have tried many times it in matlab under various headings but i am not getting proper results. I will be obliged if anyone could help me to solve it. Thanks a ton in advance. \begin{equation} \underset{p_{s},p_{r}} {\text {minimize}} \quad C = w_1.(\frac{p_{s}+p_{r}}{p_{max}}) + w_2.{\frac{\rho_{min}}{\rho}} \end{equation} \begin{equation} \begin{aligned} \text{subject to} \quad & \rho\geq \rho_{min},\quad \text{non linear constraint}\\ & p_{s}+p_{r}\leq p_{max},\quad \text{linear constraint}\\ & p_{s}\geq 0, p_{r}\geq 0. \end{aligned} \end{equation}

where , \begin{equation} \rho = \frac{\phi_{1}\phi_{2}p_{s}p_{r}}{\phi_{1}p_{s} + \phi_{2}p_{r} + 1},\\ \& \quad w_{1}+w_{2}=1,\& w_{1}=.5,w_{2}=.5\\ \phi_{1}=1000; \phi_{2}=1000.\\ \rho_{min}=10;\quad \& \quad p_{max}=100. \end{equation}

Answer 1

In Maple:

> C:= w1*(ps + pr)/pmax + w2*rhomin/rho;
> cons:= [rho >= rhomin, ps + pr <= pmax];
> rho:= phi1*phi2*ps*pr/(phi1*ps+phi2*pr+1);
> w1:= 1/2; w2:= 1/2; phi1:= 1000; phi2:= 1000;
> rhomin:= 10; pmax:= 100;
> Optimization:-Minimize(C,cons,assume=nonnegative);

$$[.0200049975024967248, [pr = 1.00049962527872, ps = 1.00049962527872]]$$