# Is there an efficient way to solve this optimization problem?

I have the following optimization problem:

Minimize $\sum{C_i}{D_{i}^{x_{i}}}$

s.t. $\forall i \quad x_i \leq S_1$ $\quad$ $\sum{x_i * N_i} \leq S_2$

where $C_i,D_i,N_i,S_1,S_2$ are all know constants, $D_i$s are real numbers between 0 and 1. $N_i$s are positive integers. $C_i$s are positive real numbers. $x_s$ are the variables to solve and $x_i$s have to be natural numbers. I believe the objective function is convex, but because the variable $x_i$s are the exponents which make it hard to solve.

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What are $r_i,N_i$? How do you say convex? Are $C_i$ positive? – Ashok Sep 11 '12 at 6:47
Sorry, $r_i$ should be $x_i$ instead, $N_i$ are positive integers. $C_i$s are positive real numbers. It is convex in the sense that the objective function is summation of a series of convex function. I know exponential function is convex and here we only change the base. – c.c. Sep 11 '12 at 7:13