Maximize $\sum_i \mathrm{rate}_i$ s.t. $\mathrm{rate}_i$

Question

Question related to optimization problems.

$$\mathrm{maximize} \sum\limits_{i=1}^{M}\log\left(1+f_i(\mathbf{x})\right)$$

$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\;\;\mathrm{subject}\, \mathrm{to}\;f_i(\mathbf{x})>\Gamma_i\,\forall\, i\in\{1, \dotsc, M\}$

Since we are maximizing a sum of increasing functions subject to a lower bound, is this problem feasible?

This kind of problem is often used in networking when the objective is maximize the sum rate subject to the individual rate of a user $i$.

There are no upper bounds on $f_i$, thus no upper bound on your objective function. Ill-posed problem. — vonbrand, Apr 23 '14 at 16:56
@vonbrand $f_i$ may has its own upper bounds, such as concave functions. — E.J., Apr 23 '14 at 23:21

score 2 · Accepted Answer · answered Apr 23 '14 at 22:42

2

It depends on how you define $f_i(x)$, if there are finite number of $x$ satisfy your constraints, then I think it's feasible. Because at least you can check every $x$ and find the maximum one.

answered Apr 23 '14 at 22:42

E.J.

939

Maximize $\sum_i \mathrm{rate}_i$ s.t. $\mathrm{rate}_i$

1 Answers1