Maximizing sum of concave functions subject to a linear constraint

Question

The specific problem I have is of the following type.

$$\begin{array}{ll} \text{maximize} & \frac{1}{1-a}x^{1-a} + \frac{1}{1-b}y^{1-b} +\frac{1}{1-c}z^{1-c}\\ \text{subject to} & x+y+z=K\end{array}$$

where $x,y,z$ are the optimization variables and $a, b, c > 1$ are constants.

I have tried using a Lagrange multiplier but it didn't work.

Is there a way to solve problems of this type? If not, is there a numerical method for approximating?

Answer 1

you will get $$x^{-a}+\lambda=0$$, $$y^{-b}+\lambda=0$$, $$z^{-c}+\lambda=0$$ and $$x+y+z=K$$ solve this system!

Answer 2

1. Represent x, y, z with $\lambda$ and substitute them into the fourth equation.

2. Then use bisection method to numerically solve for $\lambda$.